---
abstract: |
    The Shapovalov determinant for a class of pointed Hopf algebras is calculated, including quantized enveloping algebras, Lusztig’s small quantum groups, and quantized Lie superalgebras. Our main tools are root systems, Weyl groupoids, and Lusztig type isomorphisms. We elaborate powerful novel techniques for the algebras at roots of unity, and pass to the general case using a density argument.

    Key words: Hopf algebra, Nichols algebra, quantum group, representation

    MSC: 16W30; 17B37, 81R50
address:
- 'István Heckenberger, Universität zu Köln, Mathematisches Institut, Weyertal 86–90, D-50931 Köln, Germany'
- 'Hiroyuki Yamane, Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka 560-0043, Japan'
author:
- 'I. Heckenberger'
- 'H. Yamane'
title: ' Drinfel’d doubles and Shapovalov determinants'
---

Introduction
============

We study finite-dimensional representations of a large class of Hopf algebras $U(\chi )$, where $\chi $ is a bicharacter on ${\mathbb{Z}}^I$ for some finite index set $I$. These algebras emerged from a program of Andruskiewitsch and Schneider to classify pointed Hopf algebras [@a-AndrSchn98], [@p-Heck07b]. Prominent examples are quantized enveloping algebras of semisimple Lie algebras, where the deformation parameter is not a root of $1$, and Lusztig’s (finite-dimensional) small quantum groups, see Sect. \[sec:Uqg\]. Other relevant examples are quantized enveloping algebras of Lie superalgebras, see [@a-KhorTol91] and [@a-Yam99; @a-Yam99e], and Drinfeld doubles of bosonizations of Nichols algebras of diagonal type classified in [@p-Heck06b].

Our main combinatorial tools towards the study of representations are the root system and the Weyl groupoid associated to $\chi $. For quantized enveloping algebras of semisimple Lie algebras the Weyl groupoid is nothing but the Weyl group of the Lie algebra. The main concern of this paper is the determination of the Shapovalov determinants for all algebras $U(\chi )$ with finite root system. We obtain a natural analog of Shapovalov’s original formula (for complex semisimple Lie algebras) as a product of linear factors. For our approach we need that $\chi (\beta ,\beta )\not=1$ for all positive roots $\beta $. This assumption is fulfilled for the special cases mentioned above.

The generality of our setting forces us to understand the representation theory of algebras $U(\chi )$, where many values of $\chi $ are roots of $1$. We turn this bondage into a promising leading principle of our approach. We concentrate first on those bicharacters, which take values in the set of roots of $1$. In this case the positive and negative parts of $U(\chi )$ are finite-dimensional algebras. For these algebras we develop special techniques, which are very different from the usual ones for semisimple Lie algebras, based on reflections in the Weyl groupoid, With these techniques we are able to characterize easily the irreducibility of Verma modules. The characterization leads quickly to a formula for the Shapovalov determinants of Verma modules by a variant of the usual density argument. In a next step we extend our results to more general bicharacters by a new density argument. This is possible because of our good knowledge of the root system of bicharacters.

The history of Shapovalov determinants started with Shapovalov’s work [@a-Shapov72], where he defined a bilinear form on Verma modules and determinants on homogeneous subspaces to obtain information on the reducibility of Verma modules. These structures have been generalized by Kac and Kazhdan [@a-KK79] to symmetrizable Kac-Moody algebras and by Kac [@inb-Kac79] [@inp-Kac86] to Lie superalgebras with symmetrizable Cartan matrix. Shapovalov determinants have been calculated for quantized enveloping algebras by de Concini and Kac [@inp-dCK90] and for quantized Kac-Moody algebras by Joseph [@b-Joseph]. For Lusztig’s small quantum groups Kumar and Letzter [@a-KumLetz97] factorize the Shapovalov determinants under the assumption that the deformation parameter $q$ has prime order and the base field is a cyclotomic field. Shapovalov determinants have been considered recently in various contexts, see for example [@a-BrunKles02], [@inp-GorSerg05], [@a-Gorel06], [@a-Alek05], [@a-Hill08]. Our approach yields in particular an entirely new proof of the formula of de Concini and Kac. In Sect. \[sec:Uqg\] we improve Kumar’s and Letzter’s result by allowing arbitrary base fields and arbitrary orders of $q$.

The paper is organized as follows. In Sect. \[sec:prelims\] the axioms of Cartan schemes, Weyl groupoids, and root systems are recalled. The Weyl groupoid of a bicharacter fits into this framework. Besides collecting the most important facts we introduce a character $\rho ^\chi $ on ${\mathbb{Z}}^I$ which will play a similar role as the linear form $2\rho $ on the root lattice. In Sect. \[sec:DD\] the definition and properties of the Drinfeld doubles $U(\chi )$ are recalled. Sect. \[sec:Lusztig\] deals with Lusztig type isomorphisms between two (usually different) Drinfeld doubles. With Thm. \[th:PBWtau\] we establish a Lusztig type PBW basis of these algebras. Moreover, in Thm. \[th:EErel\] we develop important properties for $q$-commutators of root vectors. In Sect. \[sec:Verma\] we start to study Verma modules and special maps between them. Prop. \[pr:VTMiso\] gives a criterion for bijectivity of such maps, and Prop. \[pr:M=L\] identifies irreducible Verma modules. In Sect. \[sec:shapdet\] we study Shapovalov determinants following the approach in [@b-Joseph]. Here our main result is Thm \[th:Shapdet\], which gives a formula for the Shapovalov determinant of $U(\chi )$, where all values of $\chi $ are roots of $1$, and the root system of $\chi $ is finite. Then we pass to more general bicharacters: Thm. \[th:Shapdet2\] states a similar result for bicharacters with finite root system. We conclude the paper with the adaptation of our formulas to quantized enveloping algebras and Lusztig’s small quantum groups in Sect. \[sec:Uqg\], and with some commutative algebra in the Appendix.

Preliminaries {#sec:prelims}
=============

Let ${\Bbbk }$ be a field and ${{\Bbbk }^\times }={\Bbbk }\setminus \{0\}$. For all $n\in {\mathbb{N}}_0$ and $q\in {{\Bbbk }^\times }$ let $${(n)_{q}}=\sum _{j=0}^{n-1}q^j,\qquad {(n)^!_{q}}=\prod _{j=1}^n{(j)_{q}},$$ where ${(0)^!_{q}}=1$. For any finite set $I$ let $\{{\alpha }_i\,|\,i\in I\}$ be the standard basis of the free ${\mathbb{Z}}$-module ${\mathbb{Z}}^I$.

Cartan schemes, Weyl groupoids, and root systems {#ssec:CS}
------------------------------------------------

The combinatorics of a Drinfel’d double of a Nichols algebra of diagonal type is controlled to a large extent by its Weyl groupoid. We use the language developed in [@p-CH08]. Substantial part of the theory was obtained first in [@a-HeckYam08]. We recall the most important definitions and facts.

Let $I$ be a non-empty finite set. By [@b-Kac90 §1.1] a generalized Cartan matrix $C=(c_{ij})_{i,j\in I}$ is a matrix in ${\mathbb{Z}}^{I\times I}$ such that

1.  $c_{ii}=2$ and $c_{jk}\le 0$ for all $i,j,k\in I$ with $j\not=k$,

2.  if $i,j\in I$ and $c_{ij}=0$, then $c_{ji}=0$.

\[de:CS\] Let $I$ be a non-empty finite set, $A$ a non-empty set, ${r}_i : A \to A$ a map for all $i\in I$, and $C^a=(c^a_{jk})_{j,k \in I}$ a generalized Cartan matrix in ${\mathbb{Z}}^{I \times I}$ for all $a\in A$. The quadruple $${\mathcal{C}}= {\mathcal{C}}(I,A,({r}_i)_{i \in I}, (C^a)_{a \in A})$$ is called a *Cartan scheme* if

1.  ${r}_i^2 = {\operatorname{id}}$ for all $i \in I$,

2.  $c^a_{ij} = c^{{r}_i(a)}_{ij}$ for all $a\in A$ and $i,j\in I$.

Let $A=\{a\}$ be a set with a single element, and let $C$ be a generalized Cartan matrix. Then $r_i={\operatorname{id}}$ for all $i\in I$, and ${\mathcal{C}}$ becomes a Cartan scheme.

One says that a Cartan scheme ${\mathcal{C}}$ is *connected*, if the group $\langle {r}_i\,|\,i\in I\rangle \subset {\mathrm{Aut}}(A)$ acts transitively on $A$, that is, if for all $a,b\in A$ with $a\not=b$ there exist $n\in {\mathbb{N}}$ and $i_1,i_2,\ldots ,i_n\in I$ such that $b=r_{i_n}\cdots r_{i_2}
  r_{i_1}(a)$. Two Cartan schemes ${\mathcal{C}}={\mathcal{C}}(I,A,({r}_i)_{i\in I},(C^a)_{a\in A})$ and ${\mathcal{C}}'={\mathcal{C}}'(I',A',$ $({r}'_i)_{i\in I'},({C'}^a)_{a\in A'})$ are called *equivalent*, if there are bijections $\varphi _0:I\to I'$ and $\varphi _1:A\to A'$ such that $$\begin{aligned}
\label{eq:equivCS}
    \varphi _1({r}_i(a))={r}'_{\varphi _0(i)}(\varphi _1(a)),
    \qquad
    c^{\varphi _1(a)}_{\varphi _0(i) \varphi _0(j)}=c^a_{i j}
  \end{aligned}$$ for all $i,j\in I$ and $a\in A$.

Let ${\mathcal{C}}= {\mathcal{C}}(I,A,({r}_i)_{i \in I}, (C^a)_{a \in A})$ be a Cartan scheme. For all $i \in I$ and $a \in A$ define ${\sigma }_i^a \in
  \Aut({\mathbb{Z}}^I)$ by $$\begin{aligned}
    {\sigma }_i^a ({\alpha }_j) = {\alpha }_j - c_{ij}^a{\alpha }_i \qquad
    \text{for all $j \in I$.}
    \label{eq:sia}
  \end{aligned}$$ This map is a reflection. The *Weyl groupoid of* ${\mathcal{C}}$ is the category ${\mathcal{W}}({\mathcal{C}})$ such that ${\mathrm{Ob}}({\mathcal{W}}({\mathcal{C}}))=A$ and the morphisms are generated by the maps ${\sigma }_i^a\in {\mathrm{Hom}}(a,{r}_i(a))$ with $i\in I$, $a\in A$. Formally, for $a,b\in A$ the set ${\mathrm{Hom}}(a,b)$ consists of the triples $(b,f,a)$, where $$f={\sigma }_{i_n}^{{r}_{i_{n-1}}\cdots {r}_{i_1}(a)}\cdots
    {\sigma }_{i_2}^{{r}_{i_1}(a)}{\sigma }_{i_1}^a$$ and $b={r}_{i_n}\cdots {r}_{i_2}{r}_{i_1}(a)$ for some $n\in {\mathbb{N}}_0$ and $i_1,\ldots ,i_n\in I$. The composition is induced by the group structure of ${\mathrm{Aut}}({\mathbb{Z}}^I)$: $$(a_3,f_2,a_2)\circ (a_2,f_1,a_1) = (a_3,f_2f_1, a_1)$$ for all $(a_3,f_2,a_2),(a_2,f_1,a_1)\in {\mathrm{Hom}}({\mathcal{W}}({\mathcal{C}}))$. By abuse of notation we will write $f\in {\mathrm{Hom}}(a,b)$ instead of $(b,f,a)\in {\mathrm{Hom}}(a,b)$.

The cardinality of $I$ is termed the *rank of* ${\mathcal{W}}({\mathcal{C}})$. A Cartan scheme is called *connected* if its Weyl groupoid is connected.

Recall that a groupoid is a category such that all morphisms are isomorphisms. The Weyl groupoid ${\mathcal{W}}({\mathcal{C}})$ of a Cartan scheme ${\mathcal{C}}$ is a groupoid, see [@p-CH08]. For all $i\in I$ and $a\in A$ the inverse of ${\sigma }_i^a$ is ${\sigma }_i^{r_i(a)}$. If ${\mathcal{C}}$ and ${\mathcal{C}}'$ are equivalent Cartan schemes, then ${\mathcal{W}}({\mathcal{C}})$ and ${\mathcal{W}}({\mathcal{C}}')$ are isomorphic groupoids.

A groupoid $G$ is called *connected*, if for each $a,b\in {\mathrm{Ob}}(G)$ the class ${\mathrm{Hom}}(a,b)$ is non-empty. Hence ${\mathcal{W}}({\mathcal{C}})$ is a connected groupoid if and only if ${\mathcal{C}}$ is a connected Cartan scheme.

\[de:RSC\] Let ${\mathcal{C}}={\mathcal{C}}(I,A,({r}_i)_{i\in I},(C^a)_{a\in A})$ be a Cartan scheme. For all $a\in A$ let $R^a\subset {\mathbb{Z}}^I$, and define $m_{i,j}^a= |R^a \cap (\ndN_0{\alpha }_i + \ndN_0{\alpha }_j)|$ for all $i,j\in
  I$ and $a\in A$. We say that $${\mathcal{R}}= {\mathcal{R}}({\mathcal{C}}, (R^a)_{a\in A})$$ is a *root system of type* ${\mathcal{C}}$, if it satisfies the following axioms.

1.  $R^a=R^a_+\cup - R^a_+$, where $R^a_+=R^a\cap \ndN_0^I$, for all $a\in A$.

2.  $R^a\cap {\mathbb{Z}}{\alpha }_i=\{{\alpha }_i,-{\alpha }_i\}$ for all $i\in I$, $a\in A$.

3.  ${\sigma }_i^a(R^a) = R^{{r}_i(a)}$ for all $i\in I$, $a\in A$.

4.  If $i,j\in I$ and $a\in A$ such that $i\not=j$ and $m_{i,j}^a$ is finite, then $({r}_i{r}_j)^{m_{i,j}^a}(a)=a$.

If ${\mathcal{R}}$ is a root system of type ${\mathcal{C}}$, then ${\mathcal{W}}({\mathcal{R}})={\mathcal{W}}({\mathcal{C}})$ is the *Weyl groupoid of* ${\mathcal{R}}$. Further, ${\mathcal{R}}$ is called *connected*, if ${\mathcal{C}}$ is a connected Cartan scheme. If ${\mathcal{R}}={\mathcal{R}}({\mathcal{C}},(R^a)_{a\in A})$ is a root system of type ${\mathcal{C}}$ and ${\mathcal{R}}'={\mathcal{R}}'({\mathcal{C}}',({R'}^a_{a\in A'}))$ is a root system of type ${\mathcal{C}}'$, then we say that ${\mathcal{R}}$ and ${\mathcal{R}}'$ are *equivalent*, if ${\mathcal{C}}$ and ${\mathcal{C}}'$ are equivalent Cartan schemes given by maps $\varphi
  _0:I\to I'$, $\varphi _1:A\to A'$ as in Def. \[de:CS\], and if the map $\varphi _0^*:\ndZ^I\to \ndZ^{I'}$ given by $\varphi _0^*({\alpha }_i)={\alpha }_{\varphi _0(i)}$ satisfies $\varphi _0^*(R^a)={R'}^{\varphi _1(a)}$ for all $a\in A$.

There exist many interesting examples of root systems of type ${\mathcal{C}}$ related to semisimple Lie algebras, Lie superalgebras and Nichols algebras of diagonal type, respectively. For further details and results we refer to [@a-HeckYam08] and [@p-CH08].

\[con:uind\] In connection with Cartan schemes ${\mathcal{C}}$, upper indices usually refer to elements of $A$. Often, these indices will be omitted if they are uniquely determined by the context. In particular, for any $w,w'\in {\mathrm{Hom}}({\mathcal{W}}({\mathcal{C}}))$ and $a\in A$, the notation $1_aw$ and $w'1_a$ means that $w\in {\mathrm{Hom}}(\underline{\,\,},a)$ and $w'\in {\mathrm{Hom}}(a,\underline{\,\,})$, respectively.

A fundamental result about Weyl groupoids is the following theorem.

[@a-HeckYam08 Thm.1]\[th:Coxgr\] Let ${\mathcal{C}}={\mathcal{C}}(I,A,({r}_i)_{i\in I},(C^a)_{a\in A})$ be a Cartan scheme and ${\mathcal{R}}={\mathcal{R}}({\mathcal{C}},(R^a)_{a\in A})$ a root system of type ${\mathcal{C}}$. Let ${\mathcal{W}}$ be the abstract groupoid with ${\mathrm{Ob}}({\mathcal{W}})=A$ such that ${\mathrm{Hom}}({\mathcal{W}})$ is generated by abstract morphisms $s_i^a\in {\mathrm{Hom}}(a,{r}_i(a))$, where $i\in I$ and $a\in A$, satisfying the relations $$\begin{aligned}
    s_i s_i 1_a=1_a,\quad (s_j s_k)^{m_{j,k}^a}1_a=1_a,
    \qquad a\in A,\,i,j,k\in I,\, j\not=k,
  \end{aligned}$$ see Conv. \[con:uind\]. Here $1_a$ is the identity of the object $a$, and $(s_j s_k)^\infty 1_a$ is understood to be $1_a$. The functor ${\mathcal{W}}\to {\mathcal{W}}({\mathcal{R}})$, which is the identity on the objects, and on the set of morphisms is given by $s _i^a\mapsto \s_i^a$ for all $i\in I$, $a\in A$, is an isomorphism of groupoids.

If ${\mathcal{C}}$ is a Cartan scheme, then the Weyl groupoid ${\mathcal{W}}({\mathcal{C}})$ admits a length function $\ell :{\mathcal{W}}(\cC
)\to {\mathbb{N}}_0$ such that $$\begin{aligned}
  \ell (w)=\min \{k\in {\mathbb{N}}_0\,|\,\exists i_1,\dots ,i_k\in I,a\in A:
  w={\sigma }_{i_1}\cdots {\sigma }_{i_k}1_a\}
  \label{eq:ell}\end{aligned}$$ for all $w\in {\mathcal{W}}({\mathcal{C}})$. If there exists a root system of type ${\mathcal{C}}$, then $\ell $ has very similar properties to the well-known length function for Weyl groups, see [@a-HeckYam08].

Let ${\mathcal{C}}$ be a Cartan scheme and ${\mathcal{R}}$ a root system of type ${\mathcal{C}}$. Let $a\in A$. Then $-c^a_{ij}=\max\{m\in {\mathbb{N}}_0\,|\,{\alpha }_j+m{\alpha }_i\in
  R^a_+\}$ for all $i,j\in I$ with $i\not=j$. \[le:cm\]

By (C2) and (R3), ${\sigma }_i^{r_i(a)}({\alpha }_j)={\alpha }_j-c^a_{ij}{\alpha }_i\in R^a_+$. Hence $-c^a_{ij}\le \max\{m\in {\mathbb{N}}_0\,|\,{\alpha }_j+m{\alpha }_i\in R^a_+\}$. On the other hand, if ${\alpha }_j+m{\alpha }_i\in R^a_+$, then ${\sigma }_i^a({\alpha }_j+m\al
  _i)={\alpha }_j+(-c^a_{ij}-m){\alpha }_i\in R^{r_i(a)}_+$ by (R3) and (R1), and hence $m\le -c^a_{ij}$. This proves the lemma.

Let ${\mathcal{C}}$ be a Cartan scheme and ${\mathcal{R}}$ a root system of type ${\mathcal{C}}$. We say that ${\mathcal{R}}$ is *finite*, if $R^a$ is finite for all $a\in A$. The following lemmata are well-known for traditional root systems.

[@p-CH08 Lemma 2.11] Let ${\mathcal{C}}$ be a connected Cartan scheme and ${\mathcal{R}}$ a root system of type ${\mathcal{C}}$. The following are equivalent.

1.  ${\mathcal{R}}$ is finite.

2.  $R^a$ is finite for at least one $a\in A$.

3.  ${\mathcal{W}}({\mathcal{R}})$ is finite.

\[le:Rfincond\]

[@a-HeckYam08 Cor.5] Let ${\mathcal{C}}$ be a connected Cartan scheme and ${\mathcal{R}}$ a finite root system of type ${\mathcal{C}}$. Then for all $a\in A$ there exist unique elements $b\in A$ and $w\in {\mathrm{Hom}}(b,a)$ such that $|R^a_+|=\ell (w)\ge \ell (w')$ for all $w'\in {\mathrm{Hom}}(b',a')$, $a',b'\in A$. \[le:longestw\]

The Weyl groupoid of a bicharacter {#ssec:Weylgroupoid}
----------------------------------

Let $I$ be a non-empty finite set. Recall that a bicharacter on ${\mathbb{Z}}^I$ with values in ${{\Bbbk }^\times }$ is a map $\chi :{\mathbb{Z}}^I\times {\mathbb{Z}}^I\to {{\Bbbk }^\times }$ such that $$\begin{aligned}
    \chi (a+b,c)=&\chi (a,c)\chi (b,c),&
    \chi (c,a+b)=&\chi (c,a)\chi (c,b)
    \label{eq:bichar}\end{aligned}$$ for all $a,b,c\in {\mathbb{Z}}^I$. Then $\chi (0,a)=\chi (a,0)=1$ for all $a\in {\mathbb{Z}}^I$. Let ${\mathcal{X}}$ be the set of bicharacters on ${\mathbb{Z}}^I$. If $\chi \in {\mathcal{X}}$, then $$\begin{aligned}
  \label{eq:chiop}
  \chi {^\mathrm{op}}: & {\mathbb{Z}}^I\times {\mathbb{Z}}^I\to {{\Bbbk }^\times },&
  \chi {^\mathrm{op}}(a,b)=&\,\chi (b,a),\\
  \label{eq:chiinv}
  \chi ^{-1} : & {\mathbb{Z}}^I\times {\mathbb{Z}}^I\to {{\Bbbk }^\times },&
  \chi ^{-1}(a,b)=&\,\chi (a,b)^{-1},
  \intertext{and for all $w\in {\mathrm{Aut}}_{\mathbb{Z}}({\mathbb{Z}}^I)$ the map}
  \label{eq:w*chi}
  w^*\chi : & {\mathbb{Z}}^I\times {\mathbb{Z}}^I\to {{\Bbbk }^\times },&
  w^*\chi (a,b)=&\,\chi (w^{-1}(a),w^{-1}(b)),\end{aligned}$$ are bicharacters on ${\mathbb{Z}}^I$. The equation $$\begin{aligned}
    \label{eq:w*functor}
    (ww')^*\chi =w^*(w'{}^*\chi )\end{aligned}$$ holds for all $w,w'\in {\mathrm{Aut}}_{{\mathbb{Z}}}({\mathbb{Z}}^I)$ and all $\chi \in {\mathcal{X}}$.

\[de:Cartan\] Let $\chi \in {\mathcal{X}}$, $p\in I$, and $q_{ij}=\chi ({\alpha }_i,{\alpha }_j)$ for all $i,j\in I$. We say that $\chi $ is $p$-*finite*, if for all $j\in I$ there exists $m\in {\mathbb{N}}_0$ such that $\qnum{m+1}{q_{pp}}=0$ or $q_{pp}^m q_{pj}q_{jp}=1$.

Assume that $\chi $ is $p$-finite. Let $c_{p p}^\chi =2$, and for all $j\in I\setminus \{p\}$ let $$c_{pj}^\chi =-\min \{m\in {\mathbb{N}}_0 \,|\,
    (m+1)_{q_{pp}}(q_{pp}^m q_{pj} q_{jp}-1)=0\}.$$ If $\chi $ is $i$-finite for all $i\in I$, then the matrix $C^\chi =(c_{ij}^\chi )_{i,j\in I}$ is called the *Cartan matrix* associated to $\chi $. It is a generalized Cartan matrix, see Sect. \[ssec:CS\].

For all $p\in I$ and $\chi \in {\mathcal{X}}$, where $\chi $ is $p$-finite, let ${\sigma }_p^\chi \in {\mathrm{Aut}}_{\mathbb{Z}}({\mathbb{Z}}^I)$, $$\begin{aligned}
  {\sigma }_p^\chi ({\alpha }_j)={\alpha }_j-c_{pj}^\chi {\alpha }_p
  \quad \text{for all $j\in I$.}\end{aligned}$$ Towards the definition of the Weyl groupoid of a bicharacter, we define bijections $r_p:{\mathcal{X}}\to {\mathcal{X}}$ for all $p\in I$. Namely, let $$\begin{aligned}
  r_p: {\mathcal{X}}\to \cX,\quad
  r_p(\chi )=
  \begin{cases}
    ({\sigma }_p^\chi )^*\chi & \text{if $\chi $ is $p$-finite,}\\
    \chi & \text{otherwise.}
  \end{cases}\end{aligned}$$ Let $p\in I$, $\chi \in {\mathcal{X}}$, $q_{ij}=\chi ({\alpha }_i,{\alpha }_j)$ for all $i,j\in I$. If $\chi $ is $p$-finite, then $$\begin{aligned}
    r_p(\chi )({\alpha }_p,{\alpha }_p)=&q_{p p}, &
    r_p(\chi )({\alpha }_p,{\alpha }_j)=&q_{p j}^{-1}q_{p p}^{c_{pj}^\chi },\\
    r_p(\chi )({\alpha }_i,{\alpha }_p)=&q_{i p}^{-1}q_{p p}^{c_{pi}^\chi },&
    r_p(\chi )({\alpha }_i,{\alpha }_j)=&q_{i j} q_{i p}^{-c_{p j}^\chi }
    q_{p j}^{-c_{p i}^\chi } q_{p p}^{c_{pi}^\chi c_{p j}^\chi }
  \end{aligned}
  \label{eq:rpchi}$$ for all $i,j\in I\setminus \{p\}$. It is a small exercise to check that then $({\sigma }_p^\chi )^*\chi $ is $p$-finite, and $$\begin{aligned}
\label{eq:rp2}
  c_{pj}^{r_p(\chi )}=c_{pj}^\chi \quad \text{for all $j\in I$},
  \qquad r_p^2(\chi )=\chi .\end{aligned}$$ The reflections $r_p$, $p\in I$, generate a subgroup $$\begin{aligned}
  {\mathcal{G}}=\langle r_p\,|\, p\in I\rangle\end{aligned}$$ of the group of bijections of the set ${\mathcal{X}}$. For all $\chi \in {\mathcal{X}}$ let ${\mathcal{G}}(\chi )$ denote the ${\mathcal{G}}$-orbit of $\chi $ under the action of ${\mathcal{G}}$.

Let $\chi \in {\mathcal{X}}$ such that $\chi '$ is $p$-finite for all $\chi '\in {\mathcal{G}}(\chi )$ and $p\in I$. By Eq.  we obtain that $${\mathcal{C}}(\chi )=
{\mathcal{C}}(I,{\mathcal{G}}(\chi ),(r_p)_{p\in I},
(C^{\chi '})_{\chi '\in {\mathcal{G}}(\chi )})$$ is a connected Cartan scheme. The Weyl groupoid of $\chi $ is then the Weyl groupoid of the Cartan scheme ${\mathcal{C}}(\chi )$ and is denoted by ${\mathcal{W}}(\chi )$. Clearly, ${\mathcal{C}}(\chi )={\mathcal{C}}(\chi ')$ and ${\mathcal{W}}(\chi )={\mathcal{W}}(\chi ')$ for all $\chi '\in {\mathcal{G}}(\chi )$.

\[ex:Cartan\] Let $C=(c_{i j})_{i,j\in I}$ be a generalized Cartan matrix. Let $\chi \in {\mathcal{X}}$, $q_{ij}=\chi ({\alpha }_i,{\alpha }_j)$ for all $i,j\in I$, and assume that $q_{ii}^{c_{ij}}=q_{ij}q_{ji}$ for all $i,j\in I$, and that $\qnum{m+1}{q_{ii}}\not=0$ for all $i\in I$ and $m\in {\mathbb{N}}_0$ with $m<\max \{-c_{ij}\,|\,j\in I\setminus \{i\}\}$. (The latter is not an essential assumption, since if it fails, then one can replace $C$ by another generalized Cartan matrix $\tilde{C}$, such that $\chi $ has this property with respect to $\tilde{C}$.) One says that $\chi $ is of *Cartan type*. Then $\chi $ is $i$-finite for all $i\in I$, and $c_{ij}^\chi =c_{ij}$ for all $i,j\in I$. Eq.  gives that $$\begin{aligned}
    r_p(\chi )({\alpha }_i,{\alpha }_i)=&q_{ii}=\chi ({\alpha }_i,{\alpha }_i),\\
    r_p(\chi )({\alpha }_i,{\alpha }_j)\,r_p(\chi )({\alpha }_j,{\alpha }_i)=&q_{ij}q_{ji}=
    r_p(\chi )({\alpha }_i,{\alpha }_i)^{c_{pi}}
  \end{aligned}$$ for all $p,i,j\in I$. Hence $r_p(\chi )$ is again of Cartan type with the same Cartan matrix $C$. Thus $\chi '$ is $i$-finite for all $\chi '\in {\mathcal{G}}(\chi )$ and $i\in I$.

Let $C=(c_{ij})_{i,j\in I}$ be a symmetrizable generalized Cartan matrix, and for all $i\in I$ let $d_i\in {\mathbb{N}}$ such that $d_ic_{ij}=d_jc_{ji}$ for all $i,j\in I$. Let $q\in {{\Bbbk }^\times }$ such that $\qnum{m+1}{q^{2d_i}}\not=0$ for all $m\in {\mathbb{N}}_0$ with $m<-c_{ij}$ for some $j\in I$. Define $\chi \in {\mathcal{X}}$ by $\chi ({\alpha }_i,{\alpha }_j)=q^{d_ic_{ij}}$. Then $\chi $ is of Cartan type, hence $\chi $ is $p$-finite for all $p\in I$. Eq.  implies that $r_p(\chi )=\chi $ for all $p\in I$, and hence ${\mathcal{G}}(\chi )$ consists of precisely one element. In this case the Weyl groupoid ${\mathcal{W}}(\chi )$ is a group, which is precisely the Weyl group associated to the generalized Cartan matrix $C$. We will study this example in Sect. \[sec:Uqg\] under the assumption that $C$ is of finite type.

Roots {#ssec:roots}
-----

Let $\chi \in {\mathcal{X}}$. There exists a canonical root system of type ${\mathcal{C}}(\chi)$ which we describe in this subsection. It is based on the construction of a restricted PBW basis of Nichols algebras of diagonal type. Nichols algebras are braided Hopf algebras defined by a universal property. More details can be found in [@inp-AndrSchn02] on braided Hopf algebras and Nichols algebras, in [@a-Khar99] on the PBW basis, and in [@a-Heck06a] on the root system.

Let $V\in { {}_{{\Bbbk }{\mathbb{Z}}^I}^{{\Bbbk }{\mathbb{Z}}^I}\mathcal{YD}}$ be a $|I|$-dimensional module of diagonal type. Let ${\delta }:V\to {\Bbbk }{\mathbb{Z}}^I{\otimes }V$ and ${\boldsymbol{\cdot}}:{\Bbbk }{\mathbb{Z}}^I{\otimes }V\to V$ denote the left coaction and the left action of ${\Bbbk }{\mathbb{Z}}^I$ on $V$, respectively. Fix a basis $\{x_i\,|\,i\in I\}$ of $V$, elements $g_i$, where $i\in I$, and a matrix $(q_{ij})_{i,j\in I}\in (\fienz)^{I\times I}$, such that $${\delta }(x_i)=g_i{\otimes }x_i,\quad g_i{\boldsymbol{\cdot}}x_j=q_{ij}x_j \quad
\text{for all $i,j\in I$.}$$ Assume that $\chi ({\alpha }_i,{\alpha }_j)=q_{ij}$ for all $i,j\in I$. For all ${\alpha }\in {\mathbb{Z}}^I$ define the “bound function” $$\begin{aligned}
  {b^{\chi}} ({\alpha })=&
  \begin{cases}
    \min \{ m\in {\mathbb{N}}\,|\,
    \qnum{m}{\chi ({\alpha },{\alpha })}=0\} &
    \text{if $\qnum{m}{\chi ({\alpha },{\alpha })}=0$}\\
    & \text{for some $m\in {\mathbb{N}}$,}\\
    \infty & \text{otherwise}.
  \end{cases}
  \label{eq:height}\end{aligned}$$ If $p\in I$ such that $\chi $ is $p$-finite, then $$\begin{aligned}
  \bfun{r_p(\chi )}({\sigma }_p^\chi ({\alpha }))={b^{\chi}} ({\alpha })\quad
  \text{for all ${\alpha }\in {\mathbb{Z}}^I$}
  \label{eq:hghtrpchi}\end{aligned}$$ by Eq. .

The tensor algebra $T(V)$ admits a universal braided Hopf algebra quotient ${\mathfrak{B}(V)}$, called the *Nichols algebra of* $V$. As an algebra, ${\mathfrak{B}(V)}$ has a unique ${\mathbb{Z}}^I$-grading $$\begin{aligned}
  {\mathfrak{B}(V)}=\oplus _{{\alpha }\in {\mathbb{Z}}^I}{\mathfrak{B}(V)} _{\alpha }\label{eq:NAVgrading}\end{aligned}$$ such that $\deg x_i={\alpha }_i$ for all $i\in I$. This is also a coalgebra grading. There exists a totally ordered index set $(L,\le )$ and a family $(y_l)_{l\in L}$ of ${\mathbb{Z}}^I$-homogeneous elements $y_l\in {\mathfrak{B}(V)}$ such that the set $$\begin{aligned}
  \{ y_{l_1}^{m_1}y_{l_2}^{m_2}\cdots y_{l_k}^{m_k}\,|\,
  &k\ge 0,\,l_1,\dots ,l_k\in L,\,l_1>l_2>\cdots >l_k,\\
  &m_i\in {\mathbb{N}},\,m_i<{b^{\chi}} (\deg y_{l_i})
  \quad \text{for all $i\in I$}\}
\end{aligned}
  \label{eq:PBWbasis}$$ forms a vector space basis of ${\mathfrak{B}(V)}$. The set $$\begin{aligned}
\label{eq:roots}
  R^\chi _+=\{\deg y_l\,|\,l\in L\}\subset {\mathbb{Z}}^I\end{aligned}$$ depends on the matrix $(q_{ij})_{i,j\in I}$, but not on the choice of the basis $\{x_i\,|\,i\in I\}$, the set $L$, and the elements $g_i$, $i\in I$, and $y_l$, $l\in L$. Let $$R^\chi =R^\chi _+\cup -R^\chi _+.$$

[@p-Heck07b Thm.3.13] \[th:rschi\] Let $\chi \in {\mathcal{X}}$ such that $\chi '$ is $p$-finite for all $p\in
  I$. $\chi '\in {\mathcal{G}}(\chi )$. Then ${\mathcal{R}}(\chi )={\mathcal{R}}({\mathcal{C}}(\chi ), (R^{\chi '})_{\chi
  '\in {\mathcal{G}}(\chi )})$ is a root system of type ${\mathcal{C}}(\chi )$.

Roots with finite bounds often play a distinguished role. For all $\chi \in {\mathcal{X}}$ let $$\begin{aligned}
  R^\chi _{+{\mathrm{fin}}}=\{\beta \in R^\chi _+\,|\,
  {b^{\chi}} (\beta )<\infty \},\quad
  R^\chi _{+\infty }=R^\chi _+\setminus R^\chi _{+{\mathrm{fin}}}.
  \label{eq:R+fin}\end{aligned}$$

We will use several finiteness properties of bicharacters. $$\begin{aligned}
  \label{eq:X1}
  {\mathcal{X}}_1=\{\chi \in {\mathcal{X}}\,|\,&
  \text{$\chi $ is $p$-finite for all $p\in I$}\},\\
  {\mathcal{X}}_2=\{\chi \in {\mathcal{X}}\,|\,&
  \text{$\chi '$ is $p$-finite for all $\chi '\in {\mathcal{G}}(\chi )$, $p\in I$}\},\\
  {\mathcal{X}}_3=\{\chi \in {\mathcal{X}}\,|\,&
  \text{$R^\chi $ is finite}\},\\
  {\mathcal{X}}_4=\{\chi \in {\mathcal{X}}\,|\,&
  \text{$R^\chi $ is finite, $R^\chi _+=R^\chi _{+{\mathrm{fin}}}$}\},\\
  \label{eq:X5}
  {\mathcal{X}}_5=\{\chi \in {\mathcal{X}}_4\,|\,&
  \text{$\chi ({\alpha },{\alpha })\not=1$ for all ${\alpha }\in R^\chi _+$}\}.\end{aligned}$$ Clearly, ${\mathcal{X}}_i\supset {\mathcal{X}}_j$ for $1\le i<j\le 5$. By Eq. , $\chi \in {\mathcal{X}}_5$ if and only if $R^\chi $ is finite and $\chi ({\alpha },{\alpha })$ is a root of $1$ different from $1$ for all ${\alpha }\in R^\chi _+$.

\[le:equalrs\] Let $\chi ,\chi '\in {\mathcal{X}}_2$.

\(i) If $R^\chi _+=R^{\chi '}_+$, then $C^{w^*\chi }=C^{w^*\chi '}$ for all $w\in {\mathrm{Hom}}(\chi ,\underline{\,\,})
  \subset {\mathrm{Hom}}({\mathcal{W}}(\chi ))$.

\(ii) Assume that $\chi ,\chi '\in {\mathcal{X}}_3$. If $C^{w^*\chi }=C^{w^*\chi '}$ for all $w\in {\mathrm{Hom}}(\chi ,\underline{\,\,})
  \subset {\mathrm{Hom}}({\mathcal{W}}(\chi ))$, then $R^\chi _+=R^{\chi '}_+$.

By Thm. \[th:rschi\], ${\mathcal{R}}(\chi )$ is a root system of type $\cC
  (\chi )$.

\(i) Assume that $R^\chi _+=R^{\chi '}_+$. Then $C^{\chi }=C^{\chi '}$ by Lemma \[le:cm\]. Therefore ${\sigma }_i^\chi ={\sigma }_i^{\chi '}$ in ${\mathrm{Aut}}({\mathbb{Z}}^I)$. Since $\chi ,\chi '\in {\mathcal{X}}_2$, by induction it follows that $\s_{i_1}\cdots {\sigma }_{i_k}^\chi ={\sigma }_{i_1}\cdots
  {\sigma }_{i_k}^{\chi '}$ in ${\mathrm{Aut}}({\mathbb{Z}}^I)$ and $C^{(\s_{i_1}\cdots {\sigma }_{i_k}^\chi )^*\chi }=
  C^{(\s_{i_1}\cdots {\sigma }_{i_k}^{\chi '})^*\chi '}$ for all $k\in {\mathbb{N}}_0$ and $i_1,\dots ,i_k\in I$. Hence $C^{w^*\chi }=C^{w^*\chi '}$ for all $w\in {\mathrm{Hom}}(\chi ,\underline{\,\,})
  \subset {\mathrm{Hom}}({\mathcal{W}}(\chi ))$.

\(ii) Since $\chi \in {\mathcal{X}}_3$, $R^\chi =\{w^{-1}({\alpha }_i)\,|\,w\in \Hom
  (\chi ,\underline{\,\,})\subset {\mathrm{Hom}}({\mathcal{W}}(\chi ))\}$ by [@p-CH08 Prop.2.12]. By assumption on the Cartan matrices, $\s_{i_1}\cdots {\sigma }_{i_k}^\chi ={\sigma }_{i_1}\cdots
  {\sigma }_{i_k}^{\chi '}$ in ${\mathrm{Aut}}({\mathbb{Z}}^I)$ for all $k\in {\mathbb{N}}_0$ and $i_1,\dots,i_k\in I$. Hence $R^\chi =R^{\chi '}$, and the lemma holds by (R1).

For our study of Drinfel’d doubles we will use an analog of the sum of fundamental weights, commonly known as $\rho $. More precisely, we define a character version of the linear form $(2\rho ,\cdot )$, where $(\cdot ,\cdot )$ is the usual bilinear form on the weight lattice.

Let ${\widehat{{\mathbb{Z}}^I}}={\mathrm{Hom}}({\mathbb{Z}}^I,{{\Bbbk }^\times })$ denote the group of characters of ${\mathbb{Z}}^I$ with values in ${{\Bbbk }^\times }$.

\[de:rhomap\] Let $\chi \in {\mathcal{X}}$. Let ${\rho ^{\chi}} \in \widehat{{\mathbb{Z}}^I}$ such that $${\rho ^{\chi}} ({\alpha }_i)=\chi ({\alpha }_i,{\alpha }_i) \quad \text{for all $i\in I$}.$$

Let $\chi \in {\mathcal{X}}$, $p\in I$, and ${b}={b^{\chi}} ({\alpha }_p)$. Assume that ${b}<\infty $. Then $\chi $ is $p$-finite and $$\chi ({\alpha }_p,\beta )^{{b}-1}\chi (\beta ,{\alpha }_p)^{{b}-1}=
  \frac{\rhomap{r_p(\chi )}({\sigma }_p^\chi (\beta ))}{\rhomap\chi (\beta )}$$ for all $\beta \in {\mathbb{Z}}^I$. \[le:rho\]

Define $\xi _1,\xi _2:{\mathbb{Z}}^I\to {{\Bbbk }^\times }$ by $$\xi _1(\beta )=\chi ({\alpha }_p,\beta )^{{b}-1}\chi (\beta ,\al
  _p)^{{b}-1},
  \qquad
  \xi _2(\beta )=\frac{\rhomap{r_p(\chi )}({\sigma }_p^\chi (\beta ))}{
  \rhomap\chi (\beta )}.$$ Then $\xi _1,\xi _2\in \ZIdual$. Thus it suffices to prove that $\xi _1({\alpha }_j)=\xi _2({\alpha }_j)$ for all $j\in I$. Let $q_{jk}=\chi ({\alpha }_j,{\alpha }_k)$ for all $j,k\in I$. Then $q_{pp}^{b}=1$ since $\qnum{{b}}{q_{pp}}=0$. Moreover, $$\begin{aligned}
    \rhomap{r_p(\chi )}({\alpha }_j)=&r_p(\chi )({\alpha }_j,{\alpha }_j)\\
    =&\chi ({\alpha }_j-c_{pj}^\chi {\alpha }_p,{\alpha }_j-c_{pj}^\chi {\alpha }_p)
    =q_{jj}(q_{pj}q_{jp})^{-c_{pj}^\chi }q_{pp}^{c_{pj}^\chi c_{pj}^\chi }
  \end{aligned}$$ for all $j\in I$.

By assumption, $\chi $ is $p$-finite, and hence for all $j\in I\setminus \{p\}$ we have $q_{pp}^{c_{pj}^\chi }=q_{pj}q_{jp}$ or $q_{pp}^{1-c_{pj}^\chi }=1$. Let $j\in I$. If $q_{pp}^{c_{pj}^\chi }=q_{pj}q_{jp}$, then $$\begin{gathered}
    \rhomap{r_p(\chi )}({\alpha }_j)=q_{jj},\\
    \xi _2({\alpha }_j)=q_{jj}^{-1}\rhomap{r_p(\chi )}({\sigma }_p^\chi ({\alpha }_j))=
    q_{jj}^{-1}\rhomap{r_p(\chi )}({\alpha }_j-c_{pj}^\chi {\alpha }_p)
    =q_{pp}^{-c_{pj}^\chi},\\
    \xi _1({\alpha }_j)=q_{pj}^{{b}-1} q_{jp}^{{b}-1}=q_{pp}^{({b}-1)c_{pj}^\chi }
    =q_{pp}^{-c_{pj}^\chi },
  \end{gathered}$$ and hence $\xi _1({\alpha }_j)=\xi _2({\alpha }_j)$. Otherwise ${b}=1-c_{pj}^\chi $, $q_{pp}^{c_{pj}^\chi }=q_{pp}$, and then $$\begin{gathered}
    \rhomap{r_p(\chi )}({\alpha }_p)=q_{pp},\quad
    \rhomap{r_p(\chi )}({\alpha }_j)=q_{jj}(q_{pj}q_{jp})^{-c_{pj}^\chi }q_{pp},\\
    \xi _2({\alpha }_j)=q_{jj}^{-1}\rhomap{r_p(\chi )}({\sigma }_p^\chi ({\alpha }_j))=
    q_{jj}^{-1}\rhomap{r_p(\chi )}({\alpha }_j-c_{pj}^\chi {\alpha }_p)
    =(q_{pj}q_{jp})^{-c_{pj}^\chi },\\
    \xi _1({\alpha }_j)=q_{pj}^{{b}-1} q_{jp}^{{b}-1}
    =(q_{pj}q_{jp})^{-c_{pj}^\chi }.
  \end{gathered}$$ Hence $\xi _1({\alpha }_j)=\xi _2({\alpha }_j)$ also in this case. This proves the lemma.

Multiparameter Drinfel’d doubles {#sec:DD}
================================

In this paper we study Verma modules for a class of Hopf algebras introduced in [@p-Heck07b]. This class contains multiparameter quantizations of semisimple Lie algebras and basic classical Lie superalgebras. The precise definition is given in Eq. . It uses the Drinfel’d double construction and the theory of Nichols algebras.

The Drinfel’d double [@b-Joseph Sect.3.2] can be defined via a skew-Hopf pairing of two Hopf algebras or as the quotient of a free associative algebra by a certain ideal, see also Rem. \[re:ideal\]. The first approach is more technical, but also more powerful. We present here the second definition. For proofs see [@p-Heck07b].

Let $I$ be a non-empty finite set, $\chi $ a bicharacter on ${\mathbb{Z}}^I$ with values in ${{\Bbbk }^\times }$, and $q_{i j}=\chi ({\alpha }_i,{\alpha }_j)$ for all $i,j\in I$. Let ${\mathcal{U}}(\chi )$ be the unital associative ${\Bbbk }$-algebra with generators $K_i$, $K_i^{-1}$, $L_i$, $L_i^{-1}$, $E_i$, and $F_i$, where $i\in I$, and defining relations $$\begin{aligned}
  XY= YX \quad & \makebox[0pt][l]{for all $X,Y\in \{K_i,K_i^{-1},
  L_i,L_i^{-1}\,|\,i\in I\}$,}
  \label{eq:KLrel}\\
  K_iK_i^{-1}=&\,1, & L_iL_i^{-1}=&\,1,
  \label{eq:KKrel}
\\
  K_iE_jK_i^{-1}=&\,q_{ij}E_j, & L_iE_jL_i^{-1}=&\,q_{ji}^{-1}E_j,
  \label{eq:KErel}
\\
  K_iF_jK_i^{-1}=&\,q_{ij}^{-1}F_j, & L_iF_jL_i^{-1}=&\,q_{ji}F_j,
  \label{eq:KFrel}\\
  E_iF_j&\makebox[0pt][l]{$-F_jE_i=\delta _{i,j}(K_i-L_i)$,}
  \label{eq:EFrel}\end{aligned}$$ where $i,j\in I$, and $\delta _{i,j}$ denotes Kronecker’s $\delta $. The algebra ${\mathcal{U}}(\chi )$ can be given a Hopf algebra structure in many different ways. We will use the unique Hopf algebra structure determined by $$\begin{gathered}
  \left\{
  \begin{aligned}
    {\varepsilon }(K_i)=&\,1,\quad {\varepsilon }(E_i)=0, &
    {\varepsilon }(L_i)=&\,1,\quad {\varepsilon }(F_i)=0, \\
    {\varDelta }(K_i)=&\,K_i\otimes K_i,&
    {\varDelta }(L_i)=&\,L_i\otimes L_i,\\
    {\varDelta }(K_i^{-1})=&\,K_i^{-1}\otimes K_i^{-1},&
    {\varDelta }(L_i^{-1})=&\,L_i^{-1}\otimes L_i^{-1},\\
    {\varDelta }(E_i)=&\,E_i\otimes 1+K_i\otimes E_i,&
    {\varDelta }(F_i)=&\,1\otimes F_i+F_i\otimes L_i
  \end{aligned}
  \right.
  \label{eq:coprcU}\end{gathered}$$ for all $i\in I$.

Let ${\mathcal{U}}^{+0},{\mathcal{U}}^{-0}$, and ${{\mathcal{U}}^0}$ denote the commutative cocommutative Hopf subalgebras of ${\mathcal{U}}(\chi )$ generated by $\{K_i,K_i^{-1}\,|\,i\in I\}$, $\{L_i,L_i^{-1}\,|\,i\in I\}$, and $\{K_i,K_i^{-1},L_i,L_i^{-1}\,|\,i\in I\}$, respectively. They are isomorphic to the ring of Laurent polynomials in $|I|$, $|I|$, and $2|I|$ variables, respectively, in the natural way. For any ${\alpha }=\sum _{i\in I}m_i{\alpha }_i\in {\mathbb{Z}}^I$ let $K_{\alpha }=\prod _{i\in I}
K_i^{m_i}$ and $L_{\alpha }=\prod _{i\in I}L_i^{m_i}$.

Let ${\mathcal{U}}^+(\chi )$, ${\mathcal{V}}^+(\chi )$, ${\mathcal{U}}^-(\chi )$, and ${\mathcal{V}}^-(\chi )$ denote the subalgebras of ${\mathcal{U}}(\chi )$ generated by $\{E_i\,|\,i\in I\}$, $\{E_i,K_i,K_i^{-1}\,|\,i\in I\}$, $\{F_i\,|\,i\in I\}$, and $\{F_i,L_i,L_i^{-1}\,|\,i\in I\}$, respectively. Then ${\mathcal{V}}^+(\chi )$ and ${\mathcal{V}}^-(\chi )$ are Hopf subalgebras of ${\mathcal{U}}(\chi )$.

The algebra ${\mathcal{U}}(\chi )$ admits a unique ${\mathbb{Z}}^I$-grading $$\begin{gathered}
  {\mathcal{U}}(\chi )=\oplus _{\beta \in {\mathbb{Z}}^I}{\mathcal{U}}(\chi )_\beta ,\\
  1\in {\mathcal{U}}(\chi )_0,\quad
  {\mathcal{U}}(\chi )_\beta {\mathcal{U}}(\chi )_\gamma \subset
  {\mathcal{U}}(\chi )_{\beta +\gamma } \quad
  \text{for all $\beta ,\gamma \in {\mathbb{Z}}^I$,}
\end{gathered}
  \label{eq:Zngrading}$$ such that $K_i,K_i^{-1},L_i,L_i^{-1}\in {\mathcal{U}}(\chi )_0$, $E_i\in {\mathcal{U}}(\chi )_{{\alpha }_i}$, and $F_i\in {\mathcal{U}}(\chi )_{-{\alpha }_i}$ for all $i\in I$. Let $${\mathbb{N}}_0^I=\Big\{\sum _{i\in I}a_i{\alpha }_i\,|\,a_i\in {\mathbb{N}}_0\Big\}\subset
   {\mathbb{Z}}^I,$$ and for any subspace ${\mathcal{U}}'\subset {\mathcal{U}}(\chi )$ and any $\beta \in {\mathbb{Z}}^I$ let ${\mathcal{U}}'_\beta ={\mathcal{U}}'\cap {\mathcal{U}}(\chi )_\beta $. Then $$\begin{aligned}
  {\mathcal{U}}^+(\chi )=&\oplus _{\beta \in {\mathbb{N}}_0^I}{\mathcal{U}}^+(\chi )_\beta ,&
  {\mathcal{U}}^-(\chi )=&\oplus _{\beta \in {\mathbb{N}}_0^I}{\mathcal{U}}^-(\chi )_{-\beta }.\end{aligned}$$

For all $\beta \in {\mathbb{Z}}^I$ let $$\begin{aligned}
  |\beta |=\sum _{i\in I} a_i\in {\mathbb{Z}},\quad \text{ where }
  \beta =\sum _{i\in I}a_i{\alpha }_i.
  \label{eq:abs}\end{aligned}$$ The decomposition $${\mathcal{U}}(\chi )=\oplus _{m\in {\mathbb{Z}}}{\mathcal{U}}(\chi )_m,\quad
    \text{where}\quad
    {\mathcal{U}}(\chi )_m=\oplus _{\beta :|\beta |=m}{\mathcal{U}}(\chi )_\beta ,
  \label{eq:Zgrading}$$ gives a ${\mathbb{Z}}$-grading of ${\mathcal{U}}(\chi )$ called the *standard grading*.

\[pr:algiso\] Let $\chi \in {\mathcal{X}}$.

\(1) Let ${\underline{a}}=(a_i\,|\,i\in I)\in ({{\Bbbk }^\times })^I$. Then there exists a unique algebra automorphism $\varphi _{{\underline{a}}}$ of ${\mathcal{U}}(\chi )$ such that $$\varphi _{{\underline{a}}}(K_i)=K_i,\,\,
    \varphi _{{\underline{a}}}(L_i)=L_i,\,\,
    \varphi _{{\underline{a}}}(E_i)=a_iE_i,\,\,
    \varphi _{{\underline{a}}}(F_i)=a_i^{-1}F_i.
  \label{eq:cUauto1}$$

\(2) There is a unique algebra antiautomorphism ${\Omega }$ of ${\mathcal{U}}(\chi )$ such that $$\begin{aligned}
    {\Omega }(K_i)=&K_i,& {\Omega }(L_i)=&L_i,&
    {\Omega }(E_i)=&F_i,& {\Omega }(F_i)=&E_i.
    \label{eq:cUantiauto}\end{aligned}$$ It satisfies the relation ${\Omega }^2={\operatorname{id}}$.

\[le:commEFi\] For all $i\in I$ there exist unique linear maps ${\partial ^K}_i,{\partial ^L}_i\in {\mathrm{End}}_{\Bbbk }({\mathcal{U}}^+(\chi ))$ such that $$\begin{aligned}
        [E,F_i]={\partial ^K}_i(E)K_i-L_i{\partial ^L}_i(E)\quad
        \text{for all $E\in {\mathcal{U}}^+(\chi )$.}
    \end{aligned}$$ The maps ${\partial ^K}_i,{\partial ^L}_i\in {\mathrm{End}}_{\Bbbk }({\mathcal{U}}^+(\chi ))$ are skew-derivations. More precisely, $$\begin{gathered}
      {\partial ^K}_i(1)={\partial ^L}_i(1)=0,\quad 
      {\partial ^K}_i(E_j)={\partial ^L}_i(E_j)=\delta _{i,j},\label{eq:derKL1}\\
    \begin{aligned}
        {\partial ^K}_i(EE')=&{\partial ^K}_i(E)(K_i{\boldsymbol{.}}E')+E{\partial ^K}_i(E'),\\
        {\partial ^L}_i(EE')=&{\partial ^L}_i(E)E'+(L_i^{-1}{\boldsymbol{.}}E){\partial ^L}_i(E')
    \end{aligned}
        \label{eq:derKL2}
    \end{gathered}$$ for all $i,j\in I$ and $E,E'\in {\mathcal{U}}^+(\chi )$.

Let ${\mathcal{I}}^+(\chi )$ be the unique maximal ideal of ${\mathcal{U}}^+(\chi )$ such that ${\mathcal{I}}^+(\chi )\subset \ker \varepsilon $ and ${\partial ^K}_i({\mathcal{I}}^+(\chi ))\subset {\mathcal{I}}^+(\chi )$ for all $i\in I$. Equivalently, ${\mathcal{I}}^+(\chi )$ is the unique maximal ideal of ${\mathcal{U}}^+(\chi )$ such that ${\mathcal{I}}^+(\chi )\subset \varepsilon $ and ${\partial ^L}_i({\mathcal{I}}^+(\chi ))\subset \cI
^+(\chi )$ for all $i\in I$, see [@p-Heck07b Prop.5.4]. Let ${\mathcal{I}}^-(\chi )={\Omega }({\mathcal{I}}^+(\chi ))$. Let $$\begin{aligned}
  U^+(\chi )=&{\mathcal{U}}^+(\chi )/{\mathcal{I}}^+(\chi ), &
  U^-(\chi )=&{\mathcal{U}}^-(\chi )/{\mathcal{I}}^-(\chi ),\\
  V^+(\chi )=&{\mathcal{V}}^+(\chi )/{\mathcal{I}}^+(\chi ){\mathcal{U}}^{+0},&
  V^-(\chi )=&{\mathcal{V}}^-(\chi )/{\mathcal{I}}^-(\chi ){\mathcal{U}}^{-0},\end{aligned}$$ and $$\begin{aligned}
  U(\chi )={\mathcal{U}}(\chi )/({\mathcal{I}}^+(\chi ),{\mathcal{I}}^-(\chi )).
  \label{eq:Uchi}\end{aligned}$$ The canonical inclusions ${\mathcal{U}}^\pm (\chi )\subset {\mathcal{U}}(\chi )$, ${{\mathcal{U}}^0}\subset {\mathcal{U}}(\chi )$ induce maps $$\iota _+:U^+(\chi )\to U(\chi ),\quad
  \iota _0:{{\mathcal{U}}^0}\to U(\chi ),\quad
  \iota _-:U^-(\chi )\to U(\chi ).$$

\[re:ideal\] (i) The vector space $V=\oplus _{i\in I}{\Bbbk }E_i$ is a module over the group algebra ${\Bbbk }{\mathbb{Z}}^I\simeq {\Bbbk }[K_i,K_i^{-1}\,|\,i\in I]\subset {{\mathcal{U}}^0}$, where the left action ${\boldsymbol{\cdot}}:{\Bbbk }{\mathbb{Z}}^I\otimes V\to V$ and the left coaction ${\delta }:V\to {\Bbbk }{\mathbb{Z}}^I{\otimes }V$ are defined by $$\begin{aligned}
    K_i {\boldsymbol{\cdot}}E_j=q_{ij}E_j,\qquad {\delta }(E_i)=K_i{\otimes }E_i
  \end{aligned}$$ for all $i,j\in I$. The algebra $U^+(\chi )$ is commonly known as the *Nichols algebra* of the module $V$.

\(ii) There are various descriptions of the ideal ${\mathcal{I}}^+(\chi )$, see *e.g.* [@inp-AndrSchn02]. In case of quantized enveloping algebras, see Sect. \[sec:Uqg\], Serre relations generate the ideal ${\mathcal{I}}^+(\chi )$. A more general case is studied by Angiono [@p-Angi08]. For quantized Lie superalgebras the defining relations are determined in [@a-Yam99; @a-Yam99e]. It is in general an open problem to give a nice set of generators of ${\mathcal{I}}^+(\chi )$, see [@inp-Andr02 Question5.9].

(Triangular decomposition) The map $$\mathrm{m}(\iota _-{\otimes }\iota _0{\otimes }\iota _+):
  U^-(\chi ){\otimes }{{\mathcal{U}}^0}{\otimes }U^+(\chi )\to U(\chi )$$ is an isomorphism of ${\mathbb{Z}}^I$-graded vector spaces, where $\mathrm{m}$ denotes the multiplication map. \[pr:tridec\]

Following the convention in [@b-Joseph Sect.3.2.1], a skew-Hopf pairing ${\eta }:A\times B\to {\Bbbk }$, $(x,y)\mapsto {\eta }(x,y)$, of two Hopf algebras $A$, $B$ is a bilinear map satisfying the equations $$\begin{aligned}
  \label{eq:sHp1}
  {\eta }(1,y)=&\,{\varepsilon }(y),& {\eta }(x,1)=&\,{\varepsilon }(x),\\
  \label{eq:sHp2}
  {\eta }(xx',y)=&\,{\eta }(x',y_{(1)}){\eta }(x,y_{(2)}),&
  {\eta }(x,yy')=&\,{\eta }(x_{(1)},y){\eta }(x_{(2)},y'),\\
  \label{eq:sHp3}
  &\qquad 
  \makebox[0pt][l]{${\eta }({S}(x),y)={\eta }(x,{S}^{-1}(y))$}\end{aligned}$$ for all $x,x'\in A$ and $y,y'\in B$.

\[pr:sHpdef\] (i) There exists a unique skew-Hopf pairing ${\eta }$ of ${\mathcal{V}}^+(\chi )$ and ${\mathcal{V}}^-(\chi )$ such that for all $i,j\in I$ one has $$\begin{aligned}
    {\eta }(E_i,F_j)=-\delta _{i,j},\quad
    {\eta }(E_i,L_j)=0,\quad
    {\eta }(K_i,F_j)=0,\quad
    {\eta }(K_i,L_j)=q_{ij}.
  \end{aligned}$$

\(ii) The skew-Hopf pairing ${\eta }$ satisfies the equations $$\begin{aligned}
    {\eta }(EK,FL)={\eta }(E,F){\eta }(K,L)
  \end{aligned}$$ for all $E\in {\mathcal{U}}^+(\chi )$, $F\in {\mathcal{U}}^-(\chi )$, $K\in {\mathcal{U}}^{+0}$, and $L\in {\mathcal{U}}^{-0}$.

\(iii) If $\beta ,\gamma \in {\mathbb{N}}_0^I$ with $\beta \not=\gamma $, $E\in {\mathcal{U}}^+(\chi )_\beta $, $F\in {\mathcal{U}}^-(\chi )_{-\gamma }$, then ${\eta }(E,F)=0$.

\(iv) The restriction of ${\eta }$ to ${\mathcal{U}}^+(\chi )\times {\mathcal{U}}^-(\chi )$ induces a non-degenerate pairing ${\eta }:U ^+(\chi )\times U^-(\chi )\to {\Bbbk }$.

\(i) and (ii) are [@p-Heck07b Prop.4.3]. (iii) follows from the definition of ${\eta }$ and since ${\varDelta }$ is a ${\mathbb{Z}}^I$-homogeneous map. (iv) was proven in [@p-Heck07b Thm.5.8].

By the general theory, see [@b-Joseph 3.2.2], the pairing ${\eta }$ in Prop. \[pr:sHpdef\] can be used to describe commutation rules in ${\mathcal{U}}(\chi )$ and $U(\chi )$. Namely, $$\begin{aligned}
  \label{eq:Ucomm1}
  yx=&{\eta }(x{_{(1)}},S(y{_{(1)}}))x{_{(2)}}y{_{(2)}}{\eta }(x{_{(3)}},y{_{(3)}}),\\
  \label{eq:Ucomm2}
  xy=&{\eta }(x{_{(1)}},y{_{(1)}})y{_{(2)}}x{_{(2)}}{\eta }(x{_{(3)}},S(y{_{(3)}}))\end{aligned}$$ for all $x\in {\mathcal{V}}^+(\chi )$ and $y\in {\mathcal{V}}^-(\chi )$. Note that the second formula follows from the first one and Eqs. –.

Later we will also need some other general facts about $U(\chi )$. Some of them are collected here. Let $$\begin{aligned}
  \label{eq:kerderK}
  U^+_{i,K}(\chi )=&\ker {\partial ^K}_i\subset U^+(\chi ),&
  U^+_{i,L}(\chi )=&\ker {\partial ^L}_i\subset U^+(\chi ),\\
  \label{eq:kerderL}
  U^-_{i,K}(\chi )=&{\Omega }(U^+_{i,K}),&
  U^-_{i,L}(\chi )=&{\Omega }(U^+_{i,L}).\end{aligned}$$ Recall the definition of ${b^{\chi}} $ in Eq. .

Let $i\in I$.

\(i) Let $m\in {\mathbb{N}}$. The following are equivalent.

-   $E_i^m=0$ in $U(\chi )$,

-   $F_i^m=0$ in $U(\chi )$,

-   $m\ge {b^{\chi}} ({\alpha }_i)$.

\(ii) Let ${\Bbbk }[E_i]$ and ${\Bbbk }[F_i]$ be the subalgebras of $U(\chi )$ generated by $E_i$ and $F_i$, respectively. The multiplication maps $$\begin{aligned}
    U^+_{i,K}(\chi ) {\otimes }{\Bbbk }[E_i] \to &\,U^+(\chi ),&
    {\Bbbk }[E_i] {\otimes }U^+_{i,K}(\chi ) \to &\,U^+(\chi ),\\
    U^+_{i,L}(\chi ) {\otimes }{\Bbbk }[E_i] \to &\,U^+(\chi ),&
    {\Bbbk }[E_i] {\otimes }U^+_{i,L}(\chi ) \to &\,U^+(\chi ),\\
    U^-_{i,K}(\chi ) {\otimes }{\Bbbk }[F_i] \to &\,U^-(\chi ),&
    {\Bbbk }[F_i] {\otimes }U^-_{i,K}(\chi ) \to &\,U^-(\chi ),\\
    U^-_{i,L}(\chi ) {\otimes }{\Bbbk }[F_i] \to &\,U^-(\chi ),&
    {\Bbbk }[F_i] {\otimes }U^-_{i,L}(\chi ) \to &\,U^-(\chi ),
  \end{aligned}$$ are isomorphisms of ${\mathbb{Z}}^I$-graded algebras. \[le:Eheight\]

\(i) is standard in the theory of Nichols algebras. It follows from Eqs. , and the definitions of ${\mathcal{I}}^+(\chi )$ and ${\mathcal{I}}^-(\chi )$. The proof of (ii) for $U^+(\chi )$ can be performed as in [@a-Heck06a]. The formulas with $U^-(\chi )$ follow from those with $U^+(\chi )$ and Eqs. , .

\[le:EmFn\] Let $m,n\in {\mathbb{N}}_0$ and $p\in I$. Then $$\begin{aligned}
    E_p^m F_p^n=\sum _{i=0}^{\makebox[0pt]{\scriptsize $\min \{m,n\}$}}
    {\textstyle
    \frac{\qfact{m}{q_{p p}}\qfact{n}{q_{p p}}}{
    \qfact{i}{q_{p p}}\qfact{m-i}{q_{p p}}\qfact{n-i}{q_{p p}}}
    }
    F_p^{n-i}\prod _{j=1}^i
    (q_{p p}^{i+j-m-n}K_p-L_p)E_p^{m-i}.
  \end{aligned}$$

For $n=0$ the claim is trivial. By [@p-Heck07b Cor. 5.4], $$E_p^m F_p-F_p E_p^m=
  \qnum{m}{q_{p p}}(q_{p p}^{1-m}K_p-L_p)E_p^{m-1}.$$ Hence the lemma holds for $n=1$. It suffices to check the claim for $m\ge n$, since then it also holds for $m<n$ using the algebra antiisomorphism ${\Omega }$. The proof of the lemma for $m\ge n$ is a standard calculation by induction on $n$.

An analogue of Lusztig’s PBW basis {#sec:Lusztig}
==================================

Let $\chi \in {\mathcal{X}}$ and $p\in I$. Assume that $\chi $ is $p$-finite. Let $q_{i j}=\chi ({\alpha }_i,{\alpha }_j)$ and $c_{p i}=c_{p i}^\chi $ for all $i,j\in I$.

For all $m\in {\mathbb{N}}_0$ and $i\in I\setminus \{p\}$ define recursively $E^\pm _{i,m}\in U^+_{{\alpha }_i+m{\alpha }_p}$, $F^\pm _{i,m}\in U^-_{{\alpha }_i+m{\alpha }_p}$ by $$\begin{aligned}
  E^+_{i,0}=&E_i, &
  E^+_{i,m+1}=&\,E_p E^+_{i,m}-(K_p{\boldsymbol{.}}E^+_{i,m})E_p,\\
  E^-_{i,0}=&E_i, &
  E^-_{i,m+1}=&\,E_p E^-_{i,m}-(L_p{\boldsymbol{.}}E^-_{i,m})E_p,\\
  F^+_{i,0}=&F_i, &
  F^+_{i,m+1}=&\,F_p F^+_{i,m}-(L_p{\boldsymbol{.}}F^+_{i,m})F_p,\\
  F^-_{i,0}=&F_i, &
  F^-_{i,m+1}=&\,F_p F^-_{i,m}-(K_p{\boldsymbol{.}}F^-_{i,m})F_p.\end{aligned}$$ We also define $E^+_{i,-1}=E^-_{i,-1}=F^+_{i,-1}=F^-_{i,-1}=0$. The above definitions depend essentially on $p$. If we want to emphasize this, we will write $E^\pm _{i,m(p)}$ and $F^\pm _{i,m(p)}$ instead of $E^\pm _{i,m}$ and $F^\pm _{i,m}$, respectively.

For all $i\in I\setminus \{p\}$ define $$\lambda _i^\chi =\qfact{-c _{p i}}{q_{p p}}
\prod _{j=0}^{-c _{p i}-1}(q_{p p}^j q_{p i}q_{i p}-1).$$ Then $\lambda _i^\chi \not=0$ by definition of $c_{p i}=c^\chi _{p i}$. The next theorem was proven in [@p-Heck07b Thm.6.11].

\[th:Liso\] Let $\chi \in {\mathcal{X}}$ and $p\in I$. Assume that $\chi $ is $p$-finite. Let $c_{pi}=c_{pi}^\chi $ for all $i\in I$.

\(i) There exist unique algebra isomorphisms ${T}_p, {T}_p^-: U (\chi )\to U (r_p(\chi ))$ such that $$\begin{aligned}
    {T}_p(K_p)=&{T}_p^-(K_p)=K _p^{-1},&
    {T}_p(K_i)=&{T}_p^-(K_i)=K _iK _p^{-c_{p i}},\\
    {T}_p(L_p)=&{T}_p^-(L_p)=L _p^{-1},&
    {T}_p(L_i)=&{T}_p^-(L_i)=L _iL _p^{-c_{p i}},\\
    {T}_p(E_p)=&F _p L _p^{-1},&
    {T}_p(E_i)=&E ^+_{i,-c_{p i}},\\
    {T}_p(F_p)=&K _p^{-1}E _p,&
    {T}_p(F_i)=&\lambda _i(r_p(\chi ))^{-1}F ^+_{i,-c_{p i}},\\
    {T}_p^-(E_p)=&K _p^{-1}F _p,&
    {T}_p^-(E_i)=&\lambda _i(r_p(\chi ^{-1}))^{-1} E ^-_{i,-c_{pi}},\\
    {T}_p^-(F_p)=&E _p L _p^{-1},&
    {T}_p^-(F_i)=&(-1)^{c_{p i}} F ^-_{i,-c_{p i}}.
  \end{aligned}$$

\(ii) The maps ${T}_p$, ${T}_p^-$ satisfy ${T}_p {T}_p^-={T}_p^-{T}_p={\operatorname{id}}_{U(\chi )}$.

\(iii) There exists a unique ${\underline{a}}\in ({{\Bbbk }^\times })^I$ such that ${T}_p {\Omega }={\Omega }{T}^-_p \varphi _{{\underline{a}}}$ in ${\mathrm{Hom}}(U(\chi
  ),U(r_p(\chi )))$.

Note that ${T}_p{T}_p^-$ is an automorphism of $U(\chi )$ if one regards ${T}_p^-$ as a map from $U(\chi )$ to $U(r_p(\chi ))$ and ${T}_p$ as a map from $U(r_p(\chi ))$ to $U(r_p r_p(\chi ))=U(\chi )$.

\[pr:LTdeg\] Let $\chi \in {\mathcal{X}}$ and $p\in I$. Assume that $\chi $ is $p$-finite. Then $${T}_p(U(\chi )_{\alpha })=U(r_p(\chi ))_{{\sigma }_p^\chi ({\alpha })} \quad
  \text{for all ${\alpha }\in {\mathbb{Z}}^I$.}$$

The maps ${T}_p:U(\chi )\to U(r_p(\chi ))$ and ${T}_p^-:U(r_p(\chi ))\to
  U(\chi )$ are mutually inverse algebra isomorphisms, and send generators of degree ${\alpha }$ into the homogeneous component of degree ${\sigma }_p({\alpha })$.

\[le:TpU+U+\] Let $\chi \in {\mathcal{X}}$ and $p\in I$. Assume that $\chi $ is $p$-finite. Then $$\begin{aligned}
    {T}_p(U^+_{p,L}(\chi ))=&\,U^+_{p,K}(r_p(\chi )), &
    {T}_p(U^-_{p,K}(\chi ))=&\,U^-_{p,L}(r_p(\chi )),\\
    {T}^-_p(U^+_{p,K}(\chi ))=&\,U^+_{p,L}(r_p(\chi )), &
    {T}^-_p(U^-_{p,L}(\chi ))=&\,U^-_{p,K}(r_p(\chi )).
  \end{aligned}$$

Since $\chi $ and $r_p(\chi )$ are $p$-finite, [@p-Heck07b Prop.5.10] and [@p-Heck07b Prop.6.7(d)] give that $${T}_p(U^+_{p,L}(\chi ))\subset U^+_{p,K}(r_p(\chi )),\qquad
  {T}_p^-(U^+_{p,K}(r_p(\chi )))\subset U^+_{p,L}(\chi ).$$ Thus ${T}_p(U^+_{p,L}(\chi ))=U^+_{p,K}(r_p(\chi ))$ by Thm. \[th:Liso\](ii). Similar arguments yield that ${T}^-_p(U^+_{p,K}(\chi ))=U^+_{p,L}(r_p(\chi ))$. The remaining two equations can be obtained from these and Thm. \[th:Liso\](iii).

In the rest of the section assume that $\chi \in {\mathcal{X}}_3$. Let $n=|R_+^\chi |\in {\mathbb{N}}$. The following construction generalizes the Poincaré-Birkhoff-Witt basis of quantized enveloping algebras given by Lusztig.

Let $i_1,i_2,\dots ,i_n\in I$ such that $\ell (1_{\chi }{\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_n})=n$. For all $\nu \in \{1,2,\dots,n\}$ let $$\begin{gathered}
\label{eq:betak}
  \beta _\nu ^\chi
  =1_\chi {\sigma }_{i_1}{\sigma }_{i_2}\dots {\sigma }_{i_{\mu -1}}({\alpha }_{i_\nu }).\end{gathered}$$ Then the elements $\beta _\nu ^\chi $, $1\le \nu \le n$, are pairwise different and $$\begin{aligned}
  R^\chi _+=\{\beta _\nu ^\chi \,|\,1\le \nu \le n\}
  \label{eq:proots}\end{aligned}$$ by [@p-CH08 Prop.2.12]. For all $\nu \in \{1,2,\dots,n\}$ let $$\begin{gathered}
  \label{eq:Ebetak}
  E_{\beta _\nu }= E_{\beta _\nu }^\chi 
  ={T}_{i_1}\dots {T}_{i_{\nu -1}}(E_{i_\nu }),\quad
  F_{\beta _\nu }= F_{\beta _\nu }^\chi 
  ={T}_{i_1}\dots {T}_{i_{\nu -1}}(F_{i_\nu }),\\
  \label{eq:Ebarbetak}
  {\bar{E}}_{\beta _\nu }= {\bar{E}}_{\beta _\nu }^\chi 
  ={T}^-_{i_1}\dots {T}^-_{i_{\nu -1}}(E_{i_\nu }),\quad
  {\bar{F}}_{\beta _\nu }= {\bar{F}}_{\beta _\nu }^\chi 
  ={T}^-_{i_1}\dots {T}^-_{i_{\nu -1}}(F_{i_\nu }),\end{gathered}$$ where $E_{i_\nu },F_{i_\nu }\in U(r_{i_{\nu -1}}\dots r_{i_2}r_{i_1}(\chi ))$. Then $$\begin{aligned}
  E_{\beta _\nu },{\bar{E}}_{\beta _\nu }\in U^+(\chi )_{\beta _\nu },\qquad
  F_{\beta _\nu },{\bar{F}}_{\beta _\nu }\in U^-(\chi )_{-\beta _\nu }
  \label{eq:EbetainU+}\end{aligned}$$ for all $\nu \in \{1,\dots ,n\}$ by [@p-Heck07b Thm.6.19], Thm. \[th:Liso\](iii) and Prop. \[pr:LTdeg\].

\[le:rvrel\] Assume that $\chi \in {\mathcal{X}}_3$. Let $\nu \in \{1,2,\dots ,n\}$, and assume that ${b^{\chi}} (\beta _\nu )<\infty $. Then $E_{\beta _\nu }^{{b^{\chi}} (\beta _\nu )}=
  F_{\beta _\nu }^{{b^{\chi}} (\beta _\nu )}=0$ in $U(\chi )$.

By Eq.  and since $T_i$ is an isomorphism for each $i\in I$, it suffices to prove that $E_i^{\bfun{\chi '}({\alpha }_i)}=F_i^{\bfun{\chi '}({\alpha }_i)}=0$ in $U(\chi ')$ for all $\chi '\in {\mathcal{X}}$ and $i\in I$ with $\bfun{\chi '}({\alpha }_i)<\infty $. This follows from Lemma \[le:Eheight\](i).

\[th:PBW\] Assume that $\chi \in {\mathcal{X}}_3$. Let $n=|R_+^\chi |\in {\mathbb{N}}$. Both sets $$\begin{aligned}
    \label{eq:LusztigPBW+}
    \big\{ E_{\beta _1}^{m_1} E_{\beta _2}^{m_2}\cdots E_{\beta _n}^{m_n}\,&|\,
    0\le m_\nu <{b^{\chi}} (\beta _\nu )
    \text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\},\\
    \label{eq:LusztigPBW-}
    \big\{ {\bar{E}}_{\beta _1}^{m_1} {\bar{E}}_{\beta _2}^{m_2}\cdots
    {\bar{E}}_{\beta _n}^{m_n}\,&|\,
    0\le m_\nu <{b^{\chi}} (\beta _\nu )
    \text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\}
  \end{aligned}$$ form vector space bases of $U ^+(\chi )$.

We prove the claim for the basis in Eq. . For the other set the proof is analogous. By Eqs.  and , $$\begin{aligned}
    \dim U^+(\chi )_{\alpha }=\Big|\big\{(m_1,m_2,&\dots ,m_n)\in {\mathbb{N}}_0^n\,\big|\,
    \sum _{\nu =1}^n m_\nu \beta _\nu ={\alpha },\\
    &m_\nu <{b^{\chi}} (\beta _\nu ) \text{ for all $\nu \in \{1,\dots ,n\}$}
    \big\}\Big|
  \end{aligned}$$ for all ${\alpha }\in {\mathbb{N}}_0^I$. Since $\deg E_{\beta _\nu }=\beta _\nu $ for all $\nu \in \{1,2,\dots ,n\}$, it suffices to show that for all $\mu \in \{1,2,\dots ,n+1\}$ the elements of the set $$\begin{aligned}
    \big\{ E_{\beta _\mu }^{m_\mu } E_{\beta _{\mu +1}}^{m_{\mu +1}}\cdots
    E_{\beta _n}^{m_n}\,|\, 0\le m_\nu <{b^{\chi}} (\beta _\nu )
    \text{ for all $\nu \in \{\mu ,\mu +1 ,\dots ,n\}$} \big\}
  \end{aligned}$$ are linearly independent. We proceed by induction on $n+1-\mu $. If $\mu =n+1$, then the above set is empty, and hence its elements are linearly independent.

Let now $\mu \in \{1,2,\dots ,n\}$. For all $m_\mu ,\dots ,m_n\in {\mathbb{N}}_0$ with $m_\nu <{b^{\chi}} (\beta _\nu )$ for all $\nu \in \{\mu ,\mu +1,\dots ,n\}$ let $a_{m_\mu ,\dots ,m_n}\in {\Bbbk }$. Assume that $$\begin{aligned}
    \sum _{m_\mu ,\dots ,m_n}a _{m_\mu ,\dots ,m_n}
    E_{\beta _\mu }^{m_\mu } E_{\beta _{\mu +1}}^{m_{\mu +1}}\cdots
    E_{\beta _n}^{m_n}=0
    \label{eq:lindep}
  \end{aligned}$$ in $U^+(\chi )$. Let ${T}^-={T}^-_{i_\mu }\cdots {T}^-_{i_2}{T}^-_{i_1}$. Since ${T}^-(E_{\beta _\mu })={T}^-_{i_\mu }(E_{i_\mu })
  =K_{i_\mu }^{-1}F_{i_\mu }$, we obtain that $$\sum _{m_\mu ,\dots ,m_n}
  a _{m_\mu ,\dots ,m_n}
  (K_{i_\mu }^{-1}F_{i_\mu })^{m_\mu }{T}^-(E_{\beta _{\mu +1}})^{m_{\mu +1}}
  \cdots {T}^-(E_{\beta _n})^{m_n}=0.$$ Since ${T}^-(E_{\beta _\nu })\in U^+(r_{i_\mu }\cdots r_{i_2}r_{i_1}(\chi ))$ for all $\nu \in \{\mu +1,\mu +2,\dots ,n\}$, Prop. \[pr:tridec\] implies that $$\sum _{m_{\mu +1},\dots ,m_n}a _{m_\mu ,m_{\mu +1},\dots ,m_n}
  {T}^-(E_{\beta _{\mu +1}})^{m_{\mu +1}}\cdots {T}^-(E_{\beta _n})^{m_n}
  =0$$ for all $m_\mu \in {\mathbb{N}}_0$, $m_\mu <{b^{\chi}} (\beta _\mu )$. Therefore $$\sum _{m_{\mu +1},\dots ,m_n}a _{m_\mu ,m_{\mu +1},\dots ,m_n}
  E_{\beta _{\mu +1}}^{m_{\mu +1}}\cdots E_{\beta _n}^{m_n}=0$$ for all $m_\mu \in {\mathbb{N}}_0$, $m_\mu <{b^{\chi}} (\beta _\mu )$. Then $a_{m_\mu ,m_{\mu +1},\dots ,m_n}=0$ for all $(m_\mu ,\dots ,m_n)$ by induction hypothesis, which proves the induction step. Thus the theorem holds.

Assume that $\chi \in {\mathcal{X}}_3$. Then $\ker ({\partial ^K}_{i_1}:U^+(\chi )\to U^+(\chi ))$ coincides with the subalgebra of $U^+(\chi )$ generated by the elements $E_{\beta _\nu }$, $\nu \in \{2,3,\dots ,n\}$. The set $$\begin{aligned}
    \big\{ E_{\beta _2}^{m_2} E_{\beta _3}^{m_3}\cdots E_{\beta _n}^{m_n}\,&|\,
    0\le m_\nu <{b^{\chi}} (\beta _\nu )
    \text{ for all $\nu \in \{2,3,\dots ,n\}$} \big\}
  \end{aligned}$$ forms a vector space basis of $\ker {\partial ^K}_{i_1}$. \[le:kerderK\]

Let $\nu \in \{2,3,\dots ,n\}$. By [@p-Heck07b Lemma5.10] and Lemma \[le:rvrel\] there exist $m\in {\mathbb{N}}_0$ and $x_0,x_1,\dots ,x_m\in \ker {\partial ^K}_{i_1}$ such that $m<{b^{\chi}} ({\alpha }_{i_1})$ and $E_{\beta _\nu }=\sum _{\mu =0}^m x_\mu E_{i_1}^\mu $. Then $${T}_{i_1}^-(E_{\beta _\nu })=
  \sum _{\mu =0}^m{T}^-_{i_1}(x_\mu ) (K_{i_1}^{-1}F_{i_1})^\mu .$$ Moreover, ${T}^-_{i_1}(x_\mu )\in U^+(r_{i_1}(\chi ))$ for all $\mu \in \{0,1,\dots
  ,m\}$ by [@p-Heck07b Prop.5.19,Lemma6.7(d)]. Since ${T}^-_{i_1}(E_{\beta _\nu })\in U^+(r_{i_1}(\chi ))$, triangular decomposition of $U(r_{i_1}(\chi ))$ implies that $x_\mu =0$ for all $\mu >0$. Hence $E_{\beta _\nu }=x_0\in \ker {\partial ^K}_{i_1}$. Then the claim of the lemma follows from the inclusions $$\langle E_{\beta _\kappa }\,|\,\kappa \in \{2,3,\dots ,n\}\rangle
  \subset \ker {\partial ^K}_{i_1} \subset
  \mathop{\oplus }_{ {(m_2,m_3,\dots ,m_n) \atop
  0\le m_\kappa <{b^{\chi}} (\beta _\kappa )\,\text{for all $\kappa $}}}
  {\Bbbk }E_{\beta _2}^{m_2}\cdots E_{\beta _n}^{m_n},$$ where the second inclusion is obtained from Thm. \[th:PBW\] and the formula $${\partial ^K}_{i_1}( E_{\beta _1}^{m_1}E_{\beta _2}^{m_2}\cdots
  E_{\beta _n}^{m_n})= \qnum{m_1}{q_{i_1i_1}}
  E_{\beta _1}^{m_1-1}K_{i_1}{\boldsymbol{\cdot}}(E_{\beta _2}^{m_2}\cdots
  E_{\beta _n}^{m_n}).$$

The analogous version of Lemma \[le:kerderK\] for $\ker {\partial ^L}_{i_1}$ is obtained by replacing $E_{\beta _\nu }$ by ${\bar{E}}_{\beta _\nu }$ for all $\nu \in \{2,3,\dots ,n\}$.

\[th:EErel\] Assume that $\chi \in {\mathcal{X}}_3$. Let $n=|R_+^\chi |\in {\mathbb{N}}$. Then $$\begin{aligned}
    E_{\beta _\mu }E_{\beta _\nu }-\chi (\beta _\mu ,\beta _\nu )
    E_{\beta _\nu }E_{\beta _\mu }
    \in \, & \langle E_{\beta _\kappa }\,|\,\mu <\kappa <\nu \rangle
    \subset U^+(\chi ),\\
    {\bar{E}}_{\beta _\mu }{\bar{E}}_{\beta _\nu }
    -\chi ^{-1}(\beta _\nu ,\beta _\mu ){\bar{E}}_{\beta _\nu }{\bar{E}}_{\beta _\mu }
    \in \, & \langle {\bar{E}}_{\beta _\kappa }
    \,|\,\mu <\kappa <\nu \rangle \subset U^+(\chi ),\\
    F_{\beta _\mu }F_{\beta _\nu }-\chi (\beta _\nu ,\beta _\mu )
    F_{\beta _\nu }F_{\beta _\mu }
    \in \, & \langle F_{\beta _\kappa }\,|\,\mu <\kappa <\nu \rangle
    \subset U^-(\chi ),\\
    {\bar{F}}_{\beta _\mu }{\bar{F}}_{\beta _\nu }-\chi ^{-1}(\beta _\mu ,\beta _\nu )
    {\bar{F}}_{\beta _\nu }{\bar{F}}_{\beta _\mu }
    \in \, & \langle {\bar{F}}_{\beta _\kappa }\,|\,\mu <\kappa <\nu \rangle
    \subset U^-(\chi )
  \end{aligned}$$ for all $\mu ,\nu \in \{1,2,\dots ,n\}$ with $\mu <\nu $.

We prove the first relation for $\mu =1$ and all $\nu \in \{2,3,\dots ,n\}$. Then the first relation for $\mu >1$ follows from $$E_{\beta _\mu }E_{\beta _\nu }-\chi (\beta _\mu ,\beta _\nu )
  E_{\beta _\nu }E_{\beta _\mu }
  ={T}_{i_1}\cdots {T}_{i_{\mu -1}}(
  E_{i_\mu }E'_\nu -\chi (\beta _\mu ,\beta _\nu )E'_\nu E_{i_\mu }),$$ where $E'_\nu ={T}_{i_\mu }\cdots {T}_{i_{\nu -1}}(E_{i_\nu })$, by using the case $\mu =1$, Eq.  and the first relation in . The proof of the second relation of the theorem is similar. The third and fourth relations can be obtained from the first two by applying ${\Omega }$ and using the formulas $$\begin{aligned}
    {\Omega }(E_{\beta _\kappa })\in {{\Bbbk }^\times }{\bar{F}}_{\beta _\kappa },\quad
    {\Omega }({\bar{E}}_{\beta _\kappa })\in {{\Bbbk }^\times }F _{\beta _\kappa },
    \qquad \kappa \in \{1,2,\dots ,n\},
    \label{eq:aaaU(E)}
  \end{aligned}$$ which follow from Thm. \[th:Liso\](iii).

Let $\nu \in \{2,3,\dots ,n\}$. For all $(m_1,m_2,\dots ,m_n)\in {\mathbb{N}}_0^I$ with $m_\kappa <{b^{\chi}} (\beta _\kappa )$ for all $\kappa \in \{1,2,\dots ,n\}$ let $a_{m_1,\dots ,m_n}\in {\Bbbk }$ such that $$\begin{aligned}
    E_{i_1}E_{\beta _\nu }-\chi ({\alpha }_{i_1},\beta _\nu )E_{\beta _\nu }E_{i_1}
    =\sum _{m_1,\dots ,m_n}a_{m_1,\dots ,m_n}E_{\beta _1}^{m_1}
    \cdots E_{\beta _n}^{m_n}.
    \label{eq:EE-EE}
  \end{aligned}$$ The numbers $a_{m_1,\dots ,m_n}\in {\Bbbk }$ exist and are unique by Thm. \[th:PBW\]. Let $\chi _\nu =r_{i_\nu }\cdots r_{i_2}r_{i_1}(\chi )$. Apply to Eq.  the isomorphism ${T}^-={T}^-_{i_\nu }\cdots {T}^-_{i_2}{T}^-_{i_1}\in {\mathrm{Hom}}(U(\chi ),
  U(\chi _\nu ))$. For all $\kappa \in \{1,2,\dots ,\nu \}$, $$\begin{aligned}
    {T}^-_{i_\nu }\cdots {T}^-_{i_2}{T}^-_{i_1}(E_{\beta _\kappa })
    =&{T}^-_{i_\nu }\cdots {T}^-_{i_{\kappa +1}}
    {T}^-_{i_\kappa }(E_{i_\kappa })\\
    =&{T}^-_{i_\nu }\cdots {T}^-_{i_{\kappa +1}}
    (K_{i_\kappa }^{-1}F_{i_\kappa })
    \in U^-(\chi _\nu ){{\mathcal{U}}^0}&
  \end{aligned}$$ by Eq. . Hence $$\begin{aligned}
    &\sum _{m_1,\dots ,m_n}a_{m_1,\dots ,m_n}
    {T}^-(E_{\beta _1}^{m_1}\cdots E_{\beta _\nu }^{m_\nu })
    {T}^-(E_{\beta _{\nu +1}}^{m_{\nu +1}}\cdots E_{\beta _n}^{m_n})\\
    &\quad ={T}^-(E_{i_1}E_{\beta _\nu }
    -\chi ({\alpha }_{i_1},\beta _\nu )E_{\beta _\nu }E_{i_1})\in U^-(\chi _\nu ){{\mathcal{U}}^0}.
  \end{aligned}$$ By triangular decomposition of $U(\chi )$ it follows that $a_{m_1,\dots ,m_n}=0$ for all $(m_1,\dots ,m_n)$ with $m_\kappa >0$ for some $\kappa \in \{\nu +1,\nu +2,\dots ,n\}$.

By Lemma \[le:kerderK\], $E_{\beta _\nu }\in \ker {\partial ^K}_{i_1}$. Hence $E_{i_1}E_{\beta _\nu }-\chi ({\alpha }_{i_1},\beta _\nu )E_{\beta _\nu }E_{i_1}
  \in \ker {\partial ^K}_{i_1}$ by Lemma \[le:commEFi\]. Thus Lemma \[le:kerderK\] implies that $a_{m_1,\dots ,m_n}=0$ whenever $m_1>0$.

Suppose that there exists $(m_1,\dots ,m_n)$ with $m_\nu >0$ and $a_{m_1,\dots ,m_n}\not=0$. Since $E_{i_1}E_{\beta _\nu }-\chi ({\alpha }_{i_1},\beta _\nu )E_{\beta _\nu }E_{i_1}$ is ${\mathbb{Z}}^I$-homogeneous of degree ${\alpha }_{i_1}+\beta _\nu $, the only possibility is that $m_1=1$, $m_\nu =1$, and $m_\kappa =0$ for all $\kappa \notin \{1,\nu \}$. Since $a_{m_1,\dots ,m_n}\not=0$, this is a contradiction to the previous paragraph. Thus the theorem is proven.

Next we prove a generalization of Thm. \[th:PBW\].

\[th:PBWtau\] Assume that $\chi \in {\mathcal{X}}_3$. Let $n=|R^\chi _+|$ and let $\tau $ be a permutation of the set $\{1,2,\dots ,n\}$. Then the sets $$\begin{aligned}
    \big\{ E_{\beta _{\tau (1)}}^{m_{\tau (1)}} E_{\beta _{\tau (2)}}^{m_{\tau
    (2)}}\cdots E_{\beta _{\tau (n)}}^{m_{\tau (n)}}\,&|\,
    0\le m_\nu <{b^{\chi}} (\beta _\nu )
    \text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\},\\
    \big\{ {\bar{E}}_{\beta _{\tau (1)}}^{m_{\tau (1)}}
    {\bar{E}}_{\beta _{\tau (2)}}^{m_{\tau (2)}}\cdots 
    {\bar{E}}_{\beta _{\tau (n)}}^{m_{\tau (n)}}\,&|\,
    0\le m_\nu <{b^{\chi}} (\beta _\nu )
    \text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\}
  \end{aligned}$$ form vector space bases of $U ^+(\chi )$, and the sets $$\begin{aligned}
    \big\{ F_{\beta _{\tau (1)}}^{m_{\tau (1)}} F_{\beta _{\tau (2)}}^{m_{\tau
    (2)}}\cdots F_{\beta _{\tau (n)}}^{m_{\tau (n)}}\,&|\,
    0\le m_\nu <{b^{\chi}} (\beta _\nu )
    \text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\},\\
    \big\{ {\bar{F}}_{\beta _{\tau (1)}}^{m_{\tau (1)}}
    {\bar{F}}_{\beta _{\tau (2)}}^{m_{\tau (2)}}\cdots 
    {\bar{F}}_{\beta _{\tau (n)}}^{m_{\tau (n)}}\,&|\,
    0\le m_\nu <{b^{\chi}} (\beta _\nu )
    \text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\}
  \end{aligned}$$ form vector space bases of $U ^-(\chi )$.

It suffices to prove that the first set is a basis of $U^+(\chi )$. Indeed, the proof for the second set can be obtained by using the maps ${T}^-_i$, where $i\in I$, instead of ${T}_i$. The second part of the claim follows from the first part by applying the algebra antiautomorphism ${\Omega }$ and using Eq. .

For any ${\underline{m}}=(m_1,\ldots ,m_n)\in {\mathbb{N}}_0^n$ let $|{\underline{m}}|=\sum _{\mu =1}^n
  m_\mu |\beta _\mu |$, where $|{\alpha }|=\sum _{j\in I}a_j$ for all ${\alpha }=\sum _{j\in I}a_j{\alpha }_j \in {\mathbb{N}}_0^I$. Let $N$ be the (additive) monoid ${\mathbb{N}}_0^n$ equipped with the following ordering: $${\underline{m}}'<{\underline{m}}\quad \Leftrightarrow \quad |{\underline{m}}'|<|{\underline{m}}| \quad
  \text{or} \quad |{\underline{m}}'|=|{\underline{m}}|,\,{\underline{m}}'<_{\mathrm{lex}}{\underline{m}},$$ where $<_{\mathrm{lex}}$ means lexicographical ordering. We use the convention ${\underline{m}}<_{\mathrm{lex}}{\underline{m}}$. The ordering $<$ is a total ordering. For all ${\underline{m}}\in {\mathbb{N}}_0^n$ define $${\mathcal{F}}^{{\underline{m}}}U^+(\chi )=\mathop{\oplus }_{{\underline{m}}'\in N,{\underline{m}}'<\ulm}
  {\Bbbk }E_{\beta _1}^{m'_1}
  E_{\beta _2}^{m'_2}\cdots E_{\beta _n}^{m'_n} \subset U^+(\chi ).$$ The vector spaces ${\mathcal{F}}^{{\underline{m}}}U^+(\chi )$, where ${\underline{m}}\in N$, are finite-dimensional, since the degrees of their elements are bounded. Moreover, $${\mathcal{F}}^0 U^+(\chi )={\Bbbk }1,\qquad
  {\mathcal{F}}^{{\underline{m}}}U^+(\chi ) {\mathcal{F}}^{{\underline{m}}'}U^+(\chi )\subset
  {\mathcal{F}}^{{\underline{m}}+{\underline{m}}'}U^+(\chi )$$ for all ${\underline{m}},{\underline{m}}'\in N$ by Thm. \[th:EErel\] and since $U^+(\chi )$ is ${\mathbb{Z}}^I$-graded. Thus ${\mathcal{F}}$ defines a filtration of $U^+(\chi )$ by the monoid $N$, and the corresponding graded algebra $$\mathop{\oplus }_{{\underline{m}}\in N} \Big({\mathcal{F}}^{{\underline{m}}}U^+(\chi )
  / \sum _{{\underline{m}}'<{\underline{m}},\ulm
  '\not={\underline{m}}} {\mathcal{F}}^{{\underline{m}}'}U^+(\chi )\Big)$$ is a skew-polynomial ring in $n$ variables by Thm. \[th:EErel\]. By a standard conclusion we obtain that the first set in the claim of the theorem is indeed a basis of $U^+(\chi )$.

For later purpose we define additional elements in $U(\chi )$.

For each $\nu \in \{n+1,n+2, \dots ,2n\}$ there exists a unique element $i_\nu \in I$ such that $$\begin{aligned}
    \ell (1_{r_{i_{\nu -n}}\cdots r_{i_2}r_{i_1}(\chi )}
    {\sigma }_{i_{\nu +1-n}}{\sigma }_{i_{\nu +2-n}}\cdots {\sigma }_{i_\nu })=n.
    \label{eq:ellsss}
  \end{aligned}$$ For this $i_\nu $ we get $1_{r_{i_{\nu -n}}\cdots r_{i_2}r_{i_1}(\chi )}
  {\sigma }_{i_{\nu +1-n}}{\sigma }_{i_{\nu +2-n}}\cdots {\sigma }_{i_{\nu -1}}({\alpha }_{i_\nu })
  ={\alpha }_{i_{\nu -n}}$. \[le:longlongw\]

We proceed by induction on $\nu $. First, $$\ell (1_{r_{i_{\nu -n}}\cdots r_{i_2}r_{i_1}(\chi )}
  {\sigma }_{i_{\nu +1-n}}{\sigma }_{i_{\nu +2-n}}\cdots {\sigma }_{i_{\nu -1}})=n-1$$ by induction hypothesis (if $\nu >n+1$) or by the choice of $i_1,\dots ,i_n$ (for $\nu =n+1$). Thus, by [@a-HeckYam08 Lemma8(iii)] there exists a unique positive root ${\alpha }$ such that $1_{r_{i_{\nu -n}}\cdots r_{i_2}r_{i_1}(\chi )}
  {\sigma }_{i_{\nu +1-n}}{\sigma }_{i_{\nu +2-n}}\cdots {\sigma }_{i_{\nu -1}}({\alpha })>0$. Clearly, this has to be a simple root. Let $i_\nu \in I$ such that ${\alpha }=\al
  _{i_\nu }$. Then the first claim follows from [@a-HeckYam08 Cor.3]. By Eq.  for $\nu -1$ and by [@a-HeckYam08 Cor.3] we get $1_{r_{i_{\nu -n-1}}\cdots r_{i_2}r_{i_1}(\chi )}
  {\sigma }_{i_{\nu -n}}{\sigma }_{i_{\nu +1-n}}\cdots {\sigma }_{i_{\nu -1}}({\alpha }_{i_\nu })<0$. Together with the first claim of the lemma this implies the second claim.

For each $\nu \in \{1,2,\dots,2n\}$ let $$\begin{aligned}
 \label{eq:beta2n}
  \beta _\nu =\beta _\nu ^\chi
  =&1_\chi {\sigma }_{i_1}\dots {\sigma }_{i_{\nu -1}}({\alpha }_{i_\nu }),&
  F_{\beta _\nu }=& F_{\beta _\nu }^\chi 
  ={T}_{i_1}\dots {T}_{i_{\nu -1}}(F_{i_\nu }),\end{aligned}$$ where $F_{i_\nu }\in U^-(r_{i_{\nu -1}}\cdots r_{i_2}r_{i_1}(\chi ))$ and $F_{\beta _\nu }\in U(\chi )$. Since $$\ell (1_{r_{i_n}\cdots r_{i_2}r_{i_1}(\chi )}{\sigma }_{i_{n+1}}{\sigma }_{i_{n+2}}
\cdots {\sigma }_{i_{2n}})=n,$$ Lemma \[le:longestw\] implies that $$\begin{aligned}
  1_\chi {\sigma }_{i_1}{\sigma }_{i_2}\dots {\sigma }_{i_n}=&
  {\sigma }_{i_1}{\sigma }_{i_2}\dots {\sigma }_{i_n}1_{r_{i_n}\cdots r_{i_2}r_{i_1}(\chi )}\\
  =&
  (1_{r_{i_n}\cdots r_{i_2}r_{i_1}(\chi )}
  {\sigma }_{i_{n+1}}{\sigma }_{i_{n+2}}\dots \s_{i_{2n}})^{-1}.\end{aligned}$$ Moreover, $$\begin{aligned}
  \beta _\nu ^\chi =-\beta _{\nu -n}^\chi \in - R^\chi _+
  \label{eq:betak>n}\end{aligned}$$ for all $\nu \in \{n+1,n+2,\dots ,2n\}$ by Lemma \[le:longlongw\]. In $U(\chi )$ we also get $$\begin{aligned}
  F_{\beta ^\chi _\nu }=T_{i_1}\cdots T_{i_{\nu -n}}(F_{i_{\nu -n}})\in \fienz
  K^{-1}_{\beta _{\nu -n}^\chi }E_{\beta ^\chi _{\nu -n}}
  \label{eq:Fbetak}\end{aligned}$$ for all $\nu \in \{n+1,\dots ,2n\}$ by [@p-Heck07b Thm.6.19,Prop.6.8(ii)].

Verma modules and morphisms {#sec:Verma}
===========================

We consider Verma modules for the algebras $U(\chi )$, $\chi \in {\mathcal{X}}$. We observe that the fundaments of the theory of Verma modules for quantized enveloping algebras can be carried over to a great extent to $U(\chi )$. New phenomena appear if some generators of $U(\chi )$ are nilpotent.

Let ${\mathbb{K}}$ be a field extension of ${\Bbbk }$. Let ${{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ denote the set of ${{\mathbb{K}}^\times }$-valued characters (algebra maps from ${{\mathcal{U}}^0}$ to ${{\mathbb{K}}^\times }$) of the group algebra ${{\mathcal{U}}^0}$. For all $\chi \in {\mathcal{X}}$ there is a natural group homomorphism $$\begin{aligned}
  {\zeta ^{\chi}} : {\mathbb{Z}}^I\to {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })},\quad
  {\zeta ^{\chi}} ({\alpha })(K_\beta L_{\beta '})=
  \chi (\beta ,{\alpha })\chi (\al,\beta ')^{-1}
  \label{eq:ZIch}\end{aligned}$$ for all ${\alpha },\beta ,\beta ' \in {\mathbb{Z}}^I$. If $\chi \in {\mathcal{X}}$ and $p\in I$ such that $\chi $ is $p$-finite, then $$\begin{aligned}
  \ZIch{r_p(\chi )}({\sigma }_p^\chi ({\alpha }))
  (K_{{\sigma }_p^\chi (\beta )}L_{{\sigma }_p^\chi (\beta ')})=
  {\zeta ^{\chi}} ({\alpha })(K_\beta L_{\beta '})
  \label{eq:ZIchrefl}\end{aligned}$$ by Eq. .

Let $\chi \in {\mathcal{X}}$. Given a ${{\mathbb{K}}^\times }$-valued character $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$, one can regard ${\mathbb{K}}$ as a one-dimensional $U ^+(\chi ){{\mathcal{U}}^0}$-module with generator $1_\Lambda =1$ via $$\begin{aligned}
  uE 1_\Lambda ={\varepsilon }(E)\Lambda (u)1_\Lambda \qquad
  \text{for all $E\in U ^+(\chi )$, $u\in {{\mathcal{U}}^0}$.}
  \label{eq:fieLambda}\end{aligned}$$ We write ${\mathbb{K}}_\Lambda $ for this module.

A *Verma module* of $U (\chi )$ is a $U (\chi )$-module of the form $$M^\chi (\Lambda )=U (\chi ){\otimes }_{U ^+(\chi ){{\mathcal{U}}^0}}{\mathbb{K}}_\Lambda ,\quad
  \text{where $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$.}$$ We write $v^\chi _\Lambda $ or just $v_\Lambda $ for $1{\otimes }1_\Lambda \in M^\chi (\Lambda )$.

Any Verma module is also a ${\mathbb{K}}$-module via the ${\mathbb{K}}$-module structure of the second tensor factor.

Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Triangular decomposition of $U (\chi )$ gives the following standard fact.

The map $U ^-(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}\to M^\chi (\Lambda )$, $u{\otimes }x\mapsto uxv_\Lambda =u{\otimes }x1_\Lambda $, is an isomorphism of vector spaces over ${\mathbb{K}}$. \[le:MLiso\]

The isomorphism in Lemma \[le:MLiso\] and the ${\mathbb{Z}}^I$-grading of $U^-(\chi )$ induce a unique ${\mathbb{Z}}^I$-grading on $M^\chi (\Lambda )$ such that $$\begin{aligned}
  M^\chi (\Lambda )_{\alpha }=U^-(\chi )_{\alpha }{\otimes }_{\Bbbk }{\mathbb{K}}_\Lambda
  \quad \text{for all ${\alpha }\in {\mathbb{Z}}^I$.}
  \label{eq:MLgrading}\end{aligned}$$ Then $$\begin{aligned}
  M^\chi (\Lambda )_0={\mathbb{K}}v_\Lambda ,\,\,
  U(\chi )_{\alpha }M^\chi (\Lambda )_\beta \subset M^\chi (\Lambda )_{{\alpha }+\beta }
  \,\,\text{for all ${\alpha },\beta \in {\mathbb{Z}}^I$.}
  \label{eq:MLgrading2}\end{aligned}$$ Moreover, $M^\chi (\Lambda )_{\alpha }\not=0$ implies that $-{\alpha }\in {\mathbb{N}}_0^I$.

The group algebra ${{\mathcal{U}}^0}$ acts on $M^\chi (\Lambda )$ via left multiplication. This action is given by characters: $$\begin{aligned}
  uv=(\Lambda +{\zeta ^{\chi}} ({\alpha }))(u)v \quad
  \text{for all $u\in {{\mathcal{U}}^0}$, ${\alpha }\in {\mathbb{Z}}^I$,
  $v\in M^\chi (\Lambda )_{\alpha }$,}
  \label{eq:U0M}\end{aligned}$$ see Eqs. , , , and .

Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. The family of those $U(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}$-submodules of $M^\chi (\Lambda )$, which are contained in $\oplus _{{\alpha }\not=0}M^\chi (\Lambda )_{\alpha }$, have a unique maximal element $I^\chi (\Lambda )$. Let $$\begin{aligned}
  L^\chi (\Lambda )=M^\chi (\Lambda )/I^\chi (\Lambda )
  \label{eq:LLambda}\end{aligned}$$ be the quotient $U(\chi )$-module. The maximality of $I^\chi (\Lambda )$ implies that $$I^\chi (\Lambda )=\big(I^\chi (\Lambda )\cap (U^-(\chi ){\otimes }1)\big)\fiee$$ and that $L^\chi (\Lambda )$ is ${\mathbb{Z}}^I$-graded. For all ${\alpha }\in {\mathbb{Z}}^I$ let $$I^\chi (\Lambda )_{\alpha }=M^\chi (\Lambda )_{\alpha }\cap
I^\chi (\Lambda ), \quad
L^\chi (\Lambda )_{\alpha }=M^\chi (\Lambda )_{\alpha }/I^\chi (\Lambda )_{\alpha }.$$ Since $M^\chi (\Lambda )_0={\mathbb{K}}v_\Lambda $, and any ${\mathbb{Z}}^I$-graded quotient of $M^\chi (\Lambda )$ by a $U(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}$-submodule containing $v_\Lambda $ is zero, $L^\chi (\Lambda )$ is the unique simple ${\mathbb{Z}}^I$-graded $U(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}$-module quotient of $M^\chi (\Lambda )$.

\[de:fchar\] Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ and $V$ a ${\mathbb{Z}}^I$-graded subquotient of $M^\chi (\Lambda )$. The *(formal) character of* $V$ is the sum $$\fch{V}=\sum _{{\alpha }\in {\mathbb{N}}_0^I} (\dim V_{-{\alpha }}) e^{-{\alpha }},$$ where $e$ is a formal variable.

Eq.  implies that $$\begin{aligned}
  \fch{M^\chi (\Lambda )}=\sum _{{\alpha }\in {\mathbb{N}}_0^I}\dim U^-(\chi )_{-{\alpha }}
  e^{-{\alpha }}
  \label{eq:chML}\end{aligned}$$ for all $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$.

\[re:fch\] For all ${\alpha },\beta \in {\mathbb{Z}}^I$ we let $e^{\alpha }e^\beta =e^{{\alpha }+\beta }$. Thus we can consider formal characters as elements of the ring $\cup _{{\alpha }\in {\mathbb{N}}_0^I}e^{\alpha }{\mathbb{Z}}[[e^{-{\alpha }_i}\,|\,i\in I]]$, where $e^{\alpha }{\mathbb{Z}}[[e^{-{\alpha }_i}\,|\,i\in I]]\subset
  e^{{\alpha }+\beta }{\mathbb{Z}}[[e^{-{\alpha }_i}\,|\,i\in I]]$ for all ${\alpha },\beta \in {\mathbb{N}}_0^I$ in the natural way.

[*From now on let $\chi \in {\mathcal{X}}$, $p\in I$, and ${b}={b^{\chi}} ({\alpha }_p)={b^{r_p(\chi )}}({\alpha }_p)$. Assume that ${b}<\infty $.*]{} Then $\chi $ and $r_p(\chi )$ are $p$-finite. We deduce some phenomena which arise from the finiteness assumption on ${b}$.

For all $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ define ${t}_p^\chi (\Lambda )\in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ by $$\begin{aligned}
  {t}_p^\chi (\Lambda )(K_{\alpha }L_\beta )=\Lambda (K_{{\sigma }_p^{r_p(\chi )}({\alpha })}
  L_{{\sigma }_p^{r_p(\chi )}(\beta )})
  \frac{r_p(\chi ) ({\alpha },{\alpha }_p)^{{b}-1}}{r_p(\chi )({\alpha }_p,\beta )^{{b}-1}}
  \label{eq:tpLambda}\end{aligned}$$ for all ${\alpha },\beta \in {\mathbb{Z}}^I$. By Eq.  this is equivalent to $$\begin{aligned}
  {t}_p^\chi (\Lambda )(K_{{\sigma }^\chi _p({\alpha })}L_{{\sigma }_p^\chi (\beta )})
  =\Lambda (K_{\alpha }L_\beta )
  \frac
  {\chi ({\alpha }_p,\beta )^{{b}-1}}
  {\chi ({\alpha },{\alpha }_p)^{{b}-1}}
  \label{eq:tpLambda1}\end{aligned}$$ for all ${\alpha },\beta \in {\mathbb{Z}}^I$.

Recall the definition of ${\rho ^{\chi}} $ from Def. \[de:rhomap\].

Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Then $$\begin{aligned}
    {\rho ^{r_p(\chi )}}({\sigma }_p^\chi ({\alpha }))\, {t}_p^\chi (\Lambda )
    (K_{{\sigma }_p^\chi ({\alpha })} L_{{\sigma }_p^\chi ({\alpha })}^{-1})=
    {\rho ^{\chi }}({\alpha }) \, \Lambda (K_{\alpha }L_{{\alpha }}^{-1})
  \end{aligned}$$ for all ${\alpha }\in {\mathbb{Z}}^I$ and $p\in I$. \[le:VTinv\]

Insert Eq.  and use Lemma \[le:rho\].

Let $C$ be a symmetrizable Cartan matrix and $q\in {{\Bbbk }^\times }$, $\chi
  \in {\mathcal{X}}$ as in the second part of Ex. \[ex:Cartan\]. In particular, $\chi ({\alpha }_i,{\alpha }_j)=q^{d_i c_{i j}}$. Let $p\in I$. Assume that ${b}={b^{\chi}} ({\alpha }_p)<\infty $. Then $q^{2d_p \bnd
  }=1$, and hence $q^{2{b}({\alpha },{\alpha }_p)}=1$ for all ${\alpha }\in \ndZ
  ^I$. Further, $r_p(\chi )=\chi $.

Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Assume that $\Lambda (K_{\alpha }L_\al
  ^{-1})=q^{2({\alpha },\lambda )}$ for some $\lambda $ in the weight lattice. Then $$\begin{aligned}
    {t}_p^\chi (\Lambda )(K_{\alpha }L_{\alpha }^{-1})
    =&\Lambda (K_{{\sigma }_p^\chi ({\alpha })} L_{{\sigma }_p^\chi ({\alpha })}^{-1}) q^{2(b-1)(\al
    ,{\alpha }_p)}\\
    =&q^{2({\sigma }_p^\chi ({\alpha }),\lambda )}q^{-2({\alpha },{\alpha }_p)}
    =q^{2({\alpha },{\sigma }_p^\chi (\lambda )-{\alpha }_p)},
  \end{aligned}$$ which recovers the dot action of the Weyl group on the weight lattice.

If we consider a composition ${t}_i^{\chi '}{t}_j^{\chi ''}$, where $i,j\in I$, $\chi ',\chi ''\in {\mathcal{X}}$, then we will always assume that $$\chi '=r_j(\chi '').$$ For simplicity, we will omit the upper index $\chi '$ if it is uniquely determined by another bicharacter in the expression.

Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Then ${t}_p {t}_p^\chi (\Lambda )
  =\Lambda $. \[le:VTMrel1\]

By Eqs.  and , and since $r_p^2(\chi )=\chi $, $$\begin{aligned}
    &{t}_p^{r_p(\chi )}{t}_p^\chi (\Lambda )(K_{\alpha }L_\beta )=
    {t}_p^\chi (\Lambda )(K_{{\sigma }_p^\chi ({\alpha })}L_{{\sigma }_p^\chi (\beta )})
    \frac{\chi ({\alpha },{\alpha }_p)^{{b}-1}}{\chi ({\alpha }_p,\beta )^{{b}-1}}
    =\Lambda (K_{\alpha }L_\beta )
  \end{aligned}$$ for all ${\alpha },\beta \in {\mathbb{Z}}^I$. This proves the lemma.

\[le:MLmap\] Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. There exist unique ${\mathbb{K}}$-linear maps $$\begin{gathered}
    {\hat{T}}_p={\hat{T}}^\chi _{p,\Lambda },\,
    {\hat{T}}^-_p={\hat{T}}^{\chi ,-}_{p,\Lambda }: \,\,
    M^{r_p(\chi )}({t}_p^\chi (\Lambda ))\to M^{\chi }(\Lambda ),\\
    \intertext{such that for all $u\in U(\chi )$,}
    {\hat{T}}_p(uv_{{t}_p^\chi (\Lambda )})=
    {T}_p(u)F_p^{{b}-1}v_{\Lambda },\quad
    {\hat{T}}^-_p(uv_{{t}_p^\chi (\Lambda )})=
    {T}^-_p(u)F_p^{{b}-1}v_{\Lambda }.
  \end{gathered}$$ If $V\subset M^{r_p(\chi )}({t}_p^\chi (\Lambda ))$ is a $U(r_p(\chi ))$-submodule, then ${\hat{T}}_p(V), {\hat{T}}^-_p(V)$ are $U(\chi )$-submodules of $M^{\chi }(\Lambda )$.

The uniqueness of the maps ${\hat{T}}_p$, ${\hat{T}}^-_p$ is clear. We prove that ${\hat{T}}_p$ is well-defined. The proof for ${\hat{T}}^-_p$ is analogous.

Let $\chi '=r_p(\chi )$ and $\Lambda '={t}_p^\chi (\Lambda )$. By Lemma \[le:MLiso\] and Thm. \[th:PBWtau\], $$M^{\chi }(\Lambda )_{{\alpha }_j+a{\alpha }_p}=0=
  M^{\chi }(\Lambda )_{-{b}{\alpha }_p} \quad 
  \text{for all $a\in {\mathbb{Z}}$.}$$ Thus, since $\deg {T}_p(E_j)={\sigma }_p^{\chi '}({\alpha }_j)$ for $E_j\in U(\chi ')$, $$\begin{aligned}
    {T}_p(E_j)F_p^{{b}-1}v_\Lambda \in 
    M^{\chi }(\Lambda )_{{\alpha }_j+(1-{b}-c^{\chi '}_{pj}){\alpha }_p}=0
    \quad \text{for all $j\in I$.}
  \end{aligned}$$ Moreover, Eqs. , give that $$\begin{aligned}
    K_{{\alpha }}L_{\beta }F_p^{{b}-1} v_{\Lambda }
    =\chi ({\alpha },{\alpha }_p)^{1-{b}}\, \chi ({\alpha }_p,\beta )^{{b}-1}\,
    \Lambda (K_{\alpha }L_\beta ) F_p^{{b}-1}v_{\Lambda }\qquad &\\
    =\Lambda '(K_{{\sigma }_p^\chi ({\alpha })}
    L_{{\sigma }_p^\chi (\beta )}) F_p^{{b}-1}v_\Lambda &
  \end{aligned}$$ for all ${\alpha },\beta \in {\mathbb{Z}}^I$. Hence ${T}_p(u)F_p^{{b}-1}v_{\Lambda }={\hat{T}}_p(\Lambda '(u)v_{\Lambda '})$ for all $u\in {{\mathcal{U}}^0}$. Therefore ${\hat{T}}_p$ is well-defined.

The last claim of the lemma follows from the equations $$\begin{aligned}
    {\hat{T}}_p(uv)={T}_p(u){\hat{T}}_p(v),\quad 
    {\hat{T}}^-_p(uv)={T}^-_p(u){\hat{T}}^-_p(v),
    \label{eq:Tuv}
  \end{aligned}$$ where $u\in U(r_p(\chi ))$ and $v\in M^{r_p(\chi )}({t}_p^\chi (\Lambda ))$, and from the surjectivity of the maps ${T}_p,{T}^-_p$.

\[le:VTMrel2\] Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ and $j,k\in I$ such that $j\not=k$. Assume that $m_{j,k}^\chi =|({\mathbb{N}}_0{\alpha }_j+{\mathbb{N}}_0{\alpha }_k)\cap R^\chi _+|<\infty $ and that ${b^{\chi}} ({\alpha })<\infty $ for all ${\alpha }\in ({\mathbb{N}}_0{\alpha }_j+{\mathbb{N}}_0{\alpha }_k)\cap R^\chi _+$. Then $$\begin{aligned}
    \label{eq:VTCox}
    ({t}_j{t}_k)^{m _{j,k}^\chi -1}{t}_j{t}_k^\chi (\Lambda )=\Lambda .
  \end{aligned}$$

Let $m=m^\chi _{j,k}$, and let $i_0,i_1,\dots ,i_{m_{j,k}^\chi }\in \{j,k\}$ such that $i_\nu =j$ if $\nu $ is even and $i_\nu =k$ if $\nu $ is odd. Let $\chi '=r_{i_0}r_{i_1}\cdots r_{i_{m-1}}(\chi )=
  r_{i_1}r_{i_2}\cdots r_{i_m}(\chi )$ (by (R4)), and $$\Lambda '={t}_{i_0}{t}_{i_1}\cdots {t}_{i_{m-1}}^\chi (\Lambda ),\quad
  \Lambda ''={t}_{i_1}{t}_{i_2}\cdots {t}_{i_m}^\chi (\Lambda ).$$ These definitions make sense, since ${b^{\chi}} ({\alpha })<\infty $ for all ${\alpha }\in ({\mathbb{N}}_0{\alpha }_j+{\mathbb{N}}_0{\alpha }_k)\cap R^\chi _+$. Lemma \[le:VTMrel1\] implies that the claim of the lemma is equivalent to $\Lambda '=\Lambda ''$.

Let ${\hat{T}}'={\hat{T}}_{i_0}{\hat{T}}_{i_1}\cdots {\hat{T}}_{i_{m-1}}:
  M^{\chi '}(\Lambda ')\to M^\chi (\Lambda )$ and ${\hat{T}}''={\hat{T}}_{i_1}{\hat{T}}_{i_2}\cdots {\hat{T}}_{i_m}:
  M^{\chi '}(\Lambda '')\to M^\chi (\Lambda )$. For all $\nu \in \{1,2,\dots ,m\}$ let $$\beta '_\nu =1_\chi {\sigma }_{i_0}{\sigma }_{i_1}\cdots {\sigma }_{i_{\nu -2}}
  ({\alpha }_{i_{\nu -1}}), \quad
  \beta ''_\nu =1_\chi {\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_{\nu -1}}
  ({\alpha }_{i_\nu }).$$ By definition of ${\hat{T}}_j$ and ${\hat{T}}_k$, $$\begin{aligned}
    {\hat{T}}''(v_{\Lambda ''})=&{\hat{T}}_{i_1}\cdots {\hat{T}}_{i_{m-1}}
    (F_{i_m}^{{b^{\chi '}}({\alpha }_{i_m})-1}v_{{t}_{i_m}^{\chi '}(\Lambda '')})
    =\cdots
    \\
    =&F_{\beta ''_m}^{{b^{\chi}} (\beta ''_m)-1}\cdots
    F_{\beta ''_2}^{{b^{\chi}} (\beta ''_2)-1}
    F_{\beta ''_1}^{{b^{\chi}} (\beta ''_1)-1}v_\Lambda ,\\
    {\hat{T}}'(v_{\Lambda '})=&F_{\beta '_m}^{{b^{\chi}} (\beta '_m)-1}\cdots
    F_{\beta '_2}^{{b^{\chi}} (\beta '_2)-1}
    F_{\beta '_1}^{{b^{\chi}} (\beta '_1)-1}v_\Lambda .
  \end{aligned}$$ Both expressions are nonzero by Thm. \[th:PBWtau\]. Since $$\{\beta '_\nu \,|\,1\le \nu \le m\}=\{\beta ''_\nu \,|\,1\le \nu \le m\}
  =R^\chi _+\cap ({\mathbb{N}}_0{\alpha }_j+{\mathbb{N}}_0{\alpha }_k),$$ we obtain that

-   ${\hat{T}}'({\mathbb{K}}v_{\Lambda '})$ and ${\hat{T}}''({\mathbb{K}}v_{\Lambda ''})$ are isomorphic ${{\mathcal{U}}^0}$-modules.

By Thm. \[th:Coxgr\], $${T}_{i_0}{T}_{i_1}\cdots {T}_{i_{m-1}}(u_0)=
  {T}_{i_1}{T}_{i_2}\cdots {T}_{i_m}(u_0)\quad
  \text{for all $u_0\in {{\mathcal{U}}^0}$.}$$ Hence Lemma \[le:MLmap\] yields that $$\begin{aligned}
    \Lambda '(u_0){\hat{T}}'(v_{\Lambda '})={\hat{T}}'(u_0v_{\Lambda '})
    =&{T}_{i_0}{T}_{i_1}\cdots {T}_{i_{m-1}}(u_0)
    {\hat{T}}'(v_{\Lambda '}),\\
    \Lambda ''(u_0){\hat{T}}''(v_{\Lambda ''})={\hat{T}}''(u_0v_{\Lambda ''})
    =&{T}_{i_0}{T}_{i_1}\cdots {T}_{i_{m-1}}(u_0)
    {\hat{T}}''(v_{\Lambda ''}).
  \end{aligned}$$ Thus $\Lambda '=\Lambda ''$ by ($*$). This proves the lemma.

In view of Thm. \[th:Coxgr\] and Lemmata \[le:VTMrel1\], \[le:VTMrel2\] we can say that Eq.  defines an action of the groupoid ${\mathcal{W}}(\chi )$ on ${{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Then Lemma \[le:VTinv\] says that the numbers ${\rho ^{\chi}} ({\alpha })\Lambda (K_{\alpha }L_{\alpha }^{-1})$, where ${\alpha }\in {\mathbb{Z}}^I$, are invariants of this action.

In general, the maps ${\hat{T}}_p$ and ${\hat{T}}^-_p$ are not isomorphisms.

Assume that $\Lambda (K_pL_p^{-1})\not=\chi ({\alpha }_p,{\alpha }_p)^{t-1}$ for all $t\in \{1,2,\dots ,{b}-1\}$. Then ${\hat{T}}_p,{\hat{T}}^-_p:
  \,M^{r_p(\chi )}({t}_p^\chi (\Lambda ))\to M^\chi (\Lambda )$ are isomorphisms of vector spaces over ${\mathbb{K}}$. \[pr:VTMiso\]

Let $q=\chi ({\alpha }_p,{\alpha }_p)$, $\chi '=r_p(\chi )$, $\Lambda '={t}_p^\chi (\Lambda )$, and ${\hat{T}}'={\hat{T}}^\chi _{p,\Lambda } {\hat{T}}^{\chi ',-}_{p,\Lambda '}$. By Lemma \[le:VTMrel1\] and since $r_p^2(\chi )=\chi $, ${\hat{T}}'$ is a $U(\chi )$-module endomorphism of $M^\chi (\Lambda )$. We calculate ${\hat{T}}'(v_\Lambda )$. $$\begin{aligned}
    {\hat{T}}'(v_\Lambda )
    =&{\hat{T}}_p(F_p^{{b}-1}v_{\Lambda '})
    ={T}_p(F_p^{{b}-1})F_p^{{b}-1}v_\Lambda \\
    =&(K_p^{-1}E_p)^{{b}-1}F_p^{{b}-1}v_\Lambda
    =q^{({b}-2)({b}-1)/2}\Lambda (K_p^{1-{b}})E_p^{{b}-1}F_p^{{b}-1}v_\Lambda \\
    =&q^{({b}-2)({b}-1)/2}\Lambda (K_p^{1-{b}})\qfact{{b}-1}{q}
    \prod _{t=1}^{{b}-1}(q^{t+1-{b}}\Lambda (K_p)-\Lambda (L_p))v_\Lambda
  \end{aligned}$$ by Lemma \[le:EmFn\]. By assumption, ${\hat{T}}'(v_\Lambda )\not=0$, and hence ${\hat{T}}'$ is a nonzero multiple of ${\operatorname{id}}_{M^\chi (\Lambda )}$. Therefore ${\hat{T}}_p$ is an isomorphism. The proof for ${\hat{T}}_p^-$ is analogous.

\[le:hwvector\] Let $t\in \{1,2,\dots ,{b}-1\}$. Let $q=\chi ({\alpha }_p,{\alpha }_p)$. Assume that $\Lambda (K_pL_p^{-1})=q^{t-1}$. Then in $M^\chi (\Lambda )$ $$\begin{aligned}
    E_pF_p^m v_\Lambda =\qnum{m}{q}\Lambda (L_p)(q^{t-m}-1)F_p^{m-1} v_\Lambda \quad
    \text{for all $m\in {\mathbb{N}}_0$.}
    \label{eq:EpFpmv}
  \end{aligned}$$ In particular, if $q\not=1$, then $EF_p^mv_\Lambda ={\varepsilon }(E)F_p^mv_\Lambda $ for all $E\in U^+(\chi )$ if and only if $m=0$, $m=t$ or $m\ge {b}$. If $q=1$, then $EF_p^mv_\Lambda ={\varepsilon }(E)F_p^mv_\Lambda $ for all $E\in
  U^+(\chi )$, $m\in {\mathbb{N}}_0$.

Eq.  follows from Lemma \[le:EmFn\]. By definition of ${b}={b^{\chi}} ({\alpha }_p)$, either $q\not=1$ and $q$ is a primitive ${b}$-th root of $1$, or $q=1$ and ${b}=\mathrm{char}\,{\Bbbk }$. Therefore, if $q\not=1$ and $m\in \{0,1,\dots ,{b}-1\}$, then $q^{t-m}=1$ if and only if $t=m$. If $E=E_i$ with $i\not=p$, then $EF_p^m=0$ by Eqs. , . The rest is a consequence of Lemma \[le:Eheight\](i).

\[pr:VTMker\] Assume that $\Lambda (K_pL_p^{-1})=\chi ({\alpha }_p,{\alpha }_p)^{t-1}$ for some $t\in \{1,2,\dots ,{b}-1\}$.

\(i) If $\chi ({\alpha }_p,{\alpha }_p)\not=1$, then $t$ is unique, and $$\begin{aligned}
    \ker {\hat{T}}^{\chi }_{p,\Lambda }=\ker {\hat{T}}^{\chi ,-}_{p,\Lambda }
    = & \, U^-(\chi )F_p^{{b}-t}{\otimes }{\mathbb{K}}_{{t}_p^\chi (\Lambda )},\\
    {\operatorname{Im}}{\hat{T}}^{\chi }_{p,\Lambda }={\operatorname{Im}}{\hat{T}}^{\chi ,-}_{p,\Lambda }
    = & \, U^-(\chi )F_p^t {\otimes }{\mathbb{K}}_\Lambda .
  \end{aligned}$$

\(ii) If $\chi ({\alpha }_p,{\alpha }_p)=1$, then $\mathrm{char}\,{\Bbbk }={b}>0$ and $$\begin{aligned}
    \ker {\hat{T}}^{\chi }_{p,\Lambda }=\ker {\hat{T}}^{\chi ,-}_{p,\Lambda }
    = & \, U^-(\chi )F_p{\otimes }{\mathbb{K}}_{{t}_p^\chi (\Lambda )},\\
    {\operatorname{Im}}{\hat{T}}^{\chi }_{p,\Lambda }={\operatorname{Im}}{\hat{T}}^{\chi ,-}_{p,\Lambda }
    = & \, U^-(\chi )F_p^{{b}-1} {\otimes }{\mathbb{K}}_\Lambda .
  \end{aligned}$$

Let $q=\chi ({\alpha }_p,{\alpha }_p)$, $\chi '=r_p(\chi )$, and $\Lambda '={t}_p^\chi (\Lambda )$. We prove the claims about ${\hat{T}}^\chi _{p,\Lambda }$ in (i). The rest is analogous. By Lemma \[le:EmFn\], for all $m\in \{0,1,\dots ,{b}-1\}$, $$\begin{aligned}
    {\hat{T}}^\chi _{p,\Lambda }(F_p^mv_{\Lambda '})
    =&{T}_p(F_p^m)F_p^{{b}-1}v_\Lambda
    =(K_p^{-1}E_p)^mF_p^{{b}-1}v_\Lambda \\
    =&aE_p^mF_p^{{b}-1}v_\Lambda =a'F_p^{{b}-1-m}
    \prod _{j=1}^m (q^{j+1-{b}}\Lambda (K_pL_p^{-1})-1) v_\Lambda
  \end{aligned}$$ for some $a,a'\in {{\Bbbk }^\times }$. Thus, ${\hat{T}}^\chi _{p,\Lambda }(F_p^mv_{\Lambda '})=0$ if and only if $q^j=q^{{b}-1}\Lambda (K_pL_p^{-1})^{-1}$ for some $j\in \{1,2,\dots ,m\}$. By the assumption on $\Lambda (K_pL_p^{-1})$, this is equivalent to $j={b}-t$ for some $j\in \{1,2,\dots ,m\}$. Therefore ${\hat{T}}^\chi _{p,\Lambda }(F_p^mv_{\Lambda '})=0$ if and only if $m\ge {b}-t$.

Let $F\in U^-(\chi ')$. By Lemma \[le:Eheight\](ii), there exist unique $F'_m\in U^-_{p,K}(\chi ')$, where $m\in \{0,1,\dots ,{b}-1\}$, such that $F=\sum _{m=0}^{{b}-1}F'_mF_p^m$. By the previous paragraph, and since $\Lambda (K_pL_p^{-1})\in {\Bbbk }$, for each $m\in \{0,1,\dots ,{b}-1-t\}$ there is a unique $a_m\in {{\Bbbk }^\times }$ such that $${\hat{T}}^\chi _{p,\Lambda }(F v_{\Lambda '})=\sum _{m=0}^{{b}-1-t}
  a_m{T}_p (F'_m)F_p^{{b}-1-m}v_\Lambda \in U^-(\chi ){\otimes }1_\Lambda .$$ By Lemma \[le:Eheight\](ii), the latter expression is zero if and only if ${T}_p(F'_m)=0$ for all $m\in \{0,1,\dots ,{b}-1-t\}$. Therefore, Lemma \[le:TpU+U+\] and relations $F'_m\in U^-_{i,K}(\chi ')$ imply that ${\hat{T}}^\chi _{p,\Lambda }(F v_{\Lambda '})=0$ if and only if $F'_m=0$ for all $m\in \{0,1,\dots ,{b}-1-t\}$. Hence $\ker {\hat{T}}^{\chi }_{p,\Lambda }=U^-(\chi )F_p^{{b}-t}{\otimes }{\mathbb{K}}_{\Lambda '}$ and ${\operatorname{Im}}{\hat{T}}^{\chi }_{p,\Lambda } = U^-(\chi )F_p^t {\otimes }{\mathbb{K}}_\Lambda $.

For all $w\in {\mathrm{Aut}}({\mathbb{Z}}^I)$ and ${\alpha }\in {\mathbb{Z}}^I$ let $w(e^{\alpha })=e^{w({\alpha })}$, and extend this definition linearly on formal characters. We investigate the effect of the maps ${\hat{T}}_p,{\hat{T}}^-_p$ on formal characters. For all $\chi '\in {\mathcal{G}}(\chi )$ and $i\in I$ with $\bfun{\chi '}({\alpha }_i)<\infty $ let ${\dot{\sigma }}^{\chi '}_i:{\mathbb{Z}}^I\to {\mathbb{Z}}^I$ be the affine transformation $$\begin{aligned}
  {\dot{\sigma }}_i^{\chi '}({\alpha })={\sigma }_i^{\chi '}({\alpha })
  +(1-\bfun{\chi '}({\alpha }_i)){\alpha }_i.
  \label{eq:sdot}\end{aligned}$$ Note that then ${\dot{\sigma }}_i^{r_i(\chi ')}{\dot{\sigma }}_i^{\chi '}({\alpha })={\alpha }$ for all ${\alpha }\in {\mathbb{Z}}^I$.

\[le:Tpfch\] Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ and ${\alpha }\in {\mathbb{Z}}^I$. Then $$\begin{aligned}
    \label{eq:Tpweight+}
    {\hat{T}}_p(M^{r_p(\chi )}({t}^\chi _p(\Lambda ))_{\alpha })
    \subset &M^\chi (\Lambda )_{{\dot{\sigma }}_p^{r_p(\chi )}({\alpha })},\\
    \label{eq:Tpweight-}
    {\hat{T}}^-_p(M^{r_p(\chi )}({t}^\chi _p(\Lambda ))_{\alpha })
    \subset &M^\chi (\Lambda )_{{\dot{\sigma }}_p^{r_p(\chi )}({\alpha })}.
  \end{aligned}$$ In particular, $$\begin{aligned}
    \fch{M^\chi (\Lambda )} =
    {\dot{\sigma }}_p^{r_p(\chi )}
    (\fch{M^{r_p(\chi )}({t}_p^\chi (\Lambda ))}).
    \label{eq:fchM}
  \end{aligned}$$

Let $\Lambda '={t}^\chi _p(\Lambda )$ and $u\in U(r_p(\chi ))_{\alpha }$. Then $${\hat{T}}_p(uv_{\Lambda '})={T}_p(u)F_p^{{b}-1}v_\Lambda
  \in U(\chi )_{{\sigma }_p^{r_p(\chi )}({\alpha })}U(\chi )_{(1-{b}){\alpha }_p}v_\Lambda$$ by Prop. \[pr:LTdeg\]. This proves Eq. , since $v_\Lambda \in M^\chi (\Lambda )_0$, The proof of Eq.  is similar. By Eq. , ${\mathrm{ch}\,M}^\chi (\Lambda )$ does not depend on $\Lambda $. Hence by Prop. \[pr:VTMiso\] we may assume that ${\hat{T}}_p$ is an isomorphism. Then Eq.  follows from Eq. .

\[le:chsub\] Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ and $t\in \{1,2,\dots ,{b}-1\}$. Assume that $\Lambda (K_pL_p^{-1})=\chi ({\alpha }_p,{\alpha }_p)^{t-1}$. Then $V=U^-(\chi )F_p^t {\otimes }{\mathbb{K}}_\Lambda $ is a $U(\chi ){\otimes }{\mathbb{K}}$-submodule of $M^\chi (\Lambda )$ with $$\fch{V}=\fch{M^\chi }(\Lambda )
  \frac{e^{-t{\alpha }_p}-e^{-{b}{\alpha }_p}}{1-e^{-{b}{\alpha }_p}}.$$

The formal character of the subspace $\oplus _{m=t}^{{b}-1}F_p^m$ of $M^\chi (\Lambda )$ is $$e^{-t{\alpha }_p}+e^{-(t+1){\alpha }_p}+\cdots +e^{-({b}-1){\alpha }_p}=
  \frac{e^{-t{\alpha }_p}-e^{-{b}{\alpha }_p}}{1-e^{-{\alpha }_p}}.$$ Thus the lemma is a consequence of Lemmata \[le:hwvector\], \[le:MLiso\] and Eqs. , .

*In the rest of this section assume that $\chi \in {\mathcal{X}}_4$.* Let $n=|R^\chi _+|$ and $i_1,\dots ,i_n\in I$ with $\ell (1_\chi {\sigma }_{i_1}\cdots \s_{i_n})=n$. Recall the definitions of $\beta _\nu $ and $F_{\beta _\nu }$, where $1\le \nu \le n$, from Eq. . Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. We characterize irreducible Verma modules (see also Lemma \[le:subfch\]).

\[le:Fhweight\] Let $m\in \{0,1,\dots ,n\}$, $\chi '=r_{i_m}\cdots r_{i_2}r_{i_1}(\chi )$ and $w={\sigma }_{i_m}\cdots {\sigma }_{i_2}{\sigma }_{i_1}^\chi $. Then $$\begin{aligned}
    K_{\alpha }&L_{\alpha }^{-1}
    F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}\cdots
    F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
    F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda \\
    &=\frac{{\rho ^{\chi}} ({\alpha })}{{\rho ^{\chi '}}(w({\alpha }))}
    \Lambda (K_{{\alpha }}L_{{\alpha }}^{-1})
    F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}\cdots
    F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
    F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda .
  \end{aligned}$$ for all ${\alpha }\in {\mathbb{Z}}^I$.

By Eq. , the claim is equivalent to the equation $$\begin{aligned}
    {\zeta ^{\chi}} \Big(-\sum _{k=1}^{m-1}({b^{\chi}} (\beta _k)-1)\beta _k\Big )
    (K_{\alpha }L_{\alpha }^{-1})=
    \frac{{\rho ^{\chi}} ({\alpha })}{{\rho ^{\chi '}}(w({\alpha }))}.
    \label{eq:Fhweight}
  \end{aligned}$$ For each $k\in \{1,\dots ,n\}$ let $\chi _k=r_{i_{k-1}}\cdots r_{i_2}r_{i_1}(\chi )$, $w_k={\sigma }_{i_{k-1}}\cdots {\sigma }_{i_2}{\sigma }_{i_1}^\chi $. Since $$\begin{aligned}
    \frac{{\rho ^{\chi}} ({\alpha })}{{\rho ^{\chi '}}(w({\alpha }))}=
    \prod _{k=1}^{m-1}\frac{{\rho ^{\chi _k}}(w_k({\alpha }))}{
    {\rho ^{\chi _{k+1}}}(w_{k+1}({\alpha }))},
    \label{eq:rhochiprod}
  \end{aligned}$$ Lemma \[le:rho\] gives that $$\begin{aligned}
    \frac{{\rho ^{\chi}} ({\alpha })}{{\rho ^{\chi '}}(w({\alpha }))}=
    \prod _{k=1}^{m-1} \chi _k({\alpha }_{i_k},w_k({\alpha }))
    ^{1-\bfun{\chi _k}({\alpha }_{i_k})}
    \prod _{k=1}^{m-1} \chi _k (w_k({\alpha }),{\alpha }_{i_k})
    ^{1-\bfun{\chi _k}({\alpha }_{i_k})}.
  \end{aligned}$$ Using Eqs.  and , this implies that $$\begin{aligned}
    \frac{{\rho ^{\chi}} ({\alpha })}{{\rho ^{\chi '}}(w({\alpha })))}=
    \prod _{k=1}^{m-1} \chi 
    (\beta _k,{\alpha })^{1-{b^{\chi}} (\beta _k)}
    \chi ({\alpha },\beta _k)^{1-{b^{\chi}} (\beta _k)}.
  \end{aligned}$$ Thus Eq.  holds, and the lemma is proven.

Let ${\alpha }\in {\mathbb{Z}}^I$ and $\Lambda '={t}_{i_m}\cdots {t}_{i_2}\VT
  _{i_1}^\chi (\Lambda )$. Then $$\begin{aligned}
    &{\rho ^{\chi '}}(w({\alpha }))K_{\alpha }L_{\alpha }^{-1}
    F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}\cdots
    F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
    F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda \\
    &={\hat{T}}_{i_1}\big(
    {\rho ^{\chi '}}(w({\alpha }))K_{{\sigma }_{i_1}^{\chi }({\alpha })}
    L_{{\sigma }_{i_1}^{\chi }({\alpha })}^{-1}
    {T}_{i_1}^-(F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}\cdots
    F_{\beta _2}^{{b^{\chi}} (\beta _2)-1})v_{{t}_{i_1}^\chi (\Lambda )}
    \big)\\
    &={\hat{T}}_{i_1}{\hat{T}}_{i_2}\big(
    {\rho ^{\chi '}}(w({\alpha }))K_{{\sigma }_{i_2}{\sigma }_{i_1}^{\chi }({\alpha })}
    L_{{\sigma }_{i_2}{\sigma }_{i_1}^{\chi }({\alpha })}^{-1} \times \\
    &\qquad \qquad
    {T}_{i_2}^-{T}_{i_1}^-(F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}
    \cdots F_{\beta _3}^{{b^{\chi}} (\beta _3)-1})
    v_{{t}_{i_2}{t}_{i_1}^\chi (\Lambda )} \big) = \dots\\
    &={\hat{T}}_{i_1}{\hat{T}}_{i_2}\cdots {\hat{T}}_{i_m}\big(
    {\rho ^{\chi '}}(w({\alpha }))K_{w({\alpha })}L_{w({\alpha })}^{-1}
    v_{\Lambda '} \big)\\
    &={\hat{T}}_{i_1}{\hat{T}}_{i_2}\cdots {\hat{T}}_{i_m}\big(
    {\rho ^{\chi '}}(w({\alpha })) \, \Lambda '(K_{w({\alpha })}L_{w({\alpha })}^{-1})
    v_{\Lambda '} \big).
  \end{aligned}$$ Similarly, one obtains that $$\begin{aligned}
    {\rho ^{\chi }}({\alpha })\,\Lambda (K_{\alpha }L_{\alpha }^{-1})
    F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}\cdots
    F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
    F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda \qquad &\\
    =
    {\hat{T}}_{i_1}{\hat{T}}_{i_2}\cdots {\hat{T}}_{i_m}\big(
    {\rho ^{\chi }}({\alpha }) \, \Lambda (K_{{\alpha }}L_{{\alpha }}^{-1})
    v_{\Lambda '} \big).&
  \end{aligned}$$ Now the claim of the lemma follows from Lemma \[le:VTinv\].

\[le:TEFF\] Let $\nu \in \{0,1,\dots ,n\}$. For all $j\in I$, $$\begin{aligned}
    T_{i_1}\cdots T_{i_\nu }(E_j^{r_{i_\nu }\cdots r_{i_2}r_{i_1}(\chi )})
    F_{\beta _\nu }^{{b^{\chi}} (\beta _\nu )-1}\cdots
    F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
    F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda =0
    \label{eq:TEFF}
  \end{aligned}$$ in $M^\chi (\Lambda )$. If $0\le m\le n-1$, then in $M^\chi (\Lambda )$ $$\begin{aligned}
    F_{\beta _{m+n+1}}
    F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}\cdots
    F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
    F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda =0.
  \end{aligned}$$

We proceed by induction on $m$. For $m=0$ the claim is that $E_jv_\Lambda
  =0$ for all $j\in I$, which is clear from the definition of $M^\chi (\Lambda )$. Assume now that $m\ge 1$ and that the claim holds for all smaller values of $m$. Let $j\in I$.

Suppose first that $j=i_m$. Then $$T_{i_1}\cdots T_{i_m}(E_j)= T_{i_1}\cdots T_{i_{m-1}}(F_{i_m}L_{i_m}^{-1})
  =F_{\beta _m}L_{\beta _m}^{-1}.$$ Hence Eq.  follows from Lemma \[le:rvrel\].

Suppose now that $j\not=i_m$. Let $\chi '=r_{i_{m-1}}\cdots r_{i_2}r_{i_1}(\chi )$. Then $$\begin{aligned}
\label{eq:TEFF1}
    T_{i_1}\cdots T_{i_m}(E_j) F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}
    =T_{i_1}\cdots T_{i_{m-1}}(E^+_{j,a(i_m)}
    F_{i_m}^{\bfun{\chi '}({\alpha }_{i_m})-1})
  \end{aligned}$$ where $a=-c^{\chi '}_{i_m j}$. Let $Z=\oplus _{k=0}^a\fie
  E^+_{j,k(i_m)}\subset U^+(\chi ')$. By [@p-Heck07b Cor.5.4], $$E^+_{j,k(i_m)}F_{i_m}-F_{i_m}E^+_{j,k(i_m)}\in \fie
  L_{i_m}E^+_{j,k-1}\subset Z \quad \text{for all $k\in {\mathbb{N}}_0$.}$$ Hence $ZF_{i_m}^{\bfun{\chi '}({\alpha }_{i_m})-1}\subset U(\chi ')Z$. Therefore $$\begin{aligned}
    T_{i_1}\cdots T_{i_m}(E_j) F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}
    &\in T_{i_1}\cdots T_{i_{m-1}}(U(\chi ')Z)\\
    &\subset
    \sum _{j'\in I}U(\chi )T_{i_1}\cdots T_{i_{m-1}}(E_{j'}).
  \end{aligned}$$ by Eq.  and since $Z\subset \sum _{j'\in I}U(\chi ')E_{j'}$. This and induction hypothesis imply Eq. . The last claim of the lemma follows from inserting $j=i_{m+1}$ in Eq.  and using Eq. .

\[le:MLbasis\] Let $m\in \{0,1,\dots ,n\}$ and $\tau $ a permutation of the set $\{1,2,\dots ,n\}$. For all $k\in \{1,2,\dots ,m\}$ let $\chi _k=r_{i_{k-1}}\cdots r_{i_2}r_{i_1}(\chi )$. Assume that $$\begin{aligned}
    \label{eq:MLass}
    \prod _{k=1}^m \prod _{t=1}^{{b^{\chi}} (\beta _k)-1}
    \big( {\rho ^{\chi}} (\beta _k)\Lambda (K_{\beta _k}L_{\beta _k}^{-1})
    -{\rho ^{\chi _k}}({\alpha }_{i_k})^t\big)\not=0.
  \end{aligned}$$ Then the set $$\label{eq:MLbasis}
    \begin{aligned}
      \{ F_{\beta _{m+\tau (1)}}^{l_{m+\tau (1)}}
      F_{\beta _{m+\tau (2)}}^{l_{m+\tau (2)}}\cdots
      F_{\beta _{m+\tau (n)}}^{l_{m+\tau (n)}}
      F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}
      \cdots F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
      F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda \,|&\\
      0\le l_k<{b^{\chi}} (\beta _k)\quad
      \text{for all $k\in \{m+1,m+2,\dots ,m+n\}$}&
    \end{aligned}$$ forms a vector space basis of $M^\chi (\Lambda )$.

For all $k\in \{1,2,\dots ,m\}$ let $\Lambda _k={t}_{i_{k-1}}\cdots {t}_{i_2}{t}_{i_1}^\chi (\Lambda )$. By Lemma \[le:VTinv\] and Eq. , Eq.  is equivalent to $${\rho ^{\chi _k}} ({\alpha }_{i_k})\Lambda _k(K_{i_k}L_{i_k}^{-1})\not=
  \rhomap{\chi _k}({\alpha }_{i_k},{\alpha }_{i_k})^t$$ for all $k\in \{1,2,\dots ,m\}$, $t\in \{1,2,\dots ,{b^{\chi _k}}({\alpha }_{i_k})-1\}$. Hence, by Prop. \[pr:VTMiso\], the map $${\hat{T}}_{i_m}\cdots {\hat{T}}_{i_2}{\hat{T}}_{i_1}: M^{\chi _m}(\Lambda _m)\to
  M^\chi (\Lambda )$$ is an isomorphism. Thus the claim of the lemma holds by Lemma \[le:MLiso\] for $M^{\chi _m}(\Lambda _m)$ and by Thm. \[th:PBWtau\].

\[pr:M=L\] Assume that $$\begin{aligned}
    \label{eq:MLass}
    \prod _{\nu =1}^n \prod _{t=1}^{{b^{\chi}} (\beta _\nu )-1}
    \big( {\rho ^{\chi}} (\beta _\nu )\Lambda (K_{\beta _\nu }
    L_{\beta _\nu }^{-1})-\chi (\beta _\nu ,\beta _\nu )^t\big)\not=0.
  \end{aligned}$$ Then $I^\chi (\Lambda )=0$.

For all $\nu \in \{1,2,\dots ,n\}$ let $\chi _\nu =r_{i_{\nu -1}}\cdots r_{i_2}r_{i_1}(\chi )$ and $\Lambda _\nu ={t}_{i_{\nu -1}}\cdots {t}_{i_2}
  {t}_{i_1}^\chi (\Lambda )$. By Lemma \[le:VTinv\] and Eq. , Eq.  is equivalent to $${\rho ^{\chi _\nu }} ({\alpha }_{i_\nu })\Lambda _\nu (K_{i_\nu }L_{i_\nu }^{-1})\not=
  \chi _\nu ({\alpha }_{i_\nu },{\alpha }_{i_\nu })^t$$ for all $\nu \in \{1,2,\dots ,n\}$, $t\in \{1,2,\dots ,{b^{\chi _\nu }}({\alpha }_{i_\nu })-1\}$. Hence, by Prop. \[pr:VTMiso\], the map $${\hat{T}}_{i_1}{\hat{T}}_{i_2}\cdots {\hat{T}}_{i_n}: M^{r_{i_n}(\chi _n)}
  ({t}_{i_n}(\Lambda _n))\to M^\chi (\Lambda )$$ is an isomorphism. Thus $v=F_{\beta _n}^{{b^{\chi}} (\beta _n)-1}
  \cdots F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
  F_{\beta _1}^{{b^{\chi}} (\beta _1)-1} v_\Lambda \not=0$ and $(U^+(\chi ){\otimes }_{\Bbbk }{\mathbb{K}})v=M^\chi (\Lambda )$. Since $v$ is contained in any nonzero $U(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}$-submodule of $M^\chi (\Lambda )$ by Thms. \[th:EErel\], \[th:PBWtau\], it follows that $I^\chi (\Lambda )=0$.

\[le:FFv\] Let $t\in \{1,2,\dots ,{b}-1\}$. Assume that $\Lambda (K_{i_1}L_{i_1}^{-1})=\chi ({\alpha }_{i_1},{\alpha }_{i_1})^{t-1}$. Then in $M^\chi (\Lambda )$ $$\begin{aligned}
    F_{\beta _{n+\nu -\mu }} F_{\beta _{n+1-\mu }}^{{b^{\chi}} (\beta _{n+1-\mu })-1}
    F_{\beta _{n+2-\mu }}^{{b^{\chi}} (\beta _{n+2-\mu })-1}\cdots
    F_{\beta _n}^{{b^{\chi}} (\beta _n)-1} F_{i_1}^t v_\Lambda =0
  \end{aligned}$$ for all $\mu \in \{0,1,\dots ,n-1\}$ and $\nu \in \{\mu +1,\mu +2,\dots ,n\}$.

By Eq.  and Lemma \[le:hwvector\] it suffices to prove that $$F_{\beta _{n+\nu -\mu }} F_{\beta _{n+1-\mu }}^{{b^{\chi}} (\beta _{n+1-\mu })-1}
    F_{\beta _{n+2-\mu }}^{{b^{\chi}} (\beta _{n+2-\mu })-1}\cdots
    F_{\beta _n}^{{b^{\chi}} (\beta _n)-1} \in
    \sum _{\kappa =1}^{\nu -\mu } U(\chi )F_{\beta _{n+\kappa }}$$ for all $\mu \in \{0,1,\dots ,n-1\}$ and $\nu \in \{\mu +1,\mu +2,\dots ,n\}$. By replacing $\chi $ with an appropriate element in ${\mathcal{G}}(\chi )$ and applying isomorphisms ${T}_i^-$, it suffices to show that $$F_{\beta _\nu } F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}
    F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}\cdots
    F_{\beta _\mu }^{{b^{\chi}} (\beta _\mu )-1} \in
    \sum _{\kappa =1}^{\nu -\mu } U^-(\chi )F_{\beta _{\mu +\kappa }}$$ for all $\mu \in \{0,1,\dots ,n-1\}$ and $\nu \in \{\mu +1,\mu +2,\dots ,n\}$. The latter follows from Thms. \[th:EErel\] and \[th:PBW\] by considering ${\mathbb{Z}}^I$-degrees.

\[th:hwv\] Assume that $\chi \in {\mathcal{X}}_4$. Let $\mu \in \{1,2,\dots ,n\}$, $t\in \{1,2,
  \dots ,{b^{\chi}} (\beta _\mu )-1\}$, and $\chi _\mu =r_{i_{\mu -1}}\cdots
  r_{i_2}r_{i_1}(\chi )$. Assume that $$\begin{aligned}
    {\rho ^{\chi}} (\beta _\mu )\Lambda (K_{\beta _\mu }L_{\beta _\mu }^{-1})
    ={\rho ^{\chi _\mu }}({\alpha }_{i_\mu })^t.
    \label{eq:maid1}
  \end{aligned}$$ Then $$\begin{aligned}
    E_i F_{\beta _{n+1}}^{{b^{\chi}} (\beta _{n+1})-1}
    F_{\beta _{n+2}}^{{b^{\chi}} (\beta _{n+2})-1}\cdots
    F_{\beta _{n+\mu -1}}^{{b^{\chi}} (\beta _{n+\mu -1})-1} F_{\beta _\mu }^t \times
    \qquad &
    \\
    F_{\beta _{\mu -1}}^{{b^{\chi}} (\beta _{\mu -1})-1}\cdots
    F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
    F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda =0&
  \end{aligned}$$ for all $i\in I$.

By Eq.  it suffices to prove that $$\begin{aligned}
    F_{\beta _{n+\nu }} F_{\beta _{n+1}}^{{b^{\chi}} (\beta _{n+1})-1}
    F_{\beta _{n+2}}^{{b^{\chi}} (\beta _{n+2})-1}\cdots
    F_{\beta _{n+\mu -1}}^{{b^{\chi}} (\beta _{n+\mu -1})-1} F_{\beta _\mu }^t \times
    \qquad &
    \\
    F_{\beta _{\mu -1}}^{{b^{\chi}} (\beta _{\mu -1})-1}\cdots
    F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
    F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda =0&
  \end{aligned}
    \label{eq:hwv}$$ for all $\nu \in \{1,2,\dots ,n\}$. Let first $\nu \in \{1,2,\dots ,\mu -1\}$. Then $$F_{\beta _{n+\nu }} F_{\beta _{n+1}}^{{b^{\chi}} (\beta _{n+1})-1}
  F_{\beta _{n+2}}^{{b^{\chi}} (\beta _{n+2})-1}\cdots
  F_{\beta _{n+\mu -1}}^{{b^{\chi}} (\beta _{n+\mu -1})-1} =0$$ by Thms. \[th:PBW\] and \[th:EErel\] by considering the ${\mathbb{Z}}^I$-degree. Thus Eq.  holds. Let now $\nu \in \{\mu ,\mu +1,\dots ,n\}$, $\Lambda _\mu ={t}_{i_{\mu -1}}\cdots {t}_{i_2}{t}_{i_1}^\chi (\Lambda )$, and $\beta '_\kappa =1_{\chi _\mu }{\sigma }_{i_\mu }{\sigma }_{i_{\mu +1}}\cdots
  {\sigma }_{i_{\mu +\kappa -2}}({\alpha }_{i_{\mu +\kappa -1}})$ for all $\kappa \in \{1,2,\dots ,2n\}$. Then by Lemma \[le:VTinv\], Eq.  is equivalent to $${\rho ^{\chi _\mu }}({\alpha }_{i_\mu })\Lambda _\mu (K_{i_\mu }L_{i_\mu }^{-1})
    ={\rho ^{\chi _\mu }}({\alpha }_{i_\mu })^t,$$ and ${\alpha }_{i_\mu }=\beta '_1$. Thus we can apply Lemma \[le:FFv\] (with $\Lambda _\mu $ instead of $\Lambda $, $\chi _\mu $ instead of $\chi $, $\beta '_\kappa $ instead of $\beta _\kappa $, and $\mu -1$ instead of $\mu $). We obtain that $$\begin{aligned}
    F_{\beta '_{n+\nu +1-\mu }} F_{\beta '_{n+2-\mu }}^{{b^{\chi}} (\beta '_{n+2-\mu })-1}
    F_{\beta '_{n+3-\mu }}^{{b^{\chi}} (\beta '_{n+3-\mu })-1}\cdots
    F_{\beta '_n}^{{b^{\chi}} (\beta '_n)-1} F_{i_\mu }^t v_{\Lambda _\mu }=0
  \end{aligned}$$ for all $\mu \in \{1,2,\dots ,n\}$ and $\nu \in \{\mu ,\mu +1,\dots ,n\}$. Apply on both sides of this equation the map ${\hat{T}}_{i_1}{\hat{T}}_{i_2}\cdots {\hat{T}}_{i_{\mu -1}}:
  M^{\chi _\mu }(\Lambda _\mu )\to M^\chi (\Lambda )$ to obtain Eq. .

The Shapovalov form {#sec:shapdet}
===================

We discuss the analog of the Shapovalov form for the algebras $U(\chi )$ following the construction in [@b-Joseph 3.4.10].

Let $\chi \in {\mathcal{X}}$. By Prop. \[pr:tridec\], there exists a decomposition $${\mathcal{U}}(\chi )=\Big(\sum _{i\in I}F_i{\mathcal{U}}(\chi )+\sum _{i\in I}{\mathcal{U}}(\chi )E_i
\Big)
\oplus \Uz$$ and hence a unique projection $${\theta ^{\chi}} :{\mathcal{U}}(\chi )\to \Uz$$ with kernel $\sum _{i\in I}F_i{\mathcal{U}}(\chi )+\sum _{i\in I}{\mathcal{U}}(\chi )E_i$. This map is commonly known as the *Harish-Chandra map*. By definition, ${\theta ^{\chi}} $ satisfies the property $$\begin{aligned}
  {\theta ^{\chi}} (u_-uu_+)={\varepsilon }(u_-){\theta ^{\chi}} (u){\varepsilon }(u_+)
  \label{eq:HCprop}\end{aligned}$$ for all $u_-\in U^-(\chi )$, $u\in U(\chi )$, $u_+\in U^+(\chi )$. Since ${\Omega }(u)=u$ for all $u\in {{\mathcal{U}}^0}$, $$\begin{aligned}
  {\theta ^{\chi}} ({\Omega }(u))={\theta ^{\chi}} (u)\qquad
  \text{for all $u\in {\mathcal{U}}(\chi )$.}
  \label{eq:HCprop2}\end{aligned}$$ The bilinear map $$\begin{aligned}
  {\mathrm{Sh}}: {\mathcal{U}}(\chi )\times {\mathcal{U}}(\chi )\to {{\mathcal{U}}^0},\qquad
  {\mathrm{Sh}}(u,v)={\theta ^{\chi}} ({\Omega }(u)v),
  \label{eq:Shf}\end{aligned}$$ is called the *Shapovalov form*. By Eq.  and since ${\Omega }^2={\operatorname{id}}$, $$\begin{aligned}
  {\mathrm{Sh}}(u,v)={\mathrm{Sh}}(v,u) \quad \text{for all $u,v\in {\mathcal{U}}(\chi )$.}
  \label{eq:Shfprop}\end{aligned}$$ Moreover, by definition of ${\mathrm{Sh}}$ and ${\theta ^{\chi}} $, $$\begin{aligned}
  {\mathrm{Sh}}(u,v)=0 \quad  \text{if $u\in \sum _{i\in I}{\mathcal{U}}(\chi )E_i$ or
  $v\in \sum _{i\in I}{\mathcal{U}}(\chi )E_i$.}
  \label{eq:Shfprop2}\end{aligned}$$ Recall the definitions of $U(\chi )$, ${\mathcal{I}}^+(\chi )$ and ${\mathcal{I}}^-(\chi
)$ from Sect. \[sec:DD\]. Since ${\mathcal{U}}(\chi ){\mathcal{I}}^+(\chi ){\mathcal{U}}(\chi )+
{\mathcal{U}}(\chi ){\mathcal{I}}^-(\chi ){\mathcal{U}}(\chi )\subset \ker {\theta ^{\chi}} $, ${\theta ^{\chi}} $ and ${\mathrm{Sh}}$ induce maps $${\theta ^{\chi}} : U(\chi )\to {{\mathcal{U}}^0},\qquad
{\mathrm{Sh}}: U(\chi )\times U(\chi )\to {{\mathcal{U}}^0}.$$ The map ${\theta ^{\chi}} $ is ${\mathbb{Z}}^I$-homogeneous, that is, ${\theta ^{\chi}} (u)=0$ for all $u\in U(\chi )_{\alpha }$, where ${\alpha }\in {\mathbb{Z}}^I\setminus \{0\}$. The map ${\Omega }$ reverses degrees, that is, ${\Omega }(u)\in U(\chi)_{-{\alpha }}$ for all $u\in U(\chi )_{\alpha }$, where ${\alpha }\in {\mathbb{Z}}^I$. Therefore, for all ${\alpha },\beta \in {\mathbb{Z}}^I$, where $\al
\not=\beta $, we get $$\begin{aligned}
  {\mathrm{Sh}}(u,v)=0 \quad \text{for all $u\in U(\chi )_{\alpha }$,
  $v\in U(\chi )_\beta $ \quad $({\alpha }\not=\beta )$.}
  \label{eq:Shfprop3}\end{aligned}$$

\[de:Shapdet\] The family of determinants $$\det \nolimits ^\chi _{\alpha }=
  \det {\mathrm{Sh}}(F'_i,F'_j)_{i,j\in \{1,\dots ,k\}}\in {{\mathcal{U}}^0}/{{\Bbbk }^\times },$$ where ${\alpha }\in {\mathbb{N}}_0^I$, $k=\dim U^-(\chi )_{-{\alpha }}$, and $\{F'_1,F'_2,\dots ,F'_k\}$ is a basis of $U^-(\chi )_{-{\alpha }}$, is called the *Shapovalov determinant* of $U(\chi )$.

\[re:Shdet\] Let ${\alpha }\in {\mathbb{N}}_0^I$ and $k=\dim U^-(\chi )_{-{\alpha }}$. By the above considerations, ${\mathrm{Sh}}:U^-(\chi )_{-{\alpha }}\times U^-(\chi )_{-{\alpha }}\to {{\mathcal{U}}^0}$ is a symmetric bilinear form for all ${\alpha }\in {\mathbb{N}}_0^I$. Let $F'=\{F'_1,F'_2,\dots ,F'_k\}$ be a basis of $U^-(\chi )_{-{\alpha }}$, and let $d(F')=\det {\mathrm{Sh}}(F'_i,F'_j)_{i,j\in \{1,\dots ,k\}} \in {{\mathcal{U}}^0}$. Then $d(A'F')=(\det A')^2d(F')$ for all $A'\in \mathrm{GL}(k,{\Bbbk })$, and hence $\det ^\chi _{\alpha }=d(F')/{{\Bbbk }^\times }$ does not depend on the choice of the basis $F'$ of $U^-(\chi )_{-{\alpha }}$.

\[le:Uzideal\] Let $\chi \in {\mathcal{X}}$. Let $J$ be an ideal of ${{\mathcal{U}}^0}$. Assume that $J$ is contained in the center of $U(\chi )$. Let $J_U$ be the ideal of $U(\chi )$ generated by $J$. Then $\Shf
  :U(\chi )\times U(\chi )\to {{\mathcal{U}}^0}$ induces a map ${\mathrm{Sh}}:U(\chi )/J_U\times U(\chi )/J_U\to {{\mathcal{U}}^0}/J$.

Since $J$ is contained in the center of $U(\chi )$, triangular decomposition of $U(\chi )$ yields that $J_U=U^-(\chi )JU^+(\chi )$. This and ${\Omega }(J)=J$ imply the claim of the lemma.

Let $\chi \in {\mathcal{X}}$, ${\alpha }\in {\mathbb{Z}}^I$, $E\in U^+(\chi )_{\alpha }$, and $F\in U^-(\chi )_{-{\alpha }}$. Then $${\mathrm{Sh}}(\Omega (E),F)\in
  \sum _{\beta ,\gamma \in {\mathbb{N}}_0^I,\, \beta +\gamma ={\alpha }}
  {\Bbbk }K_\beta L_\gamma$$ Moreover, the coefficients of $K_{\alpha }$ and $L_{\alpha }$ of ${\mathrm{Sh}}({\Omega }(E),F)$ are ${\eta }(E,S(F))$ and ${\eta }(E,F)$, respectively. \[le:Shfcoeffs\]

If ${\alpha }\notin {\mathbb{N}}_0^I$ or ${\alpha }=0$, then $U ^+(\chi )_{\alpha }=0$ or $U ^+(\chi )_{\alpha }={\Bbbk }$. In this case the claim of the lemma holds by definition of ${\mathrm{Sh}}$. Assume now that ${\alpha }\in {\mathbb{N}}_0^I\setminus \{0\}$. Using Eqs. –, by induction on $\beta $ and $\beta '$ one can show that $$E'F'\in \sum _{\gamma _1,\gamma _2,\gamma _3,\gamma _4\in {\mathbb{N}}_0^I,
  \gamma _4-\gamma _1=\beta -\beta ',\gamma _2+\gamma _3+\gamma _4=\beta }
  U ^-(\chi )_{-\gamma _1}K_{\gamma _2}L_{\gamma _3}U ^+(\chi )_{\gamma
  _4}$$ for all $\beta ,\beta '\in {\mathbb{N}}_0^I$ and $E'\in U ^+(\chi )_\beta $, $F'\in U ^-(\chi )_{-\beta '}$. This implies the first claim of the lemma by letting $\beta =\beta '={\alpha }$ and $E'=E$, $F'=F$. The second claim follows from $$\begin{aligned}
    {\varDelta }(E)-K_{\alpha }{\otimes }E-E{\otimes }1\in &\mathop{\oplus }
    _{\beta ,\gamma \in {\mathbb{N}}_0^I, \beta +\gamma ={\alpha },\,\beta ,\gamma \not=0}
    U ^+(\chi )_\beta K_\gamma {\otimes }U ^+(\chi )_\gamma ,\\
    {\varDelta }(F)-1 {\otimes }F-F{\otimes }L_{\alpha }\in &\mathop{\oplus }
    _{\beta ,\gamma \in {\mathbb{N}}_0^I, \beta +\gamma ={\alpha },\beta ,\gamma \not=0}
    U ^-(\chi )_{-\beta }{\otimes }U ^-(\chi )_{-\gamma }L_\beta ,
  \end{aligned}$$ and from Eq.  (with $x=E$, $y=F$) and Prop. \[pr:sHpdef\](iii).

Let ${\mathbb{K}}$ be a field extension of ${\Bbbk }$. The importance of the Shapovalov form arises from the fact that it induces a form on the Verma modules $M^\chi (\Lambda )$ and on their simple quotients $L^\chi (\Lambda )$, where $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$.

Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Define $$\begin{aligned}
  \Lambda {\mathrm{Sh}}:U(\chi )\times U(\chi )\to {\mathbb{K}},\quad
  (u,v)\mapsto \Lambda ({\mathrm{Sh}}(u,v)).
  \label{eq:LSh}\end{aligned}$$ By Eq. , $$\begin{aligned}
  \Lambda {\mathrm{Sh}}(u_-u_0u_+,v_-v_0v_+)=&
  {\varepsilon }(u_+){\varepsilon }(v_+)\Lambda (u_0{\mathrm{Sh}}(u_-,v_-)v_0)\\
  =&\Lambda (u_0)\Lambda (v_0){\varepsilon }(u_+){\varepsilon }(v_+)
  \Lambda {\mathrm{Sh}}(u_-,v_-).\end{aligned}$$ Thus, by Eq. , $\Lambda {\mathrm{Sh}}$ induces a ${\mathbb{K}}$-bilinear form on $M^\chi (\Lambda )$ by letting $$\Lambda {\mathrm{Sh}}:M(\Lambda )\times M(\Lambda )\to {\mathbb{K}},\quad
(u{\otimes }1_\Lambda , v{\otimes }1_\Lambda )\mapsto \Lambda {\mathrm{Sh}}(u,v)$$ for all $u,v\in U(\chi )$. Moreover, Eq.  gives that $$\begin{aligned}
\label{eq:LShf}
  \Lambda {\mathrm{Sh}}( u{\otimes }1_\Lambda , v)
  =\Lambda {\mathrm{Sh}}(1{\otimes }1_\Lambda ,{\Omega }(u)v) =0\end{aligned}$$ for all $u\in U(\chi )$ and $v\in I^\chi (\Lambda )$, since ${\Omega }(u)v \in I^\chi (\Lambda )\subset
\oplus _{{\alpha }\not=0}M^\chi (\Lambda )_{\alpha }$. Thus by Eq. , $\Lambda {\mathrm{Sh}}$ induces a symmetric bilinear form on $L^\chi (\Lambda )$, also denoted by $\Lambda {\mathrm{Sh}}$. The radical of this form is a ${\mathbb{Z}}^I$-graded $U(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}$-submodule of $L^\chi (\Lambda )$, but does not contain $1{\otimes }1_\Lambda $, and hence it is zero. Thus $\Lambda {\mathrm{Sh}}$ is a nondegenerate symmetric bilinear form on $L^\chi (\Lambda )$.

[*For the rest of this section let $\chi \in {\mathcal{X}}_4$, $n=|R^\chi _+|$, and $i_1,\dots ,i_n\in I$ with $\ell (1_\chi {\sigma }_{i_1}\cdots \s_{i_n})=n$.*]{} For all $\nu \in \{1,2,\dots ,n\}$ let $$\beta _\nu =1_\chi {\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_{\nu -1}}({\alpha }_{i_\nu }),\qquad
\chi _\nu =r_{i_{\nu -1}}\cdots r_{i_2}r_{i_1}(\chi ).$$ For all $\nu \in \{1,2,\dots ,n\}$, ${\alpha }\in {\mathbb{N}}_0^I$, and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu )-1\}$ let $$\begin{aligned}
  {P}^\chi ({\alpha },\beta _\nu ;t)=\Big|\Big\{(m_1,\dots ,m_n)\in {\mathbb{N}}_0^n\,\big|\,
  \sum _{\mu =1}^n m_\mu \beta _\mu ={\alpha },\,m_\nu \ge t,\quad &\\
  m_\mu <{b^{\chi}} (\beta _\mu )\quad \text{for all $\mu \in \{1,2,\dots ,n\}$}
  \Big\}\Big|.&
  \label{eq:PF}
\end{aligned}$$

We will use two important facts on the function ${P}^\chi $.

\[le:P1\] For all $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu)-1\}$, $$\sum _{{\alpha }\in {\mathbb{N}}_0^I}{P}^\chi ({\alpha },\beta _\nu ;t)e^{-{\alpha }}=
  \frac {e^{-t\beta _\nu}-e^{-{b^{\chi}} (\beta _\nu )\beta _\nu }}
  {1-e^{-\beta _\nu }}
  \prod _{\mu \in \{1,\dots ,n\},\, \mu \not=\nu }
  \frac {1-e^{-{b^{\chi}} (\beta _\mu )\beta _\mu }}
  {1-e^{-\beta _\mu }}.$$

By Thm. \[th:PBWtau\], the two sides of the equation are two different expressions for the formal character of the subspace of $U^-(\chi ){\otimes }{\mathbb{K}}$ spanned by the elements $$\prod _{ {m_1,\dots ,m_n \atop m_\nu \ge t,\, 0\le m_\mu <{b^{\chi}} (\beta _\mu )
  \,\text{for all $\mu $}}}
  F_{\beta _1}^{m_1}F_{\beta _2}^{m_2}\cdots F_{\beta _n}^{m_n}.$$

\[le:P2\] For all ${\alpha }\in {\mathbb{N}}_0^I$, $$\begin{aligned}
    {\alpha }\dim U^-(\chi )_{-{\alpha }}=
    \sum _{\nu =1}^n \sum _{t=1}^{{b^{\chi}} (\beta _\nu )-1}
    {P}^\chi ({\alpha },\beta _\nu ;t) \beta _\nu .
  \end{aligned}$$

By Thm. \[th:PBWtau\], for each ${\alpha }\in {\mathbb{N}}_0^I$ there is a basis of $U^-(\chi )_{-{\alpha }}$ parametrized by the set $$\Big\{(m_1,\dots ,m_n)\in {\mathbb{N}}_0^n\,\big|\,
    \sum _{\mu =1}^n m_\mu \beta _\mu ={\alpha },\,\,
    m_\mu <{b^{\chi}} (\beta _\mu )\,\, \text{for all $\mu $}\Big\}.
    \label{eq:dimU-}$$ Each $(m_1,\dots ,m_n)$ in this set contributes to ${P}^\chi ({\alpha },\beta _\nu ;t)$ with a summand $1$, for all $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,m_\nu \}$. Thus the claim of the lemma follows from the decomposition of the ${P}^\chi ({\alpha },\beta _\nu ;t)$ into $1+1+\cdots +1$ by reordering the summands.

\[le:subfch\] Let $\nu \in \{1,2,\dots ,n\}$, $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu)-1\}$, and $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Assume that ${\rho ^{\chi}} (\beta _\nu )\Lambda (K_{\beta _\nu }
  L_{\beta _\nu }^{-1})
  =\chi (\beta _\nu ,\beta _\nu )^t$ and $$\prod _{\mu =1}^{\nu -1} \prod _{m=1}^{{b^{\chi}} (\beta _\mu )-1}
  ({\rho ^{\chi}} (\beta _\mu )
  \Lambda (K_{\beta _\mu }L_{\beta _\mu }^{-1})-\chi (\beta _\mu ,
  \beta _\mu )^m)\not=0.$$ Then $M^\chi (\Lambda )$ contains a $U(\chi ){\otimes }{\mathbb{K}}$-submodule $V$ with $$\begin{aligned}
    {\mathrm{ch}\,V}=\sum _{{\alpha }\in {\mathbb{N}}_0^I}{P}^\chi ({\alpha },\beta _\nu ;t)e^{-{\alpha }}.
    \label{eq:fchV}
  \end{aligned}$$ In particular, $0\not=V\subset I^\chi (\Lambda )$.

We proceed by induction on $\nu $. Let first $\nu =1$. By Lemma \[le:chsub\], $V=U^-(\chi )F_{i_1}^t{\otimes }{\mathbb{K}}_\Lambda $ is a $U(\chi ){\otimes }{\mathbb{K}}$-submodule of $M^\chi (\Lambda )$. Then Eq.  follows from Thm. \[th:PBWtau\].

Assume now that $\nu \in \{2,3,\dots ,n\}$ and that the lemma holds for $\nu -1$. Let $\chi _\mu =r_{i_{\mu -1}}\cdots
  r_{i_2}r_{i_1}(\chi )$ and $\Lambda _\mu ={t}_{i_{\mu -1}}\cdots {t}_{i_2}
  {t}_{i_1}^\chi (\Lambda )$ for all $\mu \in \{1,2,\dots ,\nu \}$. By Lemma \[le:VTinv\], the assumptions on $\Lambda $ are equivalent to the relations $$\prod _{\mu =1}^{\nu -1} \prod _{m=1}^{{b^{\chi _\mu }}({\alpha }_{i_\mu })-1}
  (\Lambda _\mu (K_{i_\mu }L_{i_\mu }^{-1})
  -{\rho ^{\chi _\mu }}({\alpha }_{i_\mu })^{m-1}) \not=0$$ and $\Lambda _\nu (K_{i_\nu }L_{i_\nu }^{-1})=
  {\rho ^{\chi _\nu }}({\alpha }_{i_\nu })^{t-1}$. Let $$\beta '_\nu ={\sigma }_{i_1}^\chi (\beta _\nu )=1_{\chi _2}{\sigma }_{i_2}\s
  _{i_3}\cdots {\sigma }_{i_{\nu -1}}({\alpha }_{i_\nu }).$$ By induction hypothesis there exists a $U(\chi _2)$-submodule $V'$ of $M^{\chi _2}(\Lambda _2)$ with $$\begin{aligned}
    \fch{V'}=\sum _{{\alpha }\in {\mathbb{N}}_0^I}{P}^{\chi _2}({\alpha },\beta '_\nu ;t)e^{-{\alpha }}.
  \end{aligned}$$ Moreover, $\Lambda (K_{i_1}L_{i_1}^{-1})\not=\rho ^\chi ({\alpha }_{i_1})^{m-1}$ for all $m\in
  \{1,2,\dots ,{b^{\chi}} ({\alpha }_{i_1})-1\}$, and hence ${\hat{T}}_{i_1}:M^{\chi _2}(\Lambda _2)\to M^\chi (\Lambda )$ is an isomorphism. Let $V={\hat{T}}_{i_1}(V')$. By Lemmata \[le:MLmap\] and \[le:Tpfch\], $V$ is a $U(\chi )$-submodule of $M^\chi (\Lambda )$ and $$\begin{aligned}
    {\mathrm{ch}\,V}={\dot{\sigma }}_{i_1}^{\chi _2}(\fch{V'})=\sum _{{\alpha }\in {\mathbb{N}}_0^I}
    {\dot{\sigma }}_{i_1}^{\chi _2}({P}^{\chi _2}({\alpha },\beta '_\nu ;t)e^{-{\alpha }}).
  \end{aligned}$$ Thus, by Lemma \[le:P1\], $$\begin{aligned}
    {\mathrm{ch}\,V}=&e^{(1-{b^{\chi}} ({\alpha }_{i_1})){\alpha }_{i_1}}{\sigma }_{i_1}^{\chi _2}\Big(
    \frac {e^{-t\beta '_\nu}-e^{-\bfun{\chi _2}(\beta '_\nu )\beta '_\nu }}
    {1-e^{-\beta '_\nu }}
    \prod _{\beta \in R_+^{\chi _2}\setminus \{\beta '_\nu \}}
    \frac {1-e^{-\bfun{\chi _2}(\beta )\beta }}
    {1-e^{-\beta }}\Big).
  \end{aligned}$$ Recall that $\beta '_\nu \not={\alpha }_{i_1}$, since $\nu >1$. Moreover, $$\begin{aligned}
    &e^{(1-{b^{\chi}} ({\alpha }_{i_1})){\alpha }_{i_1}}{\sigma }_{i_1}^{\chi _2}\Big(
    \frac {1-e^{-\bfun{\chi _2}({\alpha }_{i_1}){\alpha }_{i_1}}}
    {1-e^{-{\alpha }_{i_1}}}\Big)\\
    &\quad =e^{(1-{b^{\chi}} ({\alpha }_{i_1})){\alpha }_{i_1}}
    \frac {1-e^{\bfun\chi ({\alpha }_{i_1}){\alpha }_{i_1}}}
    {1-e^{{\alpha }_{i_1}}}
    =\frac {1-e^{-\bfun\chi ({\alpha }_{i_1}){\alpha }_{i_1}}}
    {1-e^{-{\alpha }_{i_1}}}.
  \end{aligned}$$ Therefore $$\begin{aligned}
    {\mathrm{ch}\,V}=&\frac {e^{-t\beta _\nu}-e^{-\bfun{\chi }(\beta _\nu )\beta _\nu }}
    {1-e^{-\beta _\nu }}
    \prod _{\beta \in R_+^\chi \setminus \{\beta _\nu \}}
    \frac {1-e^{-\bfun{\chi }(\beta )\beta }}
    {1-e^{-\beta }}=\sum _{{\alpha }\in {\mathbb{N}}_0^I} {P}^\chi ({\alpha },\beta _\nu ;t)
    e^{-{\alpha }}
  \end{aligned}$$ by Lemma \[le:P1\]. This proves Eq. .

Since $t>0$, ${\mathrm{ch}\,V}\not={\mathrm{ch}\,M}^\chi (\Lambda )$. By assumption on $t$, $V_{t\beta _\nu }\not=0$, and hence $V\not=0$. Since $V$ is a ${\mathbb{Z}}^I$-graded $U(\chi )$-submodule of $M^\chi (\Lambda )$, the lemma is proven.

\[th:Shapdet\] Let $\chi \in {\mathcal{X}}_5$. For all ${\alpha }\in {\mathbb{N}}_0^I$, the Shapovalov determinant of $U(\chi )$ is the family $(\det ^\chi _\al
  )_{{\alpha }\in {\mathbb{N}}_0^I}$, where $$\begin{aligned}
    \label{eq:det}
    \det \nolimits ^\chi _{\alpha }=
    \prod _{\beta \in R^\chi _+} \prod _{t=1}^{{b^{\chi}} (\beta _\nu )-1}
    ({\rho ^{\chi}} (\beta )K_{\beta }
    -\chi (\beta ,\beta )^t L_{\beta })
    ^{{P}^\chi ({\alpha },\beta ;t)}.
  \end{aligned}$$

Let ${\alpha }\in {\mathbb{N}}_0^I$, $k=\dim U^-(\chi )_{-{\alpha }}$, and let $\{F'_1,F'_2,\dots ,F'_k\}$ be a basis of $U^-(\chi )_{-{\alpha }}$. Then ${\mathrm{Sh}}(F'_i,F'_j)\in \sum _{\beta ,\gamma \in {\mathbb{N}}_0^I,\beta +\gamma
  ={\alpha }}{\Bbbk }K_\beta L_\gamma $ by Lemma \[le:Shfcoeffs\], and hence $$\begin{aligned}
\label{eq:detsummands}
    \det \nolimits ^\chi _{\alpha }\in
    \sum _{\beta ,\gamma \in {\mathbb{N}}_0^I,\beta +\gamma =k{\alpha }}
    {\Bbbk }K_\beta L_\gamma .
  \end{aligned}$$ The polynomials $${\rho ^{\chi}} (\beta _\nu )K_{\beta _\nu }
  -\chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu }={T}_{i_1}\cdots
  {T}_{i_{\nu -1}}({\rho ^{\chi}} (\beta _\nu )K_{i_\nu }
  -\chi (\beta _\nu ,\beta _\nu )^t L_{i_\nu })$$ are irreducible and pairwise distinct for all $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu )-1\}$. Thus by Lemma \[le:P2\] it suffices to prove that $\det ^\chi _{\alpha }\not=0$ and that the polynomials $({\rho ^{\chi}} (\beta _\nu )K_{\beta _\nu }
  -\chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu })
  ^{{P}^\chi ({\alpha },\beta _\nu ;t)}$ are factors of $\det ^\chi _{\alpha }$.

Let ${\bar{{\Bbbk }}}$ be the algebraic closure of ${\Bbbk }$ and ${\mathbb{T}}={\mathrm{maxspec}\,}{{\mathcal{U}}^0}{\otimes }_{\Bbbk }{\bar{{\Bbbk }}}$ the algebraic torus. The points of ${\mathbb{T}}$ are just the ${{\bar{{\Bbbk }}}^\times }$-valued characters of ${{\mathcal{U}}^0}$. The equation $\det ^\chi _{\alpha }=0$ defines a closed affine subvariety ${\mathbb{T}}'_{\alpha }\subset {\mathbb{T}}$. Let $\Lambda \in {\mathbb{T}}$. By definition, $\Lambda \in {\mathbb{T}}'_{\alpha }$ if and only if $\Lambda {\mathrm{Sh}}: U^-(\chi )_{-{\alpha }}\times U^-(\chi )_{-{\alpha }}\to {\bar{{\Bbbk }}}$ is a degenerate symmetric bilinear form, that is, if $I^\chi (\Lambda )_{-{\alpha }}\not=0$. Thus, by Prop. \[pr:M=L\], ${\mathbb{T}}'_{\alpha }$ is a subset of the finite union of irreducible varieties $${\mathbb{T}}_{{\alpha },\nu ,t}={\mathrm{maxspec}\,}({{\mathcal{U}}^0}{\otimes }_{\Bbbk }{\bar{{\Bbbk }}})/
  ({\rho ^{\chi}} (\beta _\nu )K_{\beta _\nu }-
  \chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu }),$$ where $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu )-1\}$. By Eq.  $$\det \nolimits ^\chi _{\alpha }=
  f\prod _{\nu =1}^n\prod _{t=1}^{{b^{\chi}} (\beta _\nu )-1}
  ({\rho ^{\chi}} (\beta _\nu )K_{\beta _\nu }-
  \chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu })^{N_{\nu ,t}}$$ for some $N_{\nu ,t}\in {\mathbb{N}}_0$ and an element $f\in {\Bbbk }[K_i,L_i\,|\,i\in I]$ which is invertible on ${\mathbb{T}}$. In particular, $\det ^\chi _{\alpha }\not=0$. We finish the proof of the theorem by showing that $N_{\nu ,t}\ge {P}^\chi ({\alpha },\beta _\nu ;t)$ for all $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu )-1\}$. The essential ingredients will be Lemmata \[le:subfch\] and \[le:detXfactor\].

Let $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu )-1\}$. Let $w=1_\chi {\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_{\nu -1}}$. Then $${{\mathcal{U}}^0}={\Bbbk }[K_{w({\alpha }_j)},K_{w({\alpha }_j)}^{-1},
  L_{w({\alpha }_j)},L_{w({\alpha }_j)}^{-1}\,|\,j\in I]$$ and $w({\alpha }_{i_\nu })=\beta _\nu $. Let $$B ={\Bbbk }[L_{\beta _\nu },L_{\beta _\nu }^{-1},
  K_{w({\alpha }_j)},K_{w({\alpha }_j)}^{-1},
  L_{w({\alpha }_j)},L_{w({\alpha }_j)}^{-1}\,|\,j\in I \setminus \{i_\nu \}]$$ and $x={\rho ^{\chi}} (\beta _\nu )K_{\beta _\nu }-
  \chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu }$. Then $${{\mathcal{U}}^0}\simeq B[x,(x+
  \chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu })^{-1}].$$ Let $X'=(x'_{ij})_{i,j\in \{1,2,\dots ,k\}}\in ({{\mathcal{U}}^0})^{k\times k}$ with $$x'_{ij}={\mathrm{Sh}}(F'_i,F'_j) \quad \text{for all $i,j\in \{1,2,\dots ,k\}$.}$$ Let $l\in {\mathbb{Z}}$ such that $K_{\beta _\nu }^l X'\in B[x]^{k\times k}$, and let $X=K_{\beta _\nu }^l X'$. By Lemma \[le:subfch\] and Eq.  there is a non-empty open subset of the variety of $B\simeq
  {{\mathcal{U}}^0}/(x)$ such that $\mathrm{rk}\,X(0)_p\le k-{P}^\chi ({\alpha },\beta _\nu ;t)$ for all $p$ in this set. By Lemma \[le:detXfactor\], $\det X=x^{{P}^\chi ({\alpha },\beta _\nu ;t)}b'$ for some $b'\in
  B[x]$. In particular, $x^{{P}^\chi ({\alpha },\beta _\nu ;t)}$ is a factor of $\det ^\chi _{\alpha }$, and the proof of the theorem is complete.

Shapovalov determinants for bicharacters with finite root systems {#sec:shapdetgen}
=================================================================

In Sect. \[sec:shapdet\] we mainly considered bicharacters $\chi \in {\mathcal{X}}_5$. Here we extend our results to all $\chi \in {\mathcal{X}}_3$ with $\chi (\beta ,\beta )\not=1$ for all $\beta \in R^\chi _+$.

In what follows let ${\overline{{\mathcal{X}}}}$ denote the set of ${{\bar{{\Bbbk }}}^\times }$-valued bicharacters on ${\mathbb{Z}}^I$. Identify ${\overline{{\mathcal{X}}}}$ with $({{\bar{{\Bbbk }}}^\times })^{I\times I}$ via $\chi \mapsto (\chi ({\alpha }_i,{\alpha }_j) )_{i,j\in I}$ for all $\chi \in {\overline{{\mathcal{X}}}}$. For all $i\in \{1,2,3,4,5\}$ define ${\overline{{\mathcal{X}}}}_i\subset {\overline{{\mathcal{X}}}}$ in analogy to Eqs. –. Note that ${\mathcal{X}}_i={\mathcal{X}}\cap {\overline{{\mathcal{X}}}}_i$ for all $i\in \{1,2,3,4,5\}$.

For all $\beta ,\beta '\in {\mathbb{Z}}^I$ let $f_{\beta ,\beta '}$ be the rational function on the affine variety ${\overline{{\mathcal{X}}}}=({{\bar{{\Bbbk }}}^\times })^{I\times I}$ such that $$f_{\beta ,\beta '}(\chi )=\chi (\beta ,\beta ') \quad
\text{for all $\chi \in {\overline{{\mathcal{X}}}}$.}$$ Clearly, the functions $f_{\beta ,\beta '}$ with $\beta ,\beta '\in \{\al
_i,-{\alpha }_i\,|\,i\in I\}$ generate the algebra ${\bar{{\Bbbk }}}[{\overline{{\mathcal{X}}}}]$. Recall that a subset of ${\overline{{\mathcal{X}}}}$ is locally closed, if it is the intersection of an open and a closed subset of ${\overline{{\mathcal{X}}}}$.

\[pr:Vchi\] Let $\chi \in {\overline{{\mathcal{X}}}}_3$. Assume that $\chi (\beta ,\beta )\not=1$ for all $\beta \in R^\chi _+$. Let $\underline{n}=(n_\beta )_{\beta \in R^\chi _{+\infty }}$ with $n_\beta \in {\mathbb{N}}$ for all $\beta \in R^\chi _{+\infty }$. Then there exists an ideal $J\subsetneq {\bar{{\Bbbk }}}[{\overline{{\mathcal{X}}}}]$ generated by products of polynomials of the form $$q-\prod _{i,j\in I}f_{{\alpha }_i,{\alpha }_j}^{m_{ij}},\quad
  \text{$q$ is a root of $1$, $m_{ij}\in {\mathbb{Z}}$ for all $i,j\in I$,}$$ such that the set $$\begin{aligned}
      V^\chi _{\underline{n}}=\{\chi '\in {\overline{{\mathcal{X}}}}\,|\,&
      R^{\chi '}_+ =R^\chi _+ ,\,
      {b^{\chi '}}(\beta )={b^{\chi}} (\beta ) \text{ for all }
      \beta \in R^\chi _{+{\mathrm{fin}}},\\
      &\chi '(\beta ,\beta )^n\not=1
      \text{ for all $\beta \in R^\chi _{+\infty }$,
      $1\le n\le n_\beta $}
      \}
    \end{aligned}
    \label{eq:Vchin}$$ is an open subset of ${\mathrm{maxspec}\,}{\bar{{\Bbbk }}}[{\overline{{\mathcal{X}}}}]/J$.

We use Lemma \[le:equalrs\] and Def. \[de:Cartan\] to reformulate the equation $R^\chi _+=R^{\chi '}_+$.

Let $\chi '\in {\mathcal{G}}(\chi )$. Since $\chi \in {\overline{{\mathcal{X}}}}_3$, $\chi '$ is $p$-finite for all $p\in I$. Further, $\chi '({\alpha }_p,{\alpha }_p)\not=1$ since $\chi (\beta ,\beta )\not=1$ for all $\beta \in R^\chi
  _+$, see Eq. . Thus $$(\chi '({\alpha }_p,{\alpha }_p)^{-c_{pj}^{\chi '}}\chi '({\alpha }_p,{\alpha }_j)\chi '(\al
  _j,{\alpha }_p)-1)(\chi '({\alpha }_p,{\alpha }_p)^{1-c_{pj}^{\chi '}}-1)=0$$ for all $p,j\in I$ with $p\not=j$. Let $w\in {\mathrm{Hom}}(\chi ,\chi ')\subset {\mathrm{Hom}}({\mathcal{W}}(\chi ))$. Identify $w$ with the corresponding element in ${\mathrm{Aut}}({\mathbb{Z}}^I)$ in the usual way. Then $\chi '=w^*\chi $ and hence $$\begin{aligned}
\label{eq:chirel}
    (\chi (\gamma _p,\gamma _p)^{-c_{pj}^{\chi '}}\chi (\gamma _p,\gamma _j)
    \chi (\gamma _j,\gamma _p)-1)(\chi (\gamma _p,\gamma _p)
    ^{1-c_{pj}^{\chi '}}-1)=0
  \end{aligned}$$ for all $p,j\in I$ with $p\not=j$, where $\gamma _p=w^{-1}({\alpha }_p)$ and $\gamma _j=w^{-1}({\alpha }_j)$. Let $$\begin{aligned}
    &\begin{aligned}
      &J'=\big(
      (f_{\gamma _p,\gamma _p}^{-c_{pj}^{w^*\chi }}
      f_{\gamma _p,\gamma _j}
      f_{\gamma _j,\gamma _p}-1)
      (f_{\gamma _p,\gamma _p}^{1-c_{pj}^{w^*\chi }}-1)\,|\\
      &\quad j,p\in I,j\not=p,\,
      w\in {\mathrm{Hom}}(\chi ,\underline{\,\,}),\,\gamma _p
      =w^{-1}({\alpha }_p),\,\gamma _j=w^{-1}({\alpha }_j)\big)
    \end{aligned}
    \label{eq:J'}
  \end{aligned}$$ and $$\begin{aligned}
    J=J'+\big(
    f_{\beta ,\beta }^{{b^{\chi}} (\beta )}-1\,|\,
    \beta \in R^\chi _{+{\mathrm{fin}}} \big).
    \label{eq:J}
  \end{aligned}$$ Then, by Lemma \[le:equalrs\], Def. \[de:Cartan\], and Eq. , $V^\chi _{\underline{n}}$ is the set of points $\chi ''\in \maxspec
  {\bar{{\Bbbk }}}[{\overline{{\mathcal{X}}}}]/J$ such that

-   $f_{\beta ,\beta }^n(\chi '')\not=1$ for all $\beta \in R^\chi _{+\infty }$, $1\le n\le n_\beta $ and

-   $(f_{\gamma _p,\gamma _p}^m f_{\gamma _p,\gamma _j}
          f_{\gamma _j,\gamma _p}-1)(\chi '')\,
          (f_{\gamma _p,\gamma _p}^{m+1} -1)(\chi
          '')\not=0$ for all $j,p\in I$, $w\in {\mathrm{Hom}}(\chi ,\underline{\,\,})$, and $m\in \{0,1,\dots ,-c^{w^*\chi }_{pj}-1\}$, where $j\not=p$ and $\gamma _p=w^{-1}({\alpha }_p)$, $\gamma _j=w^{-1}({\alpha }_j)$.

This is clearly an open subset, which proves the proposition.

\[pr:X5dense\] Let $\chi \in {\overline{{\mathcal{X}}}}_3$. Assume that $\chi (\beta ,\beta )\not=1$ for all $\beta \in R^\chi _+$. Let $\underline{n}=(n_\beta )_{\beta \in
  R^\chi _{+\infty }}$, where $n_\beta \in {\mathbb{N}}$ for all $\beta \in
  R^\chi _{+\infty }$. Let $V^\chi _{\underline{n}}$ be as in Prop. \[pr:Vchi\]. Then ${\overline{{\mathcal{X}}}}_5\cap V^\chi _{\underline{n}}$ is Zariski dense in $V^\chi _{\underline{n}}$.

Prop. \[pr:Vchi\] gives that $V^\chi _{\underline{n}}\subset {\overline{{\mathcal{X}}}}_3$ satisfies the conditions on $V$ in Lemma \[le:gendensity\], where $k=|I|^2$ and $\{x_i\,|\,i=1,2,\dots,k\}=\{f_{{\alpha }_i,{\alpha }_j}\,|\,i,j\in
  I\}$. Since ${\overline{{\mathcal{X}}}}_4$ contains all finite sets $V_{n_1,\dots ,n_k}$ in Lemma \[le:gendensity\], and ${\overline{{\mathcal{X}}}}_5\cap V^\chi _{\underline{n}}=
  {\overline{{\mathcal{X}}}}_4\cap V^\chi _{\underline{n}}$ by definition of $V^\chi _{\underline{n}}$, the proof is completed.

Similarly to Eq.  define ${P}^\chi ({\alpha },\beta _\nu ;t)$ for all $\chi \in {\overline{{\mathcal{X}}}}_3$, ${\alpha }\in {\mathbb{N}}_0^I$, $\beta _\nu \in R^\chi _+$, and $t\in
{\mathbb{N}}$ with $t<{b^{\chi}} (\beta _\nu )$ by $$\begin{aligned}
  {P}^\chi ({\alpha },\beta _\nu ;t)=\Big|\Big\{(m_1,\dots ,m_n)\in {\mathbb{N}}_0^n\,\big|\,
  \sum _{\mu =1}^n m_\mu \beta _\mu ={\alpha },\,m_\nu \ge t,\quad &\\
  m_\mu <{b^{\chi}} (\beta _\mu )\quad \text{for all $\mu \in \{1,2,\dots ,n\}$}
  \Big\}\Big|.&
  \label{eq:PF2}
\end{aligned}$$

\[th:Shapdet2\] Let $\chi \in {\mathcal{X}}_3$. Assume that $\chi (\beta ,\beta )\not=1$ for all $\beta \in R^\chi _+$. The Shapovalov determinant of $U(\chi )$ is the family $(\det ^\chi _{\alpha })_{{\alpha }\in {\mathbb{N}}_0^I}$, where $$\begin{aligned}
    \label{eq:det2}
    \det \nolimits ^\chi _{\alpha }=
    \prod _{\beta \in R^\chi _+}
    \prod _{t=1}^{{b^{\chi}} (\beta )-1}
    ({\rho ^{\chi}} (\beta )K_{\beta }
    -\chi (\beta ,\beta )^t L_{\beta })
    ^{{P}^\chi ({\alpha },\beta ;t)}.
  \end{aligned}$$

Let ${\alpha }\in {\mathbb{N}}_0^I$. Choose a basis $\{F'_1,\dots ,F'_k\}$ of $U^-(\chi )_{-{\alpha }}$ consisting of monomials $F_{i_1}F_{i_2}\cdots
  F_{i_l}$, where $k,l\in {\mathbb{N}}_0$ and $i_1,\dots ,i_l\in I$. Identify $\oplus _{\beta ,\gamma \in {\mathbb{N}}_0^I,\,\beta +\gamma ={\alpha }}\Fie
  K_\beta L_\gamma $ with ${\bar{{\Bbbk }}}^N$ for an appropriate $N\in {\mathbb{N}}$. By the commutation relations – and the definition of ${\mathrm{Sh}}$, the map $$d:{\overline{{\mathcal{X}}}}\to {\bar{{\Bbbk }}}^N, \quad \chi '\mapsto \det ({\mathrm{Sh}}(F'_i,F'_j))_{i,j\in
  \{1,2,\dots ,k\}}$$ is a morphism of affine varieties. Further, $d(\chi )\not=0$ by Lemma \[le:Shfcoeffs\], the choice of $\{F'_1,\dots ,F'_k\}$, and the nondegeneracy of the pairing $\eta $, see Prop. \[pr:sHpdef\](iv). Recall the definition of $|\beta |$, $\beta \in {\mathbb{Z}}^I$, from Eq. . Restrict $d$ to the set $V^\chi _{\underline{n}}$ defined in Prop. \[pr:Vchi\], with $n_\beta =|{\alpha }|/|\beta |$ for all $\beta \in R^\chi _{+\infty }$. The set $$V'=\{\chi '\in V^\chi _{\underline{n}}\,|\,d(\chi ')\not=0 \}$$ is open in $V^\chi _{\underline{n}}$ and contains $\chi $. Thus by Prop. \[pr:X5dense\] the set $$V''=\{\chi '\in {\overline{{\mathcal{X}}}}_5\cap V^\chi _{\underline{n}}\,|
  \,d(\chi ')\not=0\}$$ is Zariski dense in all irreducible components of $V^\chi
  _{\underline{n}}$ containing $\chi $. The definition of $V^\chi _{\underline{n}}$ and the choice of $\underline{n}$ yield that $R^{\chi '}_+=R^\chi
  _+$ and ${b^{\chi '}}(\beta )\le {b^{\chi}} (\beta )$ for all $\chi '\in
  V^\chi _{\underline{n}}$ and $\beta \in R^\chi _+$. Thus $\dim U(\chi ')_{-{\alpha }}\le \dim U(\chi )_{-{\alpha }}$ for all $\chi '
  \in V^\chi _{\underline{n}}$ by Eqs. , . Hence $d(\chi ')$ is a multiple of $\det ^{\chi '}_{\alpha }$ for all $\chi '\in V^\chi _{\underline{n}}$. By Thm. \[th:Shapdet\], $$\begin{aligned}
    \label{eq:d}
    d(\chi ')=a(\chi ')
    \prod _{\beta \in R^\chi _+} \prod _{t=1}^{{b^{\chi}} (\beta _\nu )-1}
    ({\rho ^{\chi}} (\beta )K_{\beta }
    -\chi (\beta ,\beta )^t L_{\beta })
    ^{{P}^\chi ({\alpha },\beta ;t)}
  \end{aligned}$$ for all $\chi '\in V''$, where $a(\cdot )$ is some regular function on ${\overline{{\mathcal{X}}}}$ which does not vanish on $V''$. By the density of $V''$, Eq.  holds for all $\chi '\in V'$ in the irreducible components of $V^\chi _{\underline{n}}$ containing $\chi $, and $a(\chi ')\not=0$ for all $\chi '\in V'$ by definition of $V'$. In particular, Eq.  holds for $\chi '=\chi $. Thus the theorem is proven.

Quantized enveloping algebras {#sec:Uqg}
=============================

We adapt our main result to quantized enveloping algebras.

Let $I$ be a finite set and let $C=(c_{ij})_{i,j\in I}$ be a symmetrizable Cartan matrix of finite type. Let ${\mathfrak{g}}$ be the associated semisimple Lie algebra and $R_+$ the set of positive roots. For all $i\in I$ let $d_i\in {\mathbb{N}}$ such that $d_ic_{ij}=d_jc_{ji}$ for all $i,j\in I$. Assume that the numbers $d_i$, where $i\in I$, are relatively prime. Identify ${\mathbb{Z}}^I$ with the root lattice by considering $\{{\alpha }_i\,|\,i\in I\}$ as the set of simple roots. Let $(\cdot ,\cdot ):{\mathbb{Z}}^I\to {\mathbb{Z}}^I$ be the (positive definite) symmetric bilinear form defined by $({\alpha }_i,{\alpha }_j)=d_ic_{ij}$. Let $\rho :{\mathbb{Z}}^I\to {\mathbb{Z}}$ be the linear form defined by $\rho ({\alpha }_i)=d_i$ for all $i\in I$.

Let ${\Bbbk }$ be a field, and let $q\in {{\Bbbk }^\times }$. Assume that $q^{2m}\not=1$ for all $m\in {\mathbb{N}}$ with $m\le \max \{d_i\,|\,i\in I\}$. The quantized enveloping algebra of ${\mathfrak{g}}$ is the associative algebra $U_q({\mathfrak{g}})$ generated by the elements $E_i$, $F_i$, $K_i$, and $K_i^{-1}$, where $i\in I$, and defined by the relations $$\begin{gathered}
  K_iK_i^{-1}=K_i^{-1}K_i=1,\quad K_iK_j=K_jK_i,\\
  K_iE_jK_i^{-1}=q^{d_ic_{ij}}E_j,\quad
  K_iF_jK_i^{-1}=q^{-d_ic_{ij}}F_j,\\
  E_iF_j-F_jE_i=\delta_{ij}(K_i-K_i^{-1}),\\
  ({\mathrm{ad}}E_i)^{1-c_{ij}}(E_j)=0,\quad
  ({\mathrm{ad}}F_i)^{1-c_{ij}}(F_j)=0 \quad (i\not=j)\end{gathered}$$ for all $i,j\in I$. Here ${\mathrm{ad}}$ denotes adjoint action: $$\begin{aligned}
({\mathrm{ad}}E_i)(x)=E_ix-K_ixK_i^{-1}E_i,\quad
({\mathrm{ad}}F_i)(x)=xF_i-F_iK_i^{-1}xK_i\end{aligned}$$ for all $i\in I$ and $x\in \langle E_j,F_j,K_j,K_j^{-1}\,|\,j\in I\rangle $. Traditionally, in the third line of the defining relations of $U_q({\mathfrak{g}})$ one inserts a denominator $q^{d_i}-q^{-d_i}$ on the right hand side, but this denominator can be eliminated by rescaling *e.g.* the variables $E_i$, $i\in I$.

Assume first that $q$ is not a root of $1$. Then, by [@b-Lusztig93 Ch.1] and [@inp-AndrSchn02 Prop.2.10], $U_q({\mathfrak{g}})\simeq U(\chi )/(K_iL_i-1\,|\,i\in I)$, where $\chi \in {\mathcal{X}}$ with $\chi ({\alpha }_i,{\alpha }_j)=q^{d_ic_{ij}}$ for all $i,j\in I$.

Assume now that $q$ is a root of $1$. Let again $\chi \in {\mathcal{X}}$ with $\chi ({\alpha }_i,{\alpha }_j)=q^{d_ic_{ij}}$ for all $i,j\in I$. Then $$U(\chi )/(K_iL_i-1,K_\beta ^{{b^{}}(\beta )}-1\,|\,i\in I,
\beta \in R^\chi _+),$$ where ${b^{}}(\beta )$ is the order of $q^{(\beta ,\beta )}$ for all $\beta \in R_+$, is isomorphic to Lusztig’s small quantum group $u_q({\mathfrak{g}})$. This was observed *e.g.* in [@inp-AndrSchn02 Thm.4.3] by referring to results of Lusztig, de Concini, Procesi, Rosso, and Müller.

Similarly to Eq.  and Def. \[de:Shapdet\] one defines the Shapovalov form and the Shapovalov determinant $(\det _{\alpha })_{{\alpha }\in {\mathbb{N}}_0^I}$ of $U_q({\mathfrak{g}})$ and $u_q({\mathfrak{g}})$, respectively. Alternatively, since $K_iL_i$ for $i\in I$ and $K_\beta ^{\bfun{}(\beta )}$ for $\beta \in R^\chi _+$ (the latter only if $q$ is a root of $1$) are central elements in $U(\chi )$ for all $i\in I$, the Shapovalov form can also be obtained from the definition in Sect. \[sec:shapdet\] via Lemma \[le:Uzideal\].

\[th:ShapdetUqg\] Let $I$, $C$, $(d_i)_{i\in I}$, and ${\mathfrak{g}}$ as above. Let $q\in {{\Bbbk }^\times }$. Assume that $q^{2m}\not=1$ for all $m\in {\mathbb{N}}$ with $m\le \max \{d_i\,|\,i\in I\}$.

\(i) [@inp-dCK90] If $q$ is not a root of $1$, then the Shapovalov determinant of $U_q({\mathfrak{g}})$ is the family $(\det _{\alpha })_{{\alpha }\in {\mathbb{N}}_0^I}$, where $$\begin{aligned}
    \label{eq:detUqg}
    \det \nolimits _{\alpha }=
    \prod _{\beta \in R_+}
    \prod _{t=1}^\infty
    (q^{2\rho (\beta )}K_{\beta }
    -q^{t(\beta ,\beta )} K_{\beta }^{-1})
    ^{{P}({\alpha },\beta ;t)}.
  \end{aligned}$$

\(ii) Assume that $q$ is a root of $1$. Then the Shapovalov determinant of $u_q({\mathfrak{g}})$ is the family $(\det _{\alpha })_{{\alpha }\in {\mathbb{N}}_0^I}$, where $$\begin{aligned}
    \label{eq:detuqg}
    \det \nolimits _{\alpha }=
    \prod _{\beta \in R_+}
    \prod _{t=1}^{\bfun{}(\beta )-1}
    (q^{2\rho (\beta )}K_{\beta }
    -q^{t(\beta ,\beta )} K_{\beta }^{-1})
    ^{{P}({\alpha },\beta ;t)}.
  \end{aligned}$$

Let $\chi \in {\mathcal{X}}$ with $\chi ({\alpha }_i,{\alpha }_j)=q^{d_ic_{ij}}$ for all $i,j\in I$. Choose the ideal $J$ in Lemma \[le:Uzideal\] as explained above. Then one gets the Shapovalov determinants of $U_q({\mathfrak{g}})$ and $u_q({\mathfrak{g}})$ from the one of $U(\chi )$ in Thm. \[th:Shapdet2\].

The second part of Thm. \[th:ShapdetUqg\] was proved in [@a-KumLetz97] in the case when the order of $q$ is prime and ${\Bbbk }$ is the cyclotomic field $\mathbb{Q}[q]$.

Appendix
========

For the proofs of Thms. \[th:Shapdet\] and \[th:Shapdet2\] we need some commutative algebra which is considered here. Let ${\bar{{\Bbbk }}}$ be an algebraically closed field.

\[le:rankX\] Let $B$ be an integral domain, $x$ an indeterminate, $k\in {\mathbb{N}}$, and $X\in B[x]^{k\times k}$. Then there exist $s\in \{0,1,\dots ,k\}$, $D_1,D_2\in
  B^{k\times k}$, $D_0\in B[x]^{k\times k}$ and $b\in B\setminus \{0\}$ such that $\det D_1,\det D_2\not=0$, $$\begin{aligned}
    D_1XD_2=xD_0+b\,\mathrm{diag} (\underbrace{1,\dots ,1}_s,0,\dots ,0).
    \label{eq:rankX}
  \end{aligned}$$

Let ${\mathrm{Frac}}(B)$ be the field of fractions of $B$. Then there exist $s\in \{0,1,\dots ,k\}$ and $D'_1,D'_2\in {\mathrm{Frac}}(B)^{k\times k}$ such that $\det D'_1,\det D'_2\not=0$ and $$D'_1X(0)D'_2=\mathrm{diag} (\underbrace{1,\dots ,1}_s,0,\dots ,0).$$ Let $b_1,b_2\in B\setminus \{0\}$ such that $b_1D'_1,b_2D'_2\in B[x]^{k\times k}$. Let $b=b_1b_2$, $D_1=b_1D'_1$, and $D_2=b_2D'_2$. Then $$D_1X(0)D_2=b\,\mathrm{diag} (\underbrace{1,\dots ,1}_s,0,\dots ,0),$$ and hence the lemma holds for $D_0=D_1X'D_2$, where $X'\in B[x]^{k\times k}$ such that $X=X(0)+xX'$.

\[le:detXfactor\] Let $B$ be a finitely generated integral domain over ${\bar{{\Bbbk }}}$, $x$ an indeterminate, $k\in {\mathbb{N}}$, $r\in \{0,1,\dots ,k\}$, and $X\in B[x]^{k\times k}$. Assume that $\mathrm{rk}\,X(0)_p\le r$ for all points $p$ in a non-empty Zariski open subset of the affine variety of $B$. Then $\det X=x^{k-r}b$ for some $b\in B[x]$.

By Lemma \[le:rankX\] there exist $s\in \{0,1,\dots ,k\}$, $b\in B\setminus \{0\}$, $D_1,D_2\in B^{k\times k}$, and $D_0\in B[x]^{k\times k}$ such that $\det D_1,\det D_2\not=0$ and Eq.  holds. Let $V$ be a non-empty Zariski open subset of the affine variety of $B\simeq B[x]/(x)$ such that $(\det D_1)_p,(\det D_2)_p,
  b_p\not=0$ and $\mathrm{rk}\,X(0)_p\le r$ for all $p\in V$. This exists by the assumption on $r$ and since the variety of $B$ is irreducible. Then $s\le r$ by Eq.  in the points $(p,0)$ of the variety of $B[x]$, where $p\in V$. Therefore $$\det D_1\, \det X \det D_2=x^{k-r}b'$$ for some $b'\in B[x]$. Since $\det D_1,\det D_2\in B$ and $B$ is an integral domain, we conclude that $\det X\in x^{k-r}B[x]$.

For all $k\in {\mathbb{N}}$ let ${\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]$ denote the ring of Laurent polynomials in $k$ variables. For all $M=(m_{ij})_{i,j\in \{1,2,\dots ,k\}}\in \mathrm{GL}(k,{\mathbb{Z}})$ let $$X^{(M)} _i=\prod _{j=1}^kx_j^{m_{ij}},\quad 1\le i\le k.$$ Then the ring endomorphism of ${\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]$ given by $x_i\mapsto X^{(M)}_i$ for all $i\in \{1,2,\dots ,k\}$ is an isomorphism with inverse map given by $x_i\mapsto X^{(M^{-1})}_i$ for all $i\in \{1,2,\dots ,k\}$.

Let $k\in {\mathbb{N}}$. Let $J\subsetneq {\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]$ be an ideal generated by elements of the form $q-\prod _{i=1}^k x_i^{m_i}$, where $m_1,\dots ,m_k\in {\mathbb{Z}}$ and $q\in {{\bar{{\Bbbk }}}^\times }$ is a root of $1$. Then $J$ is a finite intersection of ideals of the form $$\begin{aligned}
    ( X_1^{(M)}-q_1, X_2^{(M)}-q_2,\dots ,X_l^{(M)}-q_l),
    \label{eq:primeid}
  \end{aligned}$$ where $l\in \{0,1,\dots ,k\}$, $q_1,\dots ,q_l\in {{\bar{{\Bbbk }}}^\times }$ are roots of $1$, and $M\in \mathrm{GL}(k,{\mathbb{Z}})$. \[le:torusideal\]

Proceed by induction on $k$. If $J$ is empty, then the claim is true. Assume now that $q-\prod _{i=1}^kx_i^{m_i}$ is one of the generators of $J$, where $q$ is a root of $1$ and $(m_1,\dots ,m_k)\in {\mathbb{Z}}^k\setminus \{0\}$. Let $m_0=\mathrm{gcd}(m_1,\dots ,m_k)$. Let $M'\in \mathrm{GL}(k,{\mathbb{Z}})$ such that $m'_{1i}=m_i/m_0$ for all $i\in \{1,2,\dots ,k\}$. Then $(X_1^{(M')})^{m_0}-q\in J$, and hence $J$ is the intersection of the (finite number of) ideals $J+(X^{(M')}_1-q')$, where $q'\in {\bar{{\Bbbk }}}$, ${q'}^{m_0}=q$. By assumption, $J+(X^{(M')}_1-q')$ is generated by $X^{(M')}_1-q'$ and by elements of the form $q''-\prod _{i=2}^k (X^{(M')}_i)^{m'_i}$, where $m'_2,\dots ,m'_k\in {\mathbb{Z}}$ and $q''\in {{\bar{{\Bbbk }}}^\times }$ is a root of $1$. Then the claim follows by the induction hypothesis.

The ideals in Eq.  are prime ideals of ${\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]$, since the quotient ring ${\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]/J
\simeq {\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k-l]$ is an integral domain.

Let $k\in {\mathbb{N}}$. Let $J\subsetneq {\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]$ be an ideal generated by polynomials of the form $$\big(q_1-\prod _{i=1}^k x_i^{m_{1i}}\big)
  \big(q_2-\prod _{i=1}^k x_i^{m_{2i}}\big)\cdots
  \big(q_l-\prod _{i=1}^k x_i^{m_{li}}\big),$$ where $l\in {\mathbb{N}}$, $m_{j1},\dots ,m_{jk}\in {\mathbb{Z}}$ and $q_j\in {{\bar{{\Bbbk }}}^\times }$ is a root of $1$ for all $j\in \{1,2,\dots ,l\}$. Let $V\subset {\mathrm{maxspec}\,}{\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]/J$ be an open subset. Then the union of the subsets $$V_{n_1,\dots ,n_k}=\{p\in V\,|\,p_1^{n_1}=1,\dots ,p_k^{n_k}=1\},
  \quad n_1,\dots ,n_k\in {\mathbb{N}},$$ is dense in $V$ with respect to the Zariski topology. \[le:gendensity\]

We can assume that $V={\mathrm{maxspec}\,}{\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]/J$. Moreover, it suffices to prove the lemma for the irreducible components of $V$. Thus, as a first reduction, $J$ can be assumed to be as in the assumptions of Lemma \[le:torusideal\]. Then by Lemma \[le:torusideal\] we may assume that $J$ is an ideal as in Eq. , where $M\in \mathrm{GL}(k,{\mathbb{Z}})$, $l\in \{0,1,\dots ,k\}$, and $q_1,\dots ,q_l\in {{\bar{{\Bbbk }}}^\times }$ are roots of $1$. Then $V\simeq ({{\bar{{\Bbbk }}}^\times })^{k-l}$ for some $l\in \{0,1,\dots ,k\}$, and the lemma follows from the fact that infinite subsets are dense in ${{\bar{{\Bbbk }}}^\times }$.

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