---
abstract: 'We study the semidirect product of a Lie algebra with a representation up to homotopy and provide various examples coming from Courant algebroids, string Lie 2-algebras, and omni-Lie algebroids. In the end, we study the semidirect product of a Lie group with a representation up to homotopy and use it to give an integration of a certain string Lie 2-algebra.'
author:
- |
    Yunhe Sheng\
    Department of Mathematics, Jilin University, Changchun 130012, Jilin, China\
    email: ysheng888@gmail.com\
    Chenchang Zhu\
    Courant Research Centre “Higher Order Structures”, University of Göttingen\
    email:zhu@uni-math.gwdg.de
bibliography:
- '../../bib/bibz.bib'
title: ' [Semidirect products of representations up to homotopy [^1] ]{} '
---

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

This paper is the first part of our project to integrate representations up to homotopy of Lie algebras (algebroids). Our original motivation is to integrate the standard Courant algebroid $TM\oplus T^*M$ since it is this Courant algebroid that is much used in Hitchin and Gualtieri’s program of generalized complex geometry. Courant algebroids are Lie 2-algebroids in the sense of Roytenberg and Ševera [@royt; @s:funny]. The general procedure to integrate Lie $n$-algebras (algebroids) is already described in [@getzler; @henriques; @s:funny]. We want to pursue some explicit formulas for the special case of the standard Courant algebroid. It turns out that the sections of the Courant algebroid $TM \oplus T^*M$ form a semidirect product of a Lie algebra with a representation up to homotopy. Abad and Crainic [@abad-crainic:rep-homotopy] recently studied the representations up to homotopy of Lie algebras, Lie groups, and even Lie algebroids, Lie groupoids, in general. Just as one can form the semidirect product of a Lie algebra with a representation, one can form the semidirect product with representations up to homotopy too. In our case, the semidirect product coming from the standard Courant algebra is a Lie 2-algebra. But using the fact that it is also a semidirect product, the integration becomes easier. The integration result is related to the semidirect product of Lie groups with its representation up to homotopy, which will be discussed in Section \[sec:gp\]. However it turns out that the concept of representation up to homotopy of Lie groups of Abad and Crainic will not be general enough to cover all the integration results. This we will continue in a forthcoming paper [@sheng-zhu:II].

In this paper we focus on exhibiting more examples of representation up to homotopy and their semidirect products to demonstrate the importance of our integration procedure. The examples are all sorts of variations of Courant algebroids. One is Chen and Liu’s omni-Lie algebroids, which generalizes Weinstein’s omni-Lie algebras. Hence we expect to give an integration to Weinstein’s omni-Lie algebras via Lie 2-algebras in the next paper [@sheng-zhu:II].

Another example comes from the so called string Lie 2-algebra. It is essentially a Courant algebroid over a point (see Section \[sec:string\]), namely a Lie algebra with an adjoint-invariant inner product. This sort of Lie algebra is usually called a [*quadratic Lie algebra*]{}. This concept also appears in the context of Manin triples and double Lie algebras. The example ${\mathbb R}\to {\mathfrak g}\oplus {\mathfrak g}^*$ that we consider in this paper is an analogue of the standard Courant algebroid, and is basically a special case taken from [@lu-weinstein] [^2]. We give an integration of the string Lie 2-algebra ${\mathbb R}\to {\mathfrak g}\oplus {\mathfrak g}^*$ at the end.

Usually people require the base Lie algebra of a string Lie algebra to be semisimple and of compact type (see Remark \[rk:semisimple\]). For such usual sort of string Lie 2-algebras, Baez et al [@baez:2gp] have proved a no-go theorem, namely such string Lie 2-algebras can not be integrated to finite dimensional semi-strict Lie 2-groups. Here a [*semi-strict Lie 2-group*]{} is a group object in $\rm DiffCat$, where $\rm DiffCat$ is the 2-category consisting of categories, functors, and natural transformations in the category of differential manifolds, or equivalently $\rm DiffCat$ is a 2-category with Lie groupoids as objects, strict morphisms of Lie groupoids as morphisms, and 2-morphisms of Lie groupoids as 2-morphisms. Our semi-strict Lie 2-group is actually called a Lie 2-group by the authors in [@baez:2gp]. However, we call it a semi-strict Lie 2-group because compared to the Lie 2-group in the sense of Henriques [@henriques], or equivalently the stacky group in the sense of Blohmann [@blohmann], it is stricter. Basically, their Lie 2-group is a group object in the 2-category with objects as Lie groupoids, morphisms as Hilsum-Skandalas bimodules (or generalized morphisms), 2-morphisms as 2-morphisms of Lie groupoids. Schommer-Pries realizes the string 2-group as such a Lie 2-group with a finite dimensional model [@schommer:string-finite-dim] and the integration of a string Lie 2-algebra to such a model is a work in progress [@ccc:integration-string].

It is not needed in the definition of the string Lie 2-algebra for the base Lie algebra to be semisimple of compact type. One only needs a quadratic Lie algebra. As soon as we relax this condition on compactness, we find out that one can integrate ${\mathbb R}\to {\mathfrak g}\oplus {\mathfrak g}^*$ to a finite dimensional semi-strict Lie 2-group in the sense of Baez et al. The integrating object is actually a special Lie 2-group (very close to a strict Lie 2-group) in the sense of Baez et al.

Then of course, as we relax the condition, we are in danger that the class corresponding to this Lie 2-algebra in $H^3({\mathfrak g}\oplus {\mathfrak g}^*, {\mathbb R})$ might be trivial, and consequently our Lie 2-algebra might be strict. Then what we have done would not have been a big surprise because a strict Lie 2-algebra corresponds to a crossed module of Lie algebras, and it easily integrates to a strict Lie 2-group by integrating the crossed module. However, we verified that when ${\mathfrak g}$ itself (not ${\mathfrak g}\oplus {\mathfrak g}^*$) is semisimple, this Lie 2-algebra is not strict.

[**Acknowledgement:**]{} We give warmest thanks to Zhang-Ju Liu, Jiang-Hua Lu, Giorgio Trentinagia and Marco Zambon for useful comments and discussion. Y. Sheng gives his warmest thanks to Courant Research Centre “Higher Order Structures”, Göttingen University, where this work was done when he visited there.

Representations up to homotopy of Lie algebras
==============================================

In this section, we first consider the 2-term representation up to homotopy of Lie algebras. We give explicit formulas of the corresponding 2-term $L_\infty$-algebra, which is their semidirect product. Then we give several interesting examples including Courant algebroids and omni-Lie algebroids.

Representation up to homotopy of Lie algebras and their semidirect products
---------------------------------------------------------------------------

$L_\infty$-algebras, sometimes called strongly homotopy Lie algebras, were introduced by Drinfeld and Stasheff [@stasheff:shla] as a model for “Lie algebras that satisfy Jacobi up to all higher homotopies”. The following convention of $L_\infty$-algebras has the same grading as in [@henriques] and [@rw].

An $L_\infty$-algebra is a graded vector space $L=L_0\oplus
L_1\oplus\cdots$ equipped with a system $\{l_k|~1\leq k<\infty\}$ of linear maps $l_k:\wedge^kL\longrightarrow L$ with degree $\deg(l_k)=k-2$, where the exterior powers are interpreted in the graded sense and the following relation with Koszul sign “Ksgn” is satisfied for all $n\geq0$: $$\sum_{i+j=n+1}(-1)^{i(j-1)}\sum_{\sigma}{\mathrm{sgn}}(\sigma){\mathrm{Ksgn}}(\sigma)l_j(l_i(x_{\sigma(1)},\cdots,x_{\sigma(i)}),x_{\sigma(i+1)},\cdots,x_{\sigma(n)})=0,$$ where the summation is taken over all $(i,n-i)$-unshuffles with $i\geq1$.

Let $n=1$, we have $$l_1^2=0,\quad l_1:L_{i+1}\longrightarrow L_i,$$ which means that $L$ is a complex and we usually write ${\mathrm{d}}=l_1$. Let $n=2$, we have $${\mathrm{d}}l_2(x,y)=l_2({\mathrm{d}}x,y)+(-1)^pl_2(x,{\mathrm{d}}y),\quad \forall~x\in L_p,
y\in L_q,$$ which means that ${\mathrm{d}}$ is a derivation with respect to $l_2$. We usually view $l_2$ as a bracket $[\cdot,\cdot]$. However, it is not a Lie bracket, the obstruction of Jacobi identity is controlled by $l_3$: $$\begin{aligned}
 \nonumber
&&l_2(l_2(x,y),z)+(-1)^{(p+q)r}l_2(l_2(y,z),x)+(-1)^{qr+1}l_2(l_2(x,z),y)\\
&&=-{\mathrm{d}}l_3(x,y,z)-l_3({\mathrm{d}}x,y,z)+(-1)^{pq}l_3({\mathrm{d}}y,x,z)-(-1)^{(p+q)r}l_3({\mathrm{d}}z, x,y),\end{aligned}$$ where $x\in L_p,~ y\in L_q,~z\in L_q$ and $l_3$ also satisfies higher coherence laws.

In particular, if the $k$-ary brackets are zero for all $k>2$, we recover the usual notion of [**differential graded Lie algebras**]{} (DGLA). If $L$ is concentrated in degrees $<n$, $L$ is called an [**$n$-term $L_\infty$-algebra**]{}.

In this paper, we mainly consider 2-term $L_\infty$-algebras, which are equivalent to Lie 2-algebras given in [@baez:2algebras] by John Baez and Alissa Crans. In this special case, $l_4$ is always zero. Thus restricting the coherence law satisfied by $l_3$ on degree-0, we obtain $$l_3(l_2(x,y),z,w)+c.p.-\big(l_2(l_3(x,y,z),w)+c.p.\big)=0,\quad\forall~x,y,z,w\in
L_0.$$ Lie 2-algebras are categorified version of Lie algebras. In a Lie 2-algebra, the Jacobi identity is replaced by an isomorphism which is called the [ **Jacobiator**]{}. The Jacobiator satisfies a certain law of its own. Given a 2-term $L_\infty$-algebra $L_1\stackrel{{\mathrm{d}}}{\longrightarrow}L_0$, the underlying 2-vector space of the corresponding Lie 2-algebra is made up by $L_0$ as the vector space of objects and $L_0\oplus
 L_1$ as the vector space of morphism. Please see [@baez:2algebras Theorem 4.3.6] for more details.

Recall from [@baez:2algebras] that a Lie 2-algebra is skeletal if isomorphic objects are equal. Viewing a Lie 2-algebra as a 2-term $L_\infty$-algebra, explicitly we have

\[defi:skeletal L 2\] A 2-term $L_\infty$-algebra $L_1\stackrel{{\mathrm{d}}}{\longrightarrow}L_0$ is called [**skeletal**]{} if ${\mathrm{d}}=0$.

They also prove the following theorem

[[@baez:2algebras]]{}\[thm:skeletal Lie 2 a\] There is a one-to-one correspondence between 2-term skeletal $L_\infty$-algebra $L_1\stackrel{{\mathrm{d}}}{\longrightarrow}L_0$ and quadruples $({\mathfrak k}_1,{\mathfrak k}_2,\phi,\theta)$ where ${\mathfrak k}_1$ is a Lie algebra, ${\mathfrak k}_2$ is a vector space, $\phi$ is a representation of ${\mathfrak k}_1$ on ${\mathfrak k}_2$ and $\theta$ is a 3-cocycle on ${\mathfrak k}_1$ with values in ${\mathfrak k}_2$.

We briefly recall that given a 2-term skeletal $L_\infty$-algebra $L_1\stackrel{{\mathrm{d}}}{\longrightarrow}L_0$, ${\mathfrak k}_1$ is $L_0$, ${\mathfrak k}_2$ is $L_1$, the representation $\phi$ comes from $l_2$ and the 3-cocycle $\theta$ is obtained from $l_3$.

Please see [@abad-crainic:rep-homotopy; @abad] for the general theory of representation up to homotopy of Lie algebroids. In this paper we only consider the 2-term representation up to homotopy of Lie algebras.

[[@abad-crainic:rep-homotopy]]{} A 2-term representation up to homotopy of a Lie algebra ${\mathfrak g}$ consists of

-   A 2-term complex of vector spaces $V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0$.

-   Two linear maps $\mu_i: {\mathfrak g}\to End(V_i)$, which are compatible with ${\mathrm{d}}$, i.e. for any $X\in{\mathfrak g},~\xi\in V_1$, we have $$\label{eqn:d mu}
    {\mathrm{d}}\circ\mu_1(X)(\xi)=\mu_0(X)\circ{\mathrm{d}}(\xi).$$

-   A linear map $\nu:\wedge^2{\mathfrak g}\to{\mathrm{Hom}}(V_0,V_1)$ such that $$\begin{aligned}
    \label{eqn:mu0}\mu_0[X_1,X_2]-[\mu_0(X_1),\mu_0(X_2)]&=&{\mathrm{d}}\circ
    \nu(X_1,X_2),\\
    \label{eqn:mu1}\mu_1[X_1,X_2]-[\mu_1(X_1),\mu_1(X_2)]&=&
    \nu(X_1,X_2)\circ{\mathrm{d}},\end{aligned}$$ as well as $$\label{eqn:d k}
    [\mu_0(X_1) + \mu_1(X_1),\nu(X_2,X_3)]+c.p.= \nu([X_1,X_2],X_3)+c.p.,$$ where $c.p.$ stands for cyclic permutation.

We usually write $\mu=\mu_0+\mu_1$ and denote a 2-term representation up to homotopy of a Lie algebra ${\mathfrak g}$ by $(V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0,\mu,\nu)$.

In [@abad-crainic:rep-homotopy Example 3.25], the authors proved that associated to any representation up to homotopy $V_\bullet$ of a Lie algebra ${\mathfrak g}$, one can form a new $L_\infty
$-algebra ${\mathfrak g}\ltimes V_\bullet$, which is their semidirect product. Here we make this construction explicit in the 2-term case.

Let $(V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0,\mu,\nu)$ be a 2-term representation up to homotopy of ${\mathfrak g}$, then we can form a new 2-term complex $$({\mathfrak g}\ltimes
V_\bullet,{\mathrm{d}}):V_1\stackrel{{\mathrm{d}}}{\longrightarrow}({\mathfrak g}\oplus V_0).$$ Define $l_2:\wedge^2({\mathfrak g}\ltimes
V_\bullet)\longrightarrow{\mathfrak g}\ltimes V_\bullet $ by setting $$\label{2bracket}
\left\{\begin{array}{l}l_2(X+\xi,Y+\eta)=[X,Y]+\mu_0(X)(\eta)-\mu_0(Y)(\xi),\\
l_2(X+\xi,f)=\mu_1(X)(f),\\
l_2(f,g)=0,\end{array}\right.$$ for any $X+\xi,Y+\eta\in{\mathfrak g}\oplus V_0$ and $f,~g\in V_1$. One should note that $l_2$ is not a Lie bracket, but we have $$l_2(l_2(X+\xi,Y+\eta),Z+\gamma)+c.p.={\mathrm{d}}(\nu(X,Y)(\gamma))+c.p..$$ Define $l_3:\wedge^3({\mathfrak g}\ltimes
V_\bullet)\longrightarrow{\mathfrak g}\ltimes V_\bullet $ by setting: $$\label{3bracket}
l_3(X+\xi,Y+\eta,Z+\gamma)=-\nu(X,Y)(\gamma)+c.p.,$$ then we have

\[pro:Lie 2\] With the above notation, if $(V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0,\mu,\nu)$ is a 2-term representation up to homotopy of a Lie algebra ${\mathfrak g}$, then $(V_1\stackrel{{\mathrm{d}}}{\longrightarrow}({\mathfrak g}\oplus V_0),l_2,l_3)$ is a 2-term $L_\infty$-algebra.

Example I: Courant algebroids $TM\oplus T^*M$
---------------------------------------------

Courant algebroids were first introduced in [@lwx] to study the double of Lie bialgebroids. It is a vector bundle $E\longrightarrow
M$ equipped with a nondegenerate symmetric bilinear form ${\left\langle \cdot,\cdot\right\rangle}$ on the bundle, an antisymmetric bracket ${\left\llbracket  \cdot,\cdot\right\rrbracket }$ on the section space $\Gamma(E)$ and a bundle map $\rho:E\longrightarrow TM$ such that a set of axioms are satisfied. It can be viewed [@royt] as a Lie 2-algebroid with a “degree 2 symplectic form”. The first example is the standard Courant algebroid $({\mathcal{T}}=TM\oplus
T^*M,{\left\langle \cdot,\cdot\right\rangle},{\left\llbracket  \cdot,\cdot\right\rrbracket },\rho)$ associated to a manifold $M$, where $\rho:{\mathcal{T}}\longrightarrow TM$ is the projection, the canonical pairing ${\left\langle \cdot,\cdot\right\rangle}$ is given by $$\label{eqn:pair}
{\left\langle X+\xi,Y+\eta\right\rangle}=\frac{1}{2}\big(\xi(Y)+\eta(X)\big),\quad\forall
~X,Y\in{\mathfrak X}(M),~\xi,\eta\in\Omega^1(M),$$ the antisymmetric bracket ${\left\llbracket  \cdot,\cdot\right\rrbracket }$ is given by $$\label{eqn:bracket}
{\left\llbracket  X+\xi,Y+\eta\right\rrbracket }\triangleq[X,Y]+L_X\eta-L_Y\xi+\frac{1}{2}d(\xi(Y)-\eta(X)),\quad\forall~X+\xi,~Y+\eta\in\Gamma({\mathcal{T}}).$$ It is not a Lie bracket, but we have $$\label{eqn:jacobi}
{\left\llbracket  {\left\llbracket  e_1,e_2\right\rrbracket },e_3\right\rrbracket }+c.p.=d T(e_1,e_2,e_3),\quad
\forall~e_1,e_2,e_3\in\Gamma({\mathcal{T}}),$$ where $T(e_1,e_2,e_3) $ is given by $$\label{T}
T(e_1,e_2,e_3)=\frac{1}{3}({\left\langle {\left\llbracket  e_1,e_2\right\rrbracket },e_3\right\rangle}+c.p.).$$

Now we realize the section space of ${\mathcal{T}}$ as the semidirect product of the Lie algebra ${\mathfrak X}(M)$ of vector fields with the natural 2-term deRham complex $$\label{eq:natural-cx}
C^\infty(M)\stackrel{d}{\longrightarrow}\Omega^1(M).$$ For this we need to define a representation up to homotopy of ${\mathfrak X}(M)$ on this complex. For any $X\in{\mathfrak X}(M)$, define linear actions $\mu_0$ and $\mu_1$ by $$\begin{aligned}
\label{C 1}\mu_0(X)(\xi)&\triangleq &{\left\llbracket  X,\xi\right\rrbracket }=L_X\xi-{\frac{1}{2}}d(\xi(X)),\quad\forall ~\xi\in\Omega^1(M)\\
\label{C 2}\mu_1(X)(f)&\triangleq &{\left\langle X,d
f\right\rangle}=\frac{1}{2}X(f),\quad\forall~f\in C^\infty(M).\end{aligned}$$ Define $\nu:\wedge^2{\mathfrak X}(M)\to{\mathrm{Hom}}(\Omega^1(M),C^\infty(M))$ by $$\label{C 3}
\nu(X,Y)(\xi)=T(X,Y,\xi),\quad\forall
~X,Y\in{\mathfrak X}(M),~\xi\in\Omega^1(M).$$

\[pro:rep up to homotopy\] With the above notations, $(C^\infty(M)\stackrel{d}{\longrightarrow}\Omega^1(M),\mu=\mu_0+\mu_1,\nu)$ is a representation up to homotopy of the Lie algebra ${\mathfrak X}(M)$.

[[**Proof.**]{} ]{}For any $f\in C^\infty(M)$, we have $$\mu_0(X)(d f)=L_X df-\frac{1}{2}d X( f)=\frac{1}{2}d X( f),$$ which implies $\mu_0\circ d=d\circ\mu_1$, i.e. $\mu_i$’s are compatible with the differential $d$. By straightforward computations, we have $$\begin{aligned}
\mu_0[X,Y](\xi)-[\mu_0(X),\mu_0(Y)](\xi)&=&{\left\llbracket  {\left\llbracket  X,Y\right\rrbracket },\xi\right\rrbracket }+c.p.\\
&=&d T(X,Y,\xi) =d \big(\nu(X,Y)(\xi)\big),\\
\mu_1[X,Y](f)-[\mu_1(X),\mu_1(Y)](f)&=&{\frac{1}{2}}[X,Y](f)-\frac{1}{4}\big(X(Y(f))-Y(X(f))\big)\\
&=&\frac{1}{4}[X,Y](f),\\
\nu(X,Y)(d f)&=&T(X,Y,d
f)=\frac{1}{3}\big({\frac{1}{2}}[X,Y](f)+\frac{1}{4}\big(X(Y(f))-Y(X(f))\big)\\
&=&\frac{1}{4}[X,Y](f),\end{aligned}$$ which implies (\[eqn:mu0\]) and (\[eqn:mu1\]). At last we need to prove (\[eqn:d k\]), which is obviously equivalent to $$\label{c temp}
\mu_1(X)T(Y,Z,\xi)-T(Y,Z,\mu_0(X)(\xi))+c.p.(X,Y,Z)=T([X,Y],Z,\xi)+c.p.(X,Y,Z).$$ Observe that since $\mu_0(X)(\xi)={\left\llbracket  X,\xi\right\rrbracket }$, we have $$T(Y,Z,\mu_0(X)(\xi))+c.p.(X,Y,Z)+T([X,Y],Z,\xi)+c.p.(X,Y,Z)=T([X,Y],Z,\xi)+c.p.(X,Y,Z,\xi).$$ Furthermore, since for any $f\in {C^{\infty}(M)}$, $\mu_1(X)(f)={\left\langle X,df\right\rangle}$ and the cotangent bundle $T^*M$ is isotropic under the pairing $(\ref{eqn:pair})$, we have $$\mu_1(X)T(Y,Z,\xi)+c.p.(X,Y,Z)={\left\langle X,dT(Y,Z,\xi)\right\rangle}+c.p.(X,Y,Z,\xi).$$

Thus, (\[c temp\]) is equivalent to $${\left\langle X,d T(Y,Z,\xi)\right\rangle}+c.p.(X, Y, Z, \xi)=T([X,Y],Z,\xi)+c.p.(X, Y,
Z, \xi),$$ which holds by Lemma 4.5 in [@rw].

By Proposition \[pro:Lie 2\], we have

\[cor:courant\] $\big(C^\infty(M) \xrightarrow{{\mathrm{d}}=0\oplus d }({\mathfrak X}(M)\oplus\Omega^1(M)),l_2,l_3\big)$ is a 2-term $L_\infty$-algebra, where $l_2$ and $l_3$ are given by (\[2bracket\]) and (\[3bracket\]), in which $\mu_0$, $\mu_1$ and $\nu$ are given by (\[C 1\]), (\[C 2\]) and (\[C 3\]) respectively.

\[rmk:L infty\]

In [@rw] the authors proved that the sections of a Courant algebroid $({{\mathcal{C}}},{\left\langle \cdot,\cdot\right\rangle},{\left\llbracket  \cdot,\cdot\right\rrbracket },\rho)$ form an $L_\infty$-algebra. In the case when ${{\mathcal{C}}}={\mathcal{T}}$ the standard Courant algebroid, the 2-term $L_\infty$-algebra is given by $${C^{\infty}(M)}\xrightarrow{{\mathrm{d}}=0\oplus d } {\mathfrak X}(M) \oplus
\Omega^1(M)=\Gamma({\mathcal{T}}),$$ with brackets given by $$l_2(e_1,e_2)={\left\llbracket  e_1,e_2\right\rrbracket },\quad l_2(e_1,f)={\left\langle e_1,{\mathrm{d}}f\right\rangle},\quad l_3(e_1,e_2,e_3)=-T(e_1,e_2,e_3), \quad l_{i\ge 4} =0,$$ for any $e_1,e_2,e_3\in\Gamma({\mathcal{T}}),~f\in {C^{\infty}(M)}$. Here $T$ is defined by (\[T\]). It is easy to verify that this is the same as our 2-term $L_\infty$-algebra in Corollary \[cor:courant\].

We can also modify our complex to $$\Omega^1(M)\stackrel{{\rm{Id}}}{\longrightarrow}\Omega^1(M).$$ Following the same procedure, we obtain another representation up to homotopy of the Lie algebra ${\mathfrak X}(M)$ and therefore obtain another 2-term $L_\infty$-algebra which is also totally determined by the Courant algebroid $({\mathcal{T}},{\left\langle \cdot,\cdot\right\rangle},{\left\llbracket  \cdot,\cdot\right\rrbracket },\rho)$. More precisely, for any $X\in {\mathfrak X}(M),~\xi\in\Omega^1(M)$, $\mu_0=\mu_1$ is given by $$\mu_0(X)(\xi)\triangleq{\left\llbracket  X,\xi\right\rrbracket },$$ and $\nu:{\mathfrak X}(M):\longrightarrow\Omega^2({\mathfrak X}^2(M),{\mathrm{End}}(\Omega^1(M),\Omega^1(M)))$ is given by $$\nu(X,Y)(\xi)\triangleq d T(X,Y,\xi).$$

With the above notations, $(\Omega^1(M)\stackrel{{\rm{Id}}}{\longrightarrow}\Omega^1(M),\mu=\mu_0=\mu_1,\nu)$ is a representation up to homotopy of the Lie algebra ${\mathfrak X}(M)$.

Example : Courant algebroids over a point and string Lie 2-algebras {#sec:string}
-------------------------------------------------------------------

A Courant algebroid over a point is literally a quadratic Lie algebra, namely a Lie algebra ${\mathfrak k}$ together with nondegenerate inner product ${\left\langle \cdot,\cdot\right\rangle}$ which is invariant under the adjoint action. People often think that a Courant algebroid over a point is a string Lie 2-algebra [^3]. Here we justify this thinking.

\[defi:string-alg\] The string Lie 2-algebra associated to a quadratic Lie algebra $({\mathfrak k}, {\left\langle \cdot,\cdot\right\rangle})$, is a 2-term $L_\infty$-algebra $\mathbb R\stackrel{0}{\longrightarrow}{\mathfrak k}$, whose degree-$0$ part is ${\mathfrak k}$, degree-$1$ part is $\mathbb R$, and $l_2,~l_3$ are given by $$\begin{aligned}
~l_2(e,c)&=&0,\\
~l_2((e_1,c_1),(e_2,c_2))&=&([e_1,e_2],0),\\
~l_3((e_1,c_1),(e_2,c_2),(e_3,c_3))&=&(0,{\left\langle [e_1,e_2],e_3\right\rangle}),\end{aligned}$$ where $e,~e_1,~e_2,~e_3\in{\mathfrak k},~c,~c_1,~c_2,~c_3\in\mathbb R$.

The representation $\phi $ of ${\mathfrak k}$ on $\mathbb R$ and the 3-cocycle $\theta:\wedge^3{\mathfrak k}\longrightarrow \mathbb R$ in the corresponding quadruple in Theorem \[thm:skeletal Lie 2 a\] is given by $$\begin{aligned}
\nonumber\rho(e)(c)&=&l_2(e,c)\\
\label{3cocycle}\theta(e_1,e_2,e_3)&=&{\left\langle [e_1,e_2],e_3\right\rangle}.\end{aligned}$$

\[rk:semisimple\] In the definition of string Lie 2-algebras [@baez:string; @henriques] the base Lie algebra ${\mathfrak k}$ is usually required to be semisimple and of compact type, such that the Jacobiator gives rise to the generator of $H^3({\mathfrak k}, \mathbb Z)=\mathbb Z$. This is because Witten’s original motivation is to obtain a $3$-connected cover of $Spin(n)$, and $\mathfrak{so}(n)$ is simple and of compact type. However, to write down the structure of the string Lie 2-algebra, we only need a quadratic Lie algebra. Then $H^3({\mathfrak k},
\mathbb Z)$ is not necessarily $\mathbb Z$ for a general quadratic Lie algebra ${\mathfrak k}$. For example, for the abelian Lie algebra ${\mathbb R}$, any inner product is adjoint-invariant, and $H^3({\mathbb R}, {\mathbb Z})=0$. We thus face the danger that sometimes, the string Lie 2-algebra is trivial, that is the Jacobiator corresponds to the trivial element in $H^3({\mathfrak k}, \mathbb Z)$. Then what we have is a strict Lie 2-algebra, which is a crossed module of Lie algebras. Then the integration of a crossed module of Lie algebras is simply a crossed module of Lie groups. However, we will verify that the example we consider is not such a case.

The standard Courant algebroid motivates us to consider the case of the direct sum ${\mathfrak k}={\mathfrak g}\oplus{\mathfrak g}^*$ of a Lie algebra ${\mathfrak g}$ and its dual with the semidirect product Lie algebra structure: $$[X+\xi,Y+\eta]=[X,Y]_{\mathfrak g}+{\mathrm{ad}}_X^*\eta-{\mathrm{ad}}_Y^*\xi,$$here $[\cdot,\cdot]_{\mathfrak g}$ is the Lie bracket of ${\mathfrak g}$. The nondegenerate invariant pairing ${\left\langle \cdot,\cdot\right\rangle}$ on ${\mathfrak g}\oplus {\mathfrak g}^*$ is given by $${\left\langle X+\xi,Y+\eta\right\rangle}={\frac{1}{2}}(\eta(X)+\xi(Y)),\quad\forall~X+\xi,Y+\eta\in{\mathfrak g}\oplus{\mathfrak g}^*.$$ With these definitions, $({\mathfrak g}\oplus{\mathfrak g}^*,[\cdot,\cdot],{\left\langle \cdot,\cdot\right\rangle})$ is a quadratic Lie algebra. In fact, we have $$\begin{aligned}
{\left\langle [X_1+\xi_1,X_2+\xi_2],X_3+\xi_3\right\rangle}&=&{\left\langle [X_1,X_2]_{\mathfrak g}+{\mathrm{ad}}_X^*\xi_2-{\mathrm{ad}}_Y^*\xi_1,X_3+\xi_3\right\rangle}\\
&=&{\left\langle [X_1,X_2]_{\mathfrak g},\xi_3\right\rangle}+c.p..\end{aligned}$$ Similarly, we have $${\left\langle X_2+\xi_2,[X_1+\xi_1,X_3+\xi_3]\right\rangle}={\left\langle [X_1,X_3]_{\mathfrak g},\xi_2\right\rangle}+c.p.,$$ which implies that ${\left\langle X_2+\xi_2,[X_1+\xi_1,X_3+\xi_3]\right\rangle}+{\left\langle [X_1+\xi_1,X_2+\xi_2],X_3+\xi_3\right\rangle}=0$, i.e. the nondegenerate inner product ${\left\langle \cdot,\cdot\right\rangle}$ is invariant under the adjoint action. This example is a special case of [@lu-weinstein Theorem 1.12] with ${\mathfrak g}^*$ equipped with $0$ Lie bracket. Thus $({\mathfrak g}, {\mathfrak g}^*)$ forms a Lie bialgebra or equivalently $({\mathfrak g}\oplus {\mathfrak g}^*, {\mathfrak g}, {\mathfrak g}^*)$ is a Manin triple. However, honestly we have not found other Lie bialgebras (Manin triples) giving rise to Lie 2-algebras of the form of semidirect products.

We denote the corresponding string Lie 2-algebra of $({\mathfrak g}\oplus {\mathfrak g}^*,
[\cdot, \cdot], {\left\langle \cdot,\cdot\right\rangle})$ by $$\label{Lie 2 g g dual}
\mathbb R\stackrel{0}{\longrightarrow}{\mathfrak g}\oplus {\mathfrak g}^*.$$ The corresponding 3-cocycle (see ), which we denote by $\widetilde{\nu}:\wedge^3({\mathfrak g}\oplus {\mathfrak g}^*)\longrightarrow
\mathbb R$ is $$\begin{aligned}
\label{eqn:nu3}
\widetilde{\nu}(X_1+\xi_1,X_2+\xi_2,X_3+\xi_3)&=&{\left\langle [X_1+\xi_1,X_2+\xi_2],X_3+\xi_3\right\rangle}\\
\nonumber&=&{\left\langle [X_1,X_2]_{\mathfrak g},\xi_3\right\rangle}+c.p.$$

$(\mathbb
R\stackrel{0}{\longrightarrow} {\mathfrak g}^*,\mu_1=0,
\mu_0={\mathrm{ad}}^*,\nu=[\cdot,\cdot]_{\mathfrak g})$ is a 2-term representation up to homotopy of the Lie algebra ${\mathfrak g}$. Moreover the string Lie 2-algebra $ \mathbb R\stackrel{0}{\longrightarrow}{\mathfrak g}\oplus
{\mathfrak g}^* $ is the semidirect product of ${\mathfrak g}$ and the complex $\mathbb R\stackrel{0}{\longrightarrow} {\mathfrak g}^*$.

Since ${\mathrm{d}}=0$, we only need to verify that $\mu_0$ and $\mu_1$ are Lie algebra morphisms and . ${\mathrm{ad}}^*: {\mathfrak g}\to
{\mathrm{End}}({\mathfrak g}^*)$ and $0$ are both Lie algebra morphisms. follows from Jacobi identity of $[\cdot,\cdot]_{\mathfrak g}$.

Then it is not hard to see that the string Lie 2-algebra $\mathbb
R\stackrel{0}{\longrightarrow}{\mathfrak g}\oplus {\mathfrak g}^*$ with formulas in Definition \[defi:string-alg\] is exactly the semidirect product of ${\mathfrak g}$ with the complex $(\mathbb R\stackrel{0}{\longrightarrow}
{\mathfrak g}^*,\mu_1=0, \mu_0={\mathrm{ad}}^*,\nu=[\cdot,\cdot]_{\mathfrak g})$ with the formulas , .

\[pro:nondegenerate\] If the Lie algebra ${\mathfrak g}$ is semisimple, the Lie algebra 3-cocycle $\widetilde{\nu}$ which is given by (\[eqn:nu3\]) is not exact.

[[**Proof.**]{} ]{}Let ${\left\langle \cdot,\cdot\right\rangle}_k$ be the Killing form on ${\mathfrak g}$. The proof follows from the fact that the Cartan 3-form ${\left\langle [\cdot,\cdot],\cdot\right\rangle}_k$ on a semisimple Lie algebra is not exact. Since ${\mathfrak g}$ is semisimple, the Killing form ${\left\langle \cdot,\cdot\right\rangle}_k$ is nondegenerate. Identify ${\mathfrak g}^*$ and ${\mathfrak g}$ by using the Killing form ${\left\langle \cdot,\cdot\right\rangle}_k$ and let ${\mathcal{K}}$ be the corresponding isomorphism, $${\left\langle {\mathcal{K}}(\xi),X\right\rangle}_k={\left\langle \xi,X\right\rangle}.$$ Assume that $\widetilde{\nu}=d\phi$ for some $\phi:\wedge^2({\mathfrak g}\oplus{\mathfrak g}^*)\longrightarrow \mathbb R$, define $\varphi:\wedge^2({\mathfrak g}\oplus{\mathfrak g})\longrightarrow \mathbb R$ by $$\phi(X+\xi,Y+\eta)=\varphi(X+{\mathcal{K}}(\xi),Y+{\mathcal{K}}(\eta)),$$then we have $$\begin{aligned}
\widetilde{\nu}(X,Y,\xi)&=&d\phi(X,Y,\xi)\\
&=&-\phi([X,Y],\xi)+\phi({\mathrm{ad}}^*_X\xi, Y)-\phi({\mathrm{ad}}^*_Y\xi, X)\\
&=&-\varphi([X,Y],{\mathcal{K}}(\xi))+\varphi([X,{\mathcal{K}}(\xi)],
Y)-\varphi([Y,{\mathcal{K}}(\xi)], X)\\
&=&d\varphi(X,Y,{\mathcal{K}}(\xi)).\end{aligned}$$ On the other hand, we have $$\widetilde{\nu}(X,Y,\xi)={\left\langle [X,Y],\xi\right\rangle}={\left\langle [X,Y],{\mathcal{K}}(\xi)\right\rangle}_k,$$ which implies that the Cartan 3-from $${\left\langle [X,Y],{\mathcal{K}}(\xi)\right\rangle}_k=d\varphi(X,Y,{\mathcal{K}}(\xi)),$$ is exact. This is a contradiction.

In the last section, we will give the integration of the string Lie 2-algebra $ \mathbb R\stackrel{0}{\longrightarrow}{\mathfrak g}\oplus
{\mathfrak g}^* $ by using the semidirect product of a Lie group with its 2-term representation up to homotopy. It turns out that this string Lie 2-algebra can be integrated to a special Lie 2-group with a finite dimensional model.

Example III: Omni-Lie algebroids ${\mathfrak{D}}E\oplus {\mathfrak{J}}E$
------------------------------------------------------------------------

The notion of omni-Lie algebroids, which was introduced by Chen and Liu in [@clomni], is a generalization of Weinstein’s omni-Lie algebras. Just as Dirac structures of an omni-Lie algebra characterize Lie algebra structures on a vector space, Dirac structures of an omni-Lie algebroid characterize Lie algebroid structures on a vector bundle. See [@clsdirac] for more details. In fact, the role of omni-Lie algebroids ${\mathfrak{D}}E\oplus {\mathfrak{J}}E$ in $E$-Courant algebroids, which was introduced in [@clsecourant], is the same as the role of standard Courant algebroids $TM\oplus
T^*M$ in Courant algebroids.

Now we briefly recall the notion of omni-Lie algebroids and we will see that it gives rise to a 2-term $L_\infty$-algebra which is a semidirect product. In this subsection $E$ is a vector bundle over a smooth manifold $M$, $\Gamma(E)$ is the section space of $E$.

For a vector bundle $E$, let ${\mathfrak{D}}E$ be its covariant differential operator bundle. The associated Atiyah sequence is given by $$\label{Seq:DE}
\xymatrix@C=0.5cm{0 \ar[r] & {\mathfrak {gl}}(E)  \ar[rr]^{{\mathbbm{i}}} &&
                {\mathfrak{D}}{E}  \ar[rr]^{a} && TM \ar[r]  & 0.
                }$$ The associated $1$-jet vector bundle ${{\mathfrak{J}}} E$ is defined as follows. For any $m\in M$, $({{\mathfrak{J}}}{E})_m$ is defined as a quotient of local sections of $E$. Two local sections $u_1$ and $u_2$ are equivalent and we denote this by $u_1\sim u_2$ if $$u_1(m)=u_2(m) ~~\mbox{ and }~~ d{\left\langle u_{1
},\xi\right\rangle}_m=d{\left\langle u_{2 },\xi\right\rangle}_m, \quad\forall~ \xi\in\Gamma(E^*).$$ So any $\mu\in ({{\mathfrak{J}}}{E})_m$ has a representative $u\in\Gamma(E)$ such that $\mu=[u]_m$. Let ${\mathbbm{p}}$ be the projection which sends $[u]_m$ to $u(m)$. Then ${\mathrm{Ker}}{\mathbbm{p}}\cong  {\mathrm{Hom}}(TM,E)$ and there is a short exact sequence, called the jet sequence of $E$, $$\label{Seq:JetE}
\xymatrix@C=0.5cm{0 \ar[r] & {\mathrm{Hom}}(TM,E)  \ar[rr]^{~\qquad {\mathbbm{e}}~} &&
                {{\mathfrak{J}}}{E} \ar[rr]^{{\mathbbm{p}}} && E \ar[r]  & 0,
                }$$ from which it is straightforward to see that ${\mathfrak{J}}E$ is a finite dimensional vector bundle. Moreover, $\Gamma({\mathfrak{J}}{E})$ is isomorphic to $\Gamma(E) \oplus \Gamma (T^*M \otimes E)$ as an ${\mathbb R}$-vector space, and any $u\in \Gamma (E)$ has a lift ${\mathbbm{d}}u\in\Gamma ({{\mathfrak{J}}} E)$ by taking its equivalence class, such that $$\label{Eqt:CWMmoduleGammaE}
{\mathbbm{d}}(fu)=f {\mathbbm{d}}u+ d f\otimes u, \quad \forall~ f\in {C^{\infty}(M)}.$$ In [@clomni] the authors proved that $$\begin{aligned}
  {{\mathfrak{J}}E} \cong
{\left\{\nu\in {\mathrm{Hom}}({\mathfrak{D}}{E},E)\,|\,
\nu(\Phi)=\Phi\circ\nu({\rm{Id}}_{E}),\quad\forall~ ~ ~\Phi\in {\mathfrak {gl}}(E)\right\}}.\end{aligned}$$

Therefore, there is an $E$-pairing between ${\mathfrak{J}}{E}$ and ${\mathfrak{D}}{E}$ by setting: $$\label{Eqt:conpairingE} {\left\langle  \mu,{\mathfrak d}\right\rangle }_E~ {\triangleq}{\mathfrak d}(u),\quad\forall~ ~~ \mu\in ({\mathfrak{J}}{E})_m,~{\mathfrak d}\in({\mathfrak{D}}{E})_m,$$ where $u\in \Gamma(E)$ satisfies $\mu=[u]_m$. Particularly, one has $$\begin{aligned}
\label{conpairing1}
{\left\langle  \mu,\Phi\right\rangle }_E &=& \Phi\circ {\mathbbm{p}}(\mu),\quad\forall~ ~~ \Phi\in {\mathfrak {gl}}(E),~\mu\in{\mathfrak{J}}{E};\\
\label{conpairing2} {\left\langle  {{\mathfrak y}},{\mathfrak d}\right\rangle }_{E} &=& {{\mathfrak y}}\circ
a({\mathfrak d}),\quad\forall~ ~~ {\mathfrak y}\in {\mathrm{Hom}}(TM,E),~{\mathfrak d}\in{\mathfrak{D}}{E}.\end{aligned}$$ Furthermore, we claim that $\Gamma ({\mathfrak{J}}E)$ is an invariant subspace of the Lie derivative ${{\mathfrak L}}_{{\mathfrak d}}$ for any ${\mathfrak d}\in\Gamma({\mathfrak{D}}{E})$, which is defined by the Leibniz rule as follows: $$\begin{aligned}
\nonumber{\left\langle  {{\mathfrak L}}_{{\mathfrak d}}\mu,{\mathfrak d}{^\prime}\right\rangle }_{E}&{\triangleq}&
{\mathfrak d}{\left\langle  \mu,{\mathfrak d}{^\prime}\right\rangle }_{E}-{\left\langle  \mu,[{\mathfrak d},{\mathfrak d}{^\prime}]_{{\mathfrak{D}}}\right\rangle }_{E},
\quad\forall~ \mu \in \Gamma({\mathfrak{J}}{E}), ~
~{\mathfrak d}{^\prime}\in\Gamma({\mathfrak{D}}{E}).\end{aligned}$$

Introduce a nondegenerate symmetric $E$-valued $2$-form ${\left ( \cdot,\cdot\right )_E}$ on $ {\mathcal{E}}\triangleq{\mathfrak{D}}{E}\oplus {\mathfrak{J}}{E}$ by: $${\left ( {\mathfrak d}+\mu,{\mathfrak r}+\nu\right )_E}{\triangleq}{\frac{1}{2}}({\left\langle  {\mathfrak d},\nu\right\rangle }_E
+{\left\langle  {\mathfrak r},\mu\right\rangle }_E),\quad\forall~ ~~
{\mathfrak d},{\mathfrak r}\in{\mathfrak{D}}{E},~\mu,\nu\in{\mathfrak{J}}{E}.$$ Furthermore, we define an antisymmetric bracket ${\left\llbracket  \cdot,\cdot\right\rrbracket }$ on $\Gamma({\mathcal{E}})$ by: $$\begin{aligned}
{\left\llbracket  {\mathfrak d}+\mu,{\mathfrak r}+\nu\right\rrbracket }&{\triangleq}&
[{\mathfrak d},{\mathfrak r}]_{{\mathfrak{D}}}+{{\mathfrak L}}_{{\mathfrak d}}\nu-{{\mathfrak L}}_{{\mathfrak r}}\mu +
{\frac{1}{2}}\big({\mathbbm{d}}{\left\langle  \mu,{\mathfrak r}\right\rangle }_E-{\mathbbm{d}}{\left\langle  \nu,{\mathfrak d}\right\rangle }_E\big).\end{aligned}$$

In [@clomni] the authors call the quadruple $({\mathcal{E}},{\left\llbracket  \cdot,\cdot\right\rrbracket },{\left ( \cdot,\cdot\right )_E},{\rho_{\varepsilon}})$ the [**omni-Lie algebroid**]{} [^4] associated to the vector bundle $E$, where $\rho$ is the projection of ${\mathcal{E}}$ onto ${\mathfrak{D}}{E}$. Even though ${\left\llbracket  \cdot,\cdot\right\rrbracket }$ is antisymmetric, it is not a Lie bracket. More precisely, for any $X,Y,Z\in \Gamma({\mathcal{E}})$, we have $${\left\llbracket  {\left\llbracket  X,Y\right\rrbracket },Z\right\rrbracket }+c.p.={\mathbbm{d}}T(X,Y,Z),$$ where $T:\Gamma(\wedge^3{\mathcal{E}})\longrightarrow \Gamma(E)$ is defined by \[T\] T(X,Y,Z)=([( [X,Y]{},Z)\_E]{}+c.p.).

Now let us construct a 2-term $L_\infty$-algebra from the omni-Lie algebroid ${\mathcal{E}}$. Obviously, $\Gamma({\mathfrak{D}}E)$ is a Lie algebra and there is a natural 2-term complex $$\Gamma(E)\stackrel{{\mathrm{d}}=0\oplus{\mathbbm{d}}}{\longrightarrow}\Gamma({\mathfrak{J}}E).$$

For any ${\mathfrak d}\in\Gamma({\mathfrak{D}}E)$, define linear actions $\mu_0$ and $\mu_1$ by $$\label{O 1}\left\{\begin{array}{l}
\mu_0({\mathfrak d})(\mu)\triangleq {\left\llbracket  {\mathfrak d},\mu\right\rrbracket }={{\mathfrak L}}_{\mathfrak d}\mu-{\mathbbm{d}}{\left ( \mu,{\mathfrak d}\right )_E},\quad\forall~\mu\in \Gamma({\mathfrak{J}}E),\\
\mu_1({\mathfrak d})(u)\triangleq {\left ( {\mathfrak d},{\mathbbm{d}}u\right )_E}=\frac{1}{2}{\mathfrak d}(u),\quad\forall~u\in\Gamma(E).\end{array}\right.$$ Define $\nu:\wedge^2\Gamma({\mathfrak{D}}E)\longrightarrow{\mathrm{Hom}}(\Gamma({\mathfrak{J}}E),\Gamma( E))$ by $$\label{O 2}
\nu({\mathfrak d},{\mathfrak t})(\xi)=T({\mathfrak d},{\mathfrak t},\mu),\quad\forall
~{\mathfrak d},{\mathfrak t}\in\Gamma({\mathfrak{D}}E),~\mu\in\Gamma({\mathfrak{J}}E).$$ Similar to Proposition \[pro:rep up to homotopy\], we prove

With the above notations, $(\Gamma(E)\stackrel{{\mathrm{d}}=0\oplus{\mathbbm{d}}}{\longrightarrow}\Gamma({\mathfrak{J}}E),\mu=\mu_0+\mu_1,\nu)$ is a representation up to homotopy of the Lie algebra $\Gamma({\mathfrak{D}}E)$.

$\big(\Gamma(E)\stackrel{{\mathrm{d}}=0\oplus{\mathbbm{d}}}{\longrightarrow}\Gamma({\mathfrak{D}}E)\oplus\Gamma({\mathfrak{J}}E),l_2,l_3\big)$ is a 2-term $L_\infty$-algebra, where $l_2$ and $l_3$ are given by (\[2bracket\]) and (\[3bracket\]), in which $\mu$ and $\nu$ are given by (\[O 1\]) and (\[O 2\]) respectively.

If the base manifold $M$ is a point, i.e. $E$ is a vector space, for which we use a new notation $V$, then ${\mathfrak{D}}V={\mathfrak {gl}}(V),~{\mathfrak{J}}V=V$, and we recover the notion of omni-Lie algebras. The complex $\Gamma(V)\stackrel{0\oplus{\mathbbm{d}}}{\longrightarrow}\Gamma({\mathfrak{J}}V)$ reduces to $V\stackrel{{\rm{Id}}}{\longrightarrow}V$, which is a representation up to homotopy of ${\mathfrak {gl}}(V)$ with $(\mu_0=\mu_1, \nu)$ given by $$\label{omni mu nu}
\mu_0(A)(u)={\frac{1}{2}}Au,\quad
\nu(A,B)=\frac{1}{4}[A,B],\quad\forall~A,B\in{\mathfrak {gl}}(V),~u\in V.$$ Hence even though an omni-Lie algebra ${\mathfrak {gl}}(V)\oplus V$ is not a Lie algebra, we can extend it to a 2-term $L_\infty$-algebra, of which $L_0={\mathfrak {gl}}(V)\oplus V,~L_1=V$, $l_2$ and $l_3$ are given by (\[2bracket\]) and (\[3bracket\]), in which $\mu$ and $\nu$ are given by (\[omni mu nu\]). This 2-term $L_\infty$-algebra is a semidirect product of ${\mathfrak {gl}}(V)$ with $V\xrightarrow{{\rm{Id}}} V$.

We will study the global object of the 2-term $L_\infty$-algebra associated to an omni-Lie algebra in the forthcoming paper [@sheng-zhu:II].

Representation up to homotopy of Lie groups and semidirect product {#sec:gp}
==================================================================

The representation up to homotopy of a Lie group was introduced in [@abad]. In this section we define the semidirect product of a Lie group with a 2-term representation up to homotopy and prove that the semidirect product is a Lie 2-group. Thus we first recall some background on Lie 2-groups.

A group is a monoid where every element has an inverse. A 2-group is a monoidal category where every object has a weak inverse and every morphism has an inverse. Denote the category of smooth manifolds and smooth maps by $\rm Diff$, a semistrict[^5] Lie 2-group is a 2-group in $\rm DiffCat$, where $\rm DiffCat$ is the 2-category consisting of categories, functors, and natural transformations in $\rm Diff$. For more details, see [@baez:2gp] and here we only recall the expanded definition:

[[@baez:2gp]]{} A semistrict Lie 2-group consists of an object $C$ in $\rm
DiffCat$, i.e. $$\xymatrix{ C_1 \ar@<1ex>[r]^{s} \ar[r]_{t} & C_0},$$ where $C_1,~C_0$ are objects in $\rm Diff$, $s,~t$ are the source and target maps, and there is a vertical multiplication $\cdot_{\mathrm{v}}:C\times C\longrightarrow C$, together with

-   a functor (horizontal multiplication) $\cdot_{\mathrm{h}}:C\times C\longrightarrow C$,

-   an identity object $1$,

-   a contravariant functor ${\mathrm{inv}}:C\longrightarrow C$

and the following natural isomorphisms:

-   the [**associator**]{} $$a_{x,y,z}:(x\cdot_{\mathrm{h}}y)\cdot_{\mathrm{h}}z\longrightarrow x\cdot_{\mathrm{h}}(y\cdot_{\mathrm{h}}z),$$

-   the [**left**]{} and [**right unit**]{} $$l_x:1\cdot_{\mathrm{h}}x\longrightarrow x,\quad r_x:x\cdot_{\mathrm{h}}1\longrightarrow x,$$

-   the [**unit**]{} and [**counit**]{} $$i_x:1\longrightarrow x\cdot_{\mathrm{h}}{\mathrm{inv}}(x),\quad e_x:{\mathrm{inv}}(x)\cdot_{\mathrm{h}}x\longrightarrow 1,$$

such that the following diagrams commute:

-   the [**pentagon identity** ]{} for the associator $$\xymatrix{ & (w\cdot_{\mathrm{h}}x)\cdot_{\mathrm{h}}(y\cdot_{\mathrm{h}}z)\ar[dr]^{a_{w,x, y\cdot_{\mathrm{h}}z}}&  \\
    ((w\cdot_{\mathrm{h}}x)\cdot_{\mathrm{h}}y)\cdot_{\mathrm{h}}z\ar[ur]^{a_{(w\cdot_{\mathrm{h}}x),y,z}}\ar[dr]_{a_{w,x,y}\cdot_{\mathrm{h}}1_z}&&w\cdot_{\mathrm{h}}(x\cdot_{\mathrm{h}}(y\cdot_{\mathrm{h}}z))\\
    &(w\cdot_{\mathrm{h}}(x\cdot_{\mathrm{h}}y))\cdot_{\mathrm{h}}z\stackrel{a_{w,x\cdot_{\mathrm{h}}y,z}}{\longrightarrow} w\cdot_{\mathrm{h}}((x\cdot_{\mathrm{h}}y)\cdot_{\mathrm{h}}z)\ar[ur]^{1_w\cdot_{\mathrm{h}}a_{x,y,z}}&}$$

-   the [**triangle identity**]{} for the left and right unit lows: $$\xymatrix{
    ( x\cdot_{\mathrm{h}}1)\cdot_{\mathrm{h}}y\ar[rr]^{a_{x,1,y}}\ar[dr]^{r_x\cdot_{\mathrm{h}}1_y}&&x\cdot_{\mathrm{h}}(1\cdot_{\mathrm{h}}y)\ar[dl]^{1_x\cdot_{\mathrm{h}}l_y}\\
    &x\cdot_{\mathrm{h}}y& }$$

-   the [**first zig-zag identity**]{}: $$\xymatrix{
    &(x\cdot_{\mathrm{h}}{\mathrm{inv}}(x))\cdot_{\mathrm{h}}x\stackrel{a_{x,{\mathrm{inv}}(x),x}}{\longrightarrow}x\cdot_{\mathrm{h}}({\mathrm{inv}}(x)\cdot_{\mathrm{h}}x)\ar[dr]^{1_x\cdot_{\mathrm{h}}e_x}&\\
    1\cdot_{\mathrm{h}}x\ar[dr]^{l_x}\ar[ur]^{i_x\cdot_{\mathrm{h}}1_x}&&x\cdot_{\mathrm{h}}1\\
    &x\ar[ur]^{r_x^{-1}}&}$$

-   the [**second zig-zag identity**]{}: $$\xymatrix{
    &{\mathrm{inv}}(x)\cdot_{\mathrm{h}}(x\cdot_{\mathrm{h}}{\mathrm{inv}}(x))\stackrel{a_{{\mathrm{inv}}(x),x,{\mathrm{inv}}(x)}}{\longrightarrow}({\mathrm{inv}}(x)\cdot_{\mathrm{h}}x)\cdot_{\mathrm{h}}{\mathrm{inv}}(x)\ar[dr]^{e_x\cdot_{\mathrm{h}}1_{{\mathrm{inv}}(x)}}&\\
    {\mathrm{inv}}(x)\cdot_{\mathrm{h}}1\ar[dr]^{r_{{\mathrm{inv}}(x)}}\ar[ur]^{1_{{\mathrm{inv}}(x)}\cdot_{\mathrm{h}}i_x}&&1\cdot_{\mathrm{h}}{\mathrm{inv}}(x).\\
    &{\mathrm{inv}}(x)\ar[ur]^{l_{{\mathrm{inv}}(x)}^{-1}}&}$$

In the special case where $a_{x,y,z},~l_x,~r_x,~i_x,~e_x$ are all identity isomorphisms, we call such a Lie 2-group a [*strict Lie 2-group*]{}[^6].

In the same reference, they also defined the notion of special 2-groups, which we recall here,

\[defi:special 2 G\] A [**special Lie 2-group**]{} is a Lie 2-group of which the source and target coincide and the left unit law $l$, the right unit law $r$, the unit $i$ and the counit $e$ are identity isomorphisms.

For classification of special Lie 2-groups, we need the group cohomology with smooth cocycles, that is we consider the cochain complex with smooth morphisms $G^{\times
n}\to M$ with $G$ a Lie group, $M$ its module. And the differential is defined as usual for group cohomology. We denote this cohomology by $H_{sm}^\bullet(G, M)$.

[[@baez:2gp Theorem 8.3.7]]{}\[thm:special 2 G\] There is a one-to-one correspondence between special Lie 2-groups and quadruples $(K_1,K_2,\Phi,\Theta)$ consisting a Lie group $K_1$, an abelian group $K_2$, an action $\Phi$ of $K_1$ as automorphisms of $K_2$ and a normalized smooth 3-cocycle $\Theta:K_1^3\longrightarrow
K_2$. Two special Lie 2-groups are isomorphic if and only if they corresponds to the same[^7] $(K_1, K_2, \Phi)$ and the corresponding 3-cocycles represent the same element in $H^3_{sm}(K_1,
K_2)$.

We briefly recall that given a quadruple $(K_1,K_2,\Phi,\Theta)$ the corresponding semistrict Lie 2-group has the Lie group $K_1$ as the space of objects and the semidirect product Lie group $K_1 \ltimes_{\Phi}
K_2$ as the space of morphisms. The associator is given by $\Theta$.

A unital 2-term representation up to homotopy of a Lie group $G$ consists of

-   A 2-term complex of vector spaces $V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0$.

-   A nonassociative action $F_1$ on $V_0$ and $V_1$ satisfying $${\mathrm{d}}F_1 = F_1 {\mathrm{d}}, \quad F_1(1_G)={\rm{Id}}.$$

-   A smooth map $F_2:G\times G\longrightarrow{\mathrm{End}}(V_0,V_1)$ such that $$\label{eqn:F fail}
    F_1(g_1)\cdot F_1(g_2)-F_1(g_1\cdot g_2)=[{\mathrm{d}},F_2(g_1,g_2)],$$ as well as $$\label{eqn:F closed}
    F_1(g_1)\circ F_2(g_2,g_3)-F_2(g_1\cdot g_2,g_3)+F_2(g_1, g_2\cdot
    g_3)-F_2(g_1, g_2)\circ F_1(g_3)=0.$$

We denote this 2-term representation up to homotopy of the Lie group $G$ by $(V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0,F_1,F_2)$. One should be careful that even if $F_1$ is a usual associative action, is not equivalent to $F_2$ being a 2-cocycle. This is strangely different from the Lie algebra case (see Section \[sec:int\]). Define $\widetilde{F_2}:(G\ltimes
V_0)^3\longrightarrow V_1$ by $$\label{F2tuta}
\widetilde{F_2}((g_1,\xi_1),(g_2,\xi_2),(g_3,\xi_3))=F_2(g_1,
g_2)(\xi_3).$$ If $F_1$ is a usual associative action, we form $G\ltimes V_0$ the semidirect product. Then $V_1$ is a $G\ltimes V_0$-module with an associated action $\widetilde{F_1}$ of $G\ltimes V_0$ on $V_1$ $$\widetilde{F_1}(g,\xi)(m)=F_1(g)(m),\quad\forall~m\in V_1.$$

\[pro:3cocycle\] If $F_1$ is the usual associative action of the Lie group $G$ on the complex $V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0$, then $\widetilde{F_2}$ defined by (\[F2tuta\]) is a group 3-cocycle representing an element in $H^3_{sm}(G\ltimes V_0,V_1)$.

[[**Proof.**]{} ]{}By direct computations, we have $$\begin{aligned}
&&d\widetilde{F_2}((g_1,\xi_1),(g_2,\xi_2),(g_3,\xi_3),(g_4,\xi_4))\\&=&\widetilde{F_1}(g_1,\xi_1)\widetilde{F_2}((g_2,\xi_2),(g_3,\xi_3),(g_4,\xi_4))\\
&&-\widetilde{F_2}((g_1,\xi_1)\cdot(g_2,\xi_2),(g_3,\xi_3),(g_4,\xi_4))+\widetilde{F_2}((g_1,\xi_1),(g_2,\xi_2)\cdot(g_3,\xi_3),(g_4,\xi_4))\\
&&-\widetilde{F_2}((g_1,\xi_1),(g_2,\xi_2),(g_3,\xi_3)\cdot(g_4,\xi_4))+\widetilde{F_2}((g_1,\xi_1),(g_2,\xi_2),(g_3,\xi_3))\\
&=&F_1(g_1)F_2(g_2,g_3)(\xi_4)-F_2(g_1\cdot g_2,g_3)(\xi_4)+F_2(g_1,
g_2\cdot g_3)(\xi_4)\\
&&-F_2(g_1,g_2)(\xi_3+F_1(g_3)(\xi_4))+F_2(g_1,g_2)(\xi_3)\\
&=&\big(F_1(g_1)\circ F_2(g_2,g_3)-F_2(g_1\cdot g_2,g_3)+F_2(g_1,
g_2\cdot g_3)-F_2(g_1, g_2)\circ F_1(g_3)\big)(\xi_4).\end{aligned}$$ By (\[eqn:F closed\]), $\widetilde{F_2}$ is a Lie group 3-cocycle.

Similar to the fact that associated to any representation of a Lie group, we can form a new Lie group which is their semidirect product, for the 2-term representation up to homotopy of a Lie group, we can form a Lie 2-group.

\[thm:main 1\] Given a 2-term representation up to homotopy $(V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0,F_1,F_2)$ of a Lie group $G$, its semidirect product with $G$ is defined to be $$\begin{array}{c}
G\times V_0\times V_1\\
\vcenter{\rlap{s }}~\Big\downarrow\Big\downarrow\vcenter{\rlap{t }}\\
G\times V_0.
 \end{array}$$ Then it is a Lie 2-group with the following structure maps:

The source and target are given by \[s t\] s(g,,m)=(g,),t(g,,m)=(g,+m). The vertical multiplication $\cdot_{\mathrm{v}}$ is given by $$(h,\eta,n)\cdot_{\mathrm{v}}(g,\xi,m) =(g,\xi,m+n),\quad \mbox{where}~
h=g,\eta=\xi+{\mathrm{d}}m.$$ The horizontal multiplication $\cdot_{\mathrm{h}}$ of objects is given by \[m o\] (g\_1,)\_(g\_2,)=(g\_1g\_2,+F\_1(g\_1)()), the horizontal multiplication $\cdot_{\mathrm{h}}$ of morphisms is given by \[m m\] (g\_1,,m)\_(g\_2,,n)=(g\_1g\_2,+F\_1(g\_1)(),m+F\_1(g\_1)(n)). The inverse map ${\mathrm{inv}}$ is given by (g,)=(g\^[-1]{},-F\_1(g\^[-1]{})()). The identity object is $(1_G,0)$.\
The associator $$a_{(g_1,\xi),(g_2,\eta),(g_3,\gamma)}:\big((g_1,\xi)\cdot_{\mathrm{h}}(g_2,\eta)\big)\cdot_{\mathrm{h}}(g_3,\gamma)\longrightarrow
(g_1,\xi)\cdot_{\mathrm{h}}\big((g_2,\eta)\cdot_{\mathrm{h}}(g_3,\gamma)\big)$$ is given by \[associator\] a\_[(g\_1,),(g\_2,),(g\_3,)]{}=(g\_1g\_2g\_3,+F\_1(g\_1)()+F\_1(g\_1g\_2)(),F\_2(g\_1,g\_2)()). The unit $i_{(g,\xi)}:(1_G,0)\longrightarrow (g,\xi)\cdot_{\mathrm{h}}{\mathrm{inv}}(g,\xi)$ is given by \[unit\] i\_[(g,)]{}=(1\_G,0,-F\_2(g,g\^[-1]{})()). All the other natural isomorphisms are identity isomorphisms.

[[**Proof.**]{} ]{}By (\[s t\]), (\[m o\]) and (\[m m\]), it is straightforward to see that $$\begin{aligned}
s\big((g_1,\xi,m)\cdot_{\mathrm{h}}(g_2,\eta,n)\big)&=&s(g_1,\xi,m)\cdot_{\mathrm{h}}s(g_2,\eta,n),\\
t\big((g_1,\xi,m)\cdot_{\mathrm{h}}(g_2,\eta,n)\big)&=&t(g_1,\xi,m)\cdot_{\mathrm{h}}t(g_2,\eta,n).\end{aligned}$$ Thus the multiplication $\cdot_{\mathrm{h}}$ respects the source and target map. Furthermore, it is not hard to check that the horizontal and vertical multiplications commute, i.e. $$\big((g,\xi+{\mathrm{d}}m,n)\cdot_{\mathrm{h}}(g^\prime,\eta+{\mathrm{d}}p,q)\big)\cdot_{\mathrm{v}}\big((g,\xi,m)\cdot_{\mathrm{h}}(g^\prime,\eta,p)\big)=\big((g,\xi+{\mathrm{d}}m,n)\cdot_{\mathrm{v}}(g,\xi,m)\big)\cdot_{\mathrm{h}}\big((g^\prime,\eta+{\mathrm{d}}p,q)\cdot_{\mathrm{v}}(g^\prime,\eta,p)\big)$$

\[m commute with v\] @C+2em[ & @/\_2pc/\[l\]\_[(g,)]{}\_=“0” |[(g,+m)]{}\^=“1”\_=“1b” @/\^2pc/\[l\]\^[(g,+(m+n))]{}\^=“2” @[=&gt;]{} “0”;“1”\^[m]{} @[=&gt;]{} “1b”;“2”\^[n]{} & @/\_2pc/\[l\]\_[(g\^,)]{}\_=“3” |[(g\^,+p)]{}\^=“4”\_=“4b” @/\^2pc/\[l\]\^[(g\^,+(p+q))]{}\^=“5” @[=&gt;]{} “3”;“4”\^[p]{} @[=&gt;]{} “4b”;“5”\^[q]{} . ]{}

It follows from (\[eqn:F fail\]) that the associator $a_{(g_1,\xi),(g_2,\eta),(g_3,\gamma)}$ defined by (\[associator\]) is indeed a morphism from $\big((g_1,\xi)\cdot_{\mathrm{h}}(g_2,\eta)\big)\cdot_{\mathrm{h}}(g_3,\gamma)$ to $(g_1,\xi)\cdot_{\mathrm{h}}\big((g_2,\eta)\cdot_{\mathrm{h}}(g_3,\gamma)\big)$. To see that it is natural, we need to show that \[left\] a\_[(g\_1,+m),(g\_2,+n),(g\_3,+k)]{}\_(((g\_1,,m)\_(g\_2,,n))\_(g\_3,,k)) is equal to \[right\] ((g\_1,,m)\_((g\_2,,n)\_(g\_3,,k)))\_a\_[(g\_1,),(g\_2,),(g\_3,)]{}, i.e. the following diagram commutates: $$\xymatrix{
\big((g_1,\xi)\cdot_{\mathrm{h}}(g_2,\eta)\big)\cdot_{\mathrm{h}}(g_3,\gamma)\ar[d]\ar[r]^{a}&(g_1,\xi)\cdot_{\mathrm{h}}\big((g_2,\eta)\cdot_{\mathrm{h}}(g_3,\gamma)\big)\ar[d]\\
\big((g_1,\xi+{\mathrm{d}}m)\cdot_{\mathrm{h}}(g_2,\eta+{\mathrm{d}}n)\big)\cdot_{\mathrm{h}}(g_3,\gamma+{\mathrm{d}}k)\ar[r]^{a}&(g_1,\xi+{\mathrm{d}}m)\cdot_{\mathrm{h}}\big((g_2,\eta+{\mathrm{d}}n)\cdot_{\mathrm{h}}(g_3,\gamma+{\mathrm{d}}k)\big). }$$ By straightforward computations, we obtain that (\[left\]) is equal to $$\big(g_1\cdot g_2\cdot g_3,\xi+F_1(g_1)(\eta)+F_1(g_1\cdot
g_2)(\gamma),m+F_1(g_1)(n)+F_1(g_1\cdot
g_2)(k)+F_2(g_1,g_2)(\gamma+{\mathrm{d}}k)\big),$$ and (\[right\]) is equal to $$\big(g_1\cdot g_2\cdot g_3,\xi+F_1(g_1)(\eta)+F_1(g_1\cdot
g_2)(\gamma),m+F_1(g_1)(n)+F_1(g_1)\cdot
F_1(g_2)(k)+F_2(g_1,g_2)(\gamma)\big).$$ Hence (\[left\]) is equal to (\[right\]) by (\[eqn:F fail\]). This implies that $a_{(g_1,\xi),(g_2,\eta),(g_3,\gamma)}$ defined by (\[associator\]) is a natural isomorphism.

By (\[eqn:F fail\]) and the fact that $F_1(1_G)={\rm{Id}}$, the unit given by (\[unit\]) is indeed a morphism from $(1_G,0)$ to $(g,\xi)\cdot_{\mathrm{h}}{\mathrm{inv}}(g,\xi)$. To see that it is natural, we need to prove $$\big((g,\xi,m)\cdot_{\mathrm{h}}{\mathrm{inv}}(g,\xi,m)\big)\cdot_{\mathrm{h}}i_{(g,\xi)}=i_{(g,\xi+{\mathrm{d}}m)},$$ i.e. the following commutative diagram:

$$\xymatrix{
  &  (1_G,0)\ar[dr]^{i_{(g,\xi)}} \ar_{i_{(g,\xi+{\mathrm{d}}m)}}[dl]\\
(g,\xi+{\mathrm{d}}m)\cdot_{\mathrm{h}}{\mathrm{inv}}(g,\xi+{\mathrm{d}}m)
&&\ar[ll]_{\quad\qquad\qquad~(g,\xi,m)\cdot_{\mathrm{h}}{\mathrm{inv}}(g,\xi,m)}(g,\xi)\cdot_{\mathrm{h}}{\mathrm{inv}}(g,\xi)
}$$ This follows from $$F_2(g,g^{-1})({\mathrm{d}}m)=F_1(g)\cdot F_1(g^{-1})(m)-F_1(g\cdot
g^{-1})(m)=F_1(g)\cdot F_1(g^{-1})(m)-m,$$ which is a special case of (\[eqn:F fail\]).

Since $F(1_G)={\rm{Id}}$, we have $$(1_G,0)\cdot_{\mathrm{h}}(g,\xi)=(g,\xi),\quad
(g,\xi)\cdot_{\mathrm{h}}(1_G,0)= (g,\xi).$$ Hence the left unit and the right unit can also be taken as the identity isomorphism.

The counit $e_{(g,\xi)}:{\mathrm{inv}}(g,\xi)\cdot_{\mathrm{h}}(g,\xi)\longrightarrow (1_G,0)$ can be taken as the identity morphism since we have $${\mathrm{inv}}(g,\xi)\cdot_{\mathrm{h}}(g,\xi)=(g^{-1},-F_1(g^{-1})(\xi))\cdot_{\mathrm{h}}(g,\xi)=(1_G,0).$$ At last, we need to show

-   the pentagon identity for the associator,

-   the triangle identity for the left and right unit laws,

-   the first zig-zag identity,

-   the second zig-zag identity.

We only give the proof of the pentagon identity, the others can be proved similarly and we leave them to the readers. In fact, the pentagon identity is equivalent to $$\begin{aligned}
&&a_{(g_1,\xi),(g_2,\eta),(g_3,\gamma)\cdot_{\mathrm{h}}(g_4,\theta)}\cdot_{\mathrm{h}}a_{(g_1,\xi)\cdot_{\mathrm{h}}(g_2,\eta),(g_3,\gamma),(g_4,\theta)}=\\
&&\big((g_1,\xi)\cdot_{\mathrm{h}}a_{(g_2,\eta),(g_3,\gamma),(g_4,\theta)}\big)\cdot_{\mathrm{h}}a_{(g_1,\xi),(g_2,\eta)\cdot_{\mathrm{h}}(g_3,\gamma),(g_4,\theta)}\cdot_{\mathrm{h}}\big(a_{(g_1,\xi),(g_2,\eta),(g_3,\gamma)}\cdot_{\mathrm{h}}(g_4,\theta)\big)\end{aligned}$$ By straightforward computations, the left hand side is equal to $$\big(g_1\cdot g_2\cdot g_3\cdot g_4,\xi+F_1(g_1)(\eta)+F_1(g_1\cdot
g_2)(\gamma)+F_1(g_1\cdot g_2\cdot g_3)(\theta),F_2(g_1\cdot g_2,
g_3)(\theta)+F_2(g_1, g_2)(\gamma+F_1(g_3)(\theta))\big),$$ and the right hand side is equal to $$\begin{aligned}
&\big(g_1\cdot g_2\cdot g_3\cdot g_4,\xi+F_1(g_1)(\eta)+F_1(g_1\cdot
g_2)(\gamma)+F_1(g_1\cdot g_2\cdot g_3)(\theta),\\&F_2(g_1,
g_2)(\gamma)+F_2(g_1, g_2\cdot g_3)(\theta)+F_1(g_1)\circ F_2(g_2,
g_3)(\theta)\big).\end{aligned}$$ By (\[eqn:F closed\]), they are equal.

Integrating string Lie 2-algebra $\mathbb R\longrightarrow {\mathfrak g}\oplus {\mathfrak g}^*$ {#sec:int}
===============================================================================================

As an application of Theorem \[thm:main 1\], we consider the integration of the string Lie 2-algebra $\mathbb R\longrightarrow
{\mathfrak g}\oplus {\mathfrak g}^*$ given by (\[Lie 2 g g dual\]). Now we restrict to the case that ${\mathfrak g}$ is finite dimensional. Obviously, given a quadruple $(K_1,K_2,\Phi,\Theta)$ which represents a special Lie 2-group (see Theorem \[thm:special 2 G\]), by differentiation, we obtain a quadruple $({\mathfrak k}_1,{\mathfrak k}_2,\phi,\theta)$, which represents a 2-term skeletal $L_\infty$-algebra.

A special Lie 2-group which is represented by $(K_1,K_2,\Phi,\Theta)$ is an integration of a 2-term skeletal $L_\infty$-algebra which is represented by $({\mathfrak k}_1,{\mathfrak k}_2,\phi,\theta)$ if the differentiation of $(K_1,K_2,\Phi,\Theta)$ is $({\mathfrak k}_1,{\mathfrak k}_2,\phi,\theta)$.

If the differential ${\mathrm{d}}$ in a 2-term complex $V_1\stackrel{{\mathrm{d}}}{\longrightarrow}V_0$ is 0, a representation up to homotopy of Lie algebra ${\mathfrak g}$ on $V_1\stackrel{0}{\longrightarrow}V_0$ consists of two strict representations $\mu_1$ and $\mu_0$, and a liner map $\nu:{\mathfrak g}\wedge{\mathfrak g}\longrightarrow{\mathrm{Hom}}(V_0,V_1)$ satisfying equation (\[eqn:d k\]). This equation implies that $\nu$ is a Lie algebra 2-cocycle representing an element in $H^2({\mathfrak g},{\mathrm{Hom}}(V_0,V_1))$, with the representation $[\mu(\cdot),\cdot]$ of ${\mathfrak g}$ on ${\mathrm{Hom}}(V_0,V_1)$ defined by $$[\mu(\cdot),\cdot](X)(A)\triangleq[\mu(X), A]=\mu_1(X)\circ A-A\circ
\mu_0(X),\quad \forall ~X\in {\mathfrak g},~A\in{\mathrm{Hom}}(V_0,V_1).$$

Define $\widetilde{\nu}:\wedge^3({\mathfrak g}\oplus V_0)\longrightarrow V_1$ by $$\widetilde{\nu}(X_1+\xi_1,X_2+\xi_2,X_3+\xi_3)=\nu(X_1,X_2)(\xi_3)+c.p.$$ then $\nu$ is a 2-cocycle if and only if $\widetilde{\nu}$ is a 3-cocycle where the representation $\widetilde{\mu}$ of ${\mathfrak g}\oplus
V_0$ on $V_1$ is given by $$\widetilde{\mu}(X+\xi)(m)=\mu(X)(m).$$

[[**Proof.**]{} ]{}By direct computations, for any $X_i+\xi_i\in{\mathfrak g}\oplus
V_0,~i=1,2,3,4,$ we have $$d\widetilde{\nu}(X_1+\xi_1,X_2+\xi_2,X_3+\xi_3,X_4+\xi_4)=d\nu(X_1,X_2,X_3)(\xi_4)+c.p.,$$ which implies the conclusion.

The Lie algebra homomorphism $\mu$ from ${\mathfrak g}$ to ${\mathrm{End}}(V_0)\oplus{\mathrm{End}}(V_1)$ integrates to a Lie group homomorphism $F_1$ from the simply connected Lie group $G$ of ${\mathfrak g}$ to $GL(V_0)\oplus GL(V_1)$ with $$\mu(X)=\frac{d}{dt}\Big|_{t=0}F_1(\exp tX),\quad\forall~X\in{\mathfrak g}.$$ Consequently, ${\mathrm{Hom}}(V_0,V_1)$ is a $G$-module with $G$ action $$g\cdot A=F_1(g)\circ A \circ F_1(g)^{-1},\quad\forall ~g\in G,~A\in
{\mathrm{Hom}}(V_0,V_1).$$ The Lie algebra 2-cocycle $\nu:{\mathfrak g}\wedge{\mathfrak g}\longrightarrow{\mathrm{Hom}}(V_0,V_1)$ can integrate to a smooth Lie group 2-cocycle $\overline{F_2}:G\times G\longrightarrow
{\mathrm{Hom}}(V_0,V_1)$, satisfying $$\label{eqn:F2 closed}
F_1(g_1)\circ(\overline{F_2})(g_2,g_3)\circ
F_1(g_1)^{-1}-(\overline{F_2})(g_1\cdot
g_2,g_3)+(\overline{F_2})(g_1, g_2\cdot
g_3)-(\overline{F_2})(g_1,g_2)=0,$$ and $\overline{F_2}(1_G,1_G)=0.$ Let us explain how.

The classical theory of cohomology of discrete groups tells us that the equivalence classes of extensions of $G$ by a $G$ module $M$, one to one correspond to the elements in $H^2(G, M)$. In our case, the same theory tells us that $H^2_{sm}(G, {\mathrm{Hom}}(V_0, V_0))$ classifies the equivalence classes of splitting extensions of $G$ by the $G$-module ${\mathrm{Hom}}(V_0, V_1)$, which is a splitting short exact sequence of Lie groups, with ${\mathrm{Hom}}(V_0, V_1)$ endowed with an abelian group structure, $$\label{eq:extension} {\mathrm{Hom}}(V_0, V_1) \to \hat{G} \to G.$$ In a general extension $\hat{G}$ is a principal bundle over $G$, thus it usually does not permit a smooth lift $G
\xrightarrow{\sigma} \hat{G}$. It permits such a lift if and only if the sequence splits. However in our case, since the abelian group ${\mathrm{Hom}}(V_0, V_1) $ is a vector space, we have $H^1(X,{\mathrm{Hom}}(V_0, V_1)
)=0$ for any manifold $X$. The proof makes use of a partition of unity and similar to the proof showing that $H^1(X,
\underline{{\mathbb R}})=0$ for the sheaf cohomology. Hence all $ {\mathrm{Hom}}(V_0,
V_1)$ principal bundles are trivial. Therefore always splits. On the other hand it is well-known that when $G$ is simply connected, there is a one-to-one correspondence between extensions of $G$ [@brahic Theorem 4.15] and extensions of its Lie algebra ${\mathfrak g}$, which in turn are classified by the Lie algebra cohomology $H^2({\mathfrak g}, {\mathrm{Hom}}(V_0, V_1))$. Hence in our case the differentiation map $$H^2_{sm} (G, {\mathrm{Hom}}(V_0, V_1)) \to H^2({\mathfrak g}, {\mathrm{Hom}}(V_0, V_1))$$ is an isomorphism. Hence $\nu$ always integrates to a smooth Lie group 2-cocycle unique up to exact 2-cocycles. Then $\overline{F_2}(1_G,1_G)=0$ can be arranged too, because we can always modify the section $\sigma: G \to \hat{G}$ to satisfy $\sigma(1_G)=1_{\hat{G}}$ and the modification of sections results in an exact term. Then combined with (\[eqn:F2 closed\]), it is not hard to see that $$\label{temp1}
\overline{F_2}(1_G,g)=\overline{F_2}(g,1_G)=0,\quad\forall~g\in
G.$$ Thus $\overline{F_2}$ is a normalized 2-cocycle.

\[pro:int mu nu \] For any 2-term representation up to homotopy $(\mu,\nu)$ of a Lie algebra ${\mathfrak g}$ on the complex $V_1\stackrel{0}{\longrightarrow}V_0$, there is an associated representation up to homotopy $(F_1,F_2)$ of the Lie group $G$ on the complex $V_1\stackrel{0}{\longrightarrow}V_0$, where $F_1$ is the integration of $\mu$ and $F_2:G\times G\longrightarrow
{\mathrm{End}}(V_0,V_1)$ is defined by $$\label{eqn:F2}
F_2(g_1,g_2)=\overline{F_2}(g_1,g_2)\circ F_1(g_1\cdot g_2).$$

[[**Proof.**]{} ]{}Obviously, (\[eqn:F fail\]) is satisfied. To see (\[eqn:F closed\]) is also satisfied, combine (\[eqn:F2\]) with (\[eqn:F2 closed\]). By the fact that $F_1$ is a homomorphism, we obtain $$\begin{aligned}
&F_1(g_1)\circ F_2(g_2,g_3)\circ F_1(g_2\cdot g_3)^{-1}\circ
F_1(g_1)^{-1}-F_2(g_1\cdot g_2,g_3)\circ F_1(g_1\cdot g_2\cdot
g_3)^{-1}\\&+F_2(g_1, g_2\cdot g_3)\circ F_1(g_1\cdot g_2\cdot
g_3)^{-1}-F_2(g_1,g_2)\circ F_1(g_1\cdot g_2)^{-1}=0.\end{aligned}$$ Composed with $F_1(g_1\cdot g_2\cdot g_3)$ on the right hand side, we obtain (\[eqn:F closed\]).

By Proposition \[pro:int mu nu \] and Theorem \[thm:main 1\], we have

\[thm:main 2\] Let $G$ be the simply connected Lie group integrating ${\mathfrak g}$, then the string Lie 2-algebra $\mathbb
R\stackrel{0}{\longrightarrow}{\mathfrak g}\oplus{\mathfrak g}^*$ given by (\[Lie 2 g g dual\]) integrates to the following Lie 2-group, $$\label{2 group}\begin{array}{c}
G\times {\mathfrak g}^*\times \mathbb R\\
\vcenter{\rlap{s }}~\Big\downarrow\Big\downarrow\vcenter{\rlap{t }}\\
G\times {\mathfrak g}^*,
 \end{array}$$ in which the source and target are given by s(g,,m)=t(g,,m)=(g,)the vertical multiplication $\cdot_{\mathrm{v}}$ is given by $$(h,\eta,n)\cdot_{\mathrm{v}}(g,\xi,m) =(g,\xi,m+n),\quad \mbox{where}~
h=g,\eta=\xi,$$ the horizontal multiplication $\cdot_{\mathrm{h}}$ of objects is given by $$(g_1,\xi)\cdot_{\mathrm{h}}(g_2,\eta)=(g_1\cdot
g_2,\xi+{\mathrm{Ad}}^*_{g_1}\eta),$$ the horizontal multiplication $\cdot_{\mathrm{h}}$ of morphisms is given by $$(g_1,\xi,m)\cdot_{\mathrm{h}}(g_2,\eta,n)=(g_1\cdot
g_2,\xi+{\mathrm{Ad}}^*_{g_1}\eta,m+n),$$ the inverse map ${\mathrm{inv}}$ is given by $${\mathrm{inv}}(g,\xi)=(g^{-1},-{\mathrm{Ad}}^*_{g^{-1}}\xi),$$ The identity object is $(1_G,0)$,\
the associator $$a_{(g_1,\xi),(g_2,\eta),(g_3,\gamma)}:\big((g_1,\xi)\cdot_{\mathrm{h}}(g_2,\eta)\big)\cdot_{\mathrm{h}}(g_3,\gamma)\longrightarrow
(g_1,\xi)\cdot_{\mathrm{h}}\big((g_2,\eta)\cdot_{\mathrm{h}}(g_3,\gamma)\big)$$ is given by \[associator\] a\_[(g\_1,),(g\_2,),(g\_3,)]{}=(g\_1g\_2g\_3,+\^\*\_[g\_1]{}+\^\*\_[g\_1g\_2]{},F\_2(g\_1,g\_2)()), All the other structures are identity isomorphisms.

[[**Proof.**]{} ]{}Since $F_1$ is a usual associative action, we may modify the unit (\[unit\]) given in Theorem \[thm:main 1\] to be the identity natural transformation. It turns out that (\[2 group\]) is a special Lie 2-group and is represented by $(G\ltimes
{\mathfrak g}^*,\mathbb R,{\rm{Id}},\widetilde{F_2})$, where $G\ltimes {\mathfrak g}^*$ is the semidirect product with the coadjoint action of $G$ on ${\mathfrak g}^*$, ${\rm{Id}}$ is the constant map $G\ltimes {\mathfrak g}^* \to {\mathrm{Aut}}({\mathbb R})$ by mapping everything to ${\rm{Id}}\in {\mathrm{Aut}}({\mathbb R})$, and $\widetilde{F_2}$ is given by $$\label{eq:tf2}
\widetilde{F_2}((g_1,\xi_1),(g_2,\xi_2),(g_3,\xi_3))=F_2(g_1,
g_2)(\xi_3)=\overline{F_2}(g_1, g_2)\circ F_1(g_1\cdot g_2)(\xi_3).$$ Since $\overline{F_2}$ is normalized, $\widetilde{F_2}$ is also normalized. The string Lie 2-algebra $\mathbb
R\stackrel{0}{\longrightarrow}{\mathfrak g}\oplus{\mathfrak g}^*$ is skeletal and is represented by $({\mathfrak g}\oplus{\mathfrak g}^*,\mathbb
R,0,\widetilde{\nu})$, where $\widetilde{\nu}$ is given by (\[eqn:nu3\]). Thus to show that our Lie 2-group is an integration of the string Lie algebra, we only need to show that the differential of the Lie group 3-cocycle $\widetilde{F_2}$ is the Lie algebra 3-cocycle $\widetilde{\nu}$. By direct computations [@brylinski Lemma 7.3.9], we have $$\begin{aligned}
&&\frac{\partial^3}{\partial_{t_1}\partial_{t_2}\partial_{t_3}}\Big|_{t_i=0}\sum_{\sigma\in
S_3}\epsilon(\sigma)\widetilde{F_2}\big((e^
{t_{\sigma(1)}X_{\sigma(1)}},t_{\sigma(1)}\xi_{\sigma(1)}),(e^
{t_{\sigma(2)}X_{\sigma(2)}},t_{\sigma(2)}\xi_{\sigma(2)}),(e^
{t_{\sigma(3)}X_{\sigma(3)}},t_{\sigma(3)}\xi_{\sigma(3)})\big)\\
&=&\frac{\partial^3}{\partial_{t_1}\partial_{t_2}\partial_{t_3}}\Big|_{t_i=0}\Big(\overline{F_2}(e^{
t_1X_1},e^ {t_2X_2})\circ F_1(e^{ t_1X_1}\cdot e^
{t_2X_2})(t_3\xi_3)\Big)+c.p.\\
&=&\frac{\partial^2}{\partial_{t_1}\partial_{t_2}}\Big|_{t_i=0}\Big(\overline{F_2}(e^{
t_1X_1},e^ {t_2X_2})\circ F_1(e^{ t_1X_1}\cdot e^ {t_2X_2})(\xi_3)\Big)+c.p.\\
&=&\frac{\partial}{\partial_{t_1}}\Big|_{t_1=0}\Big(\frac{\partial}{\partial_{t_2}}\big|_{t_2=0}\overline{F_2}(e^{
t_1X_1},e^ {t_2X_2})\circ F_1(e^{
t_1X_1})(\xi_3)+\overline{F_2}(e^{t_1X_1},1_G)\circ\frac{\partial}{\partial_{t_2}}\big|_{t_2=0}F_1(e^{
t_1X_1}\cdot e^{ t_2X_2})\Big)\\&&+c.p.\\
&=&\frac{\partial}{\partial_{t_1}}\frac{\partial}{\partial_{t_2}}\Big|_{t_i=0}\overline{F_2}(e^{
t_1X_1},e^
{t_2X_2})(\xi_3)+\frac{\partial}{\partial_{t_2}}\big|_{t_2=0}\overline{F_2}(1_G,e^
{t_2X_2})\circ \frac{\partial}{\partial_{t_1}}\Big|_{t_1=0}F_1(e^{
t_1X_1})(\xi_3)\\
&&+c.p.\quad\mbox{by (\ref{temp1})}\\
&=&\nu(X_1,X_2)(\xi_3)+c.p.\\
&=&\widetilde{\nu}(X_1+\xi_1,X_2+\xi_2,X_3+\xi_3)  \quad\mbox{by
(\ref{eqn:nu3})},\end{aligned}$$ which completes the proof.

If Lie algebra ${\mathfrak g}$ is semisimple, the Lie group 3-cocycle $\widetilde{F_2}$ is not exact, i.e. $[\widetilde{F_2}]\neq0$ in $H^3_{sm}(G\ltimes {\mathfrak g}^*,\mathbb R)$.

[[**Proof.**]{} ]{}By Theorem \[thm:main 2\], the differentiation of the Lie group 3-cocycle $\widetilde{F_2}$ is the Lie algebra 3-cocycle $\widetilde{\nu}$. We only need to show that when ${\mathfrak g}$ is semisimple, the Lie algebra 3-cocycle $\widetilde{\nu}$ is not exact. This fact is proved in Proposition \[pro:nondegenerate\].

Since $G\ltimes {\mathfrak g}^*$ is a fibration over $G$, the spectral sequence with $E_2^{p,q}=H^p_{sm}(G, H^q_{sm}( {\mathfrak g}^*,
  {\mathbb R}))$ calculates the group cohomology $H^3_{sm}(G\ltimes
  {\mathfrak g}^*,\mathbb R)$. Since ${\mathfrak g}^*$ is an abelian group, $H^q_{sm}( {\mathfrak g}^*,
  {\mathbb R})=\wedge^q{\mathfrak g}^*$. Thus when $G$ is compact, $ H^p_{sm}(G, H^q_{sm}( {\mathfrak g}^*,
  {\mathbb R}))  = (\wedge^q{\mathfrak g}^*)^G$ if $p=0$ and 0 otherwise, where $(\wedge^q{\mathfrak g}^*)^G$ denotes the set of invariant elements of $\wedge^q {\mathfrak g}^*$ under the coadjoint action of $G$. Thus when $G$ is compact, $H^3_{sm}(G\ltimes {\mathfrak g}^*,\mathbb R)=(\wedge^3
g^*)^G \neq 0$ because the Cartan 3-form is an element of $(\wedge^3 g^*)^G$.

Notice that our 2-cocycle $\overline{F_2}$ is unique only up to exact terms. Hence by Theorem \[thm:special 2 G\], to verify that our construction is unique up to isomorphism, we need the following lemma,

If $\overline{F_2}= {\mathrm{d}}\alpha$ is exact, then $\widetilde{F_2}={\mathrm{d}}\beta$ is also exact with $$\beta((g_1, \xi_1), (g_2, \xi_2))=\alpha(g_1)F_1(g_1)(\xi_2).$$

It is a direct calculation. Since $\overline{F_2}= {\mathrm{d}}\alpha$, we have $$\overline{F_2}(g_1, g_2) = F_1 (g_1) \alpha(g_2) F_1(g_1)^{-1} - \alpha(g_1
g_2) + \alpha(g_1).$$ From the definition of $F_2$, we know that $$F_2(g_1, g_2)= F_1(g_1) \alpha(g_2) F_1(g_1)^{-1} F_1(g_1 g_2) -
\alpha(g_1 g_2) F_1(g_1 g_2) + \alpha(g_1) F_1(g_1g_2).$$ By , we have $$\begin{split}
\widetilde{F_2}((g_1, \xi_1), (g_2, \xi_2),(g_3, \xi_3))= &F_1(g_1) \alpha(g_2)
F_1(g_1)^{-1} F_1(g_1g_2) (\xi_3) - \alpha(g_1g_2) F_1(g_1g_2) (\xi_3)
+ \alpha(g_1) F_1(g_1g_2) (\xi_3) \\
= &{\mathrm{d}}\beta((g_1, \xi_1), (g_2, \xi_2),(g_3, \xi_3)),
\end{split}$$ since $F_1$ is a group homomorphism.

Our Lie 2-group as a stacky group has the underlying differential stack $G \times {\mathfrak g}^* \times B{\mathbb R}$. Thus it is 0,1,2-connected (i.e. it has $\pi_0=\pi_1=\pi_2=0$) since $\pi_2(B{\mathbb R})=\pi_1({\mathbb R})=0$ and $\pi_1(B{\mathbb R})=\pi_0({\mathbb R})=0$. Thus it is the unique 0,1,2-connected stacky Lie group integrating the string Lie 2-algebra ${\mathbb R}\xrightarrow{0} {\mathfrak g}\oplus {\mathfrak g}^*$ in the sense of [@z:lie2].

[^1]: Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen

[^2]: private conversation with Jiang-Hua Lu

[^3]: private conversation with John Baez and Urs Schreiber

[^4]: It is slightly different from the notion given in [@clomni], where the bracket is not skewsymmetric.

[^5]: Please see the introduction for the reason why we call it a semistrict Lie 2-group.

[^6]: The notion of strict Lie 2-groups is the same as [@baez:2gp].

[^7]: up to isomorphisms of groups of course
