Lie algebra, free

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over a ring $R$

A Lie algebra $L = L(X)$ over $R$ in which one can distinguish a free generating set $X$, a mapping from which into an arbitrary algebra $G$ over $R$ can be be extended to a homomorphism from $L$ into $G$. The cardinality of $X$ completely determines $L$ and is called its rank. A free Lie algebra is a free $R$-module (for bases of it see Basic commutator). A subalgebra $M$ of a free Lie algebra over a field is itself a free Lie algebra (Shirshov's theorem, [1]). If $R = \mathbf{Z}$, then this is true only under the condition that $L/M$ is a free Abelian group [2]. The finitely-generated subalgebras of a free Lie algebra over a field form a sublattice of the lattice of all subalgebras [3]. W. Magnus [4] established canonical connections between free Lie algebras and both free groups and free associative algebras.




To construct $L(X)$ one can start from the free associative algebra $A(X)$ generated by $X$, which is made into a Lie algebra $A(X)$ by taking as Lie product $[a,b] = a \cdot b - b \cdot a$. Then $L(X)$ is the Lie subalgebra of $A(X)$ generated by $X$, and $A(X)$ is the universal enveloping algebra of $A(X)$.

In case $R$ is a field of characteristic 0, more precise results on which elements of $A(X)$ belong to $L(X)$ are given by the Specht–Wever theorem and the Friedrichs theorem, respectively. The first one says that a homogeneous element $a$ of degree $m$ belongs to $L(X)$ if and only if $\sigma(a) = ma$, where $\sigma$ is the linear mapping defined by $$ \sigma(x_1 \ldots x_m) = [\ldots[x_1,x_2],\ldots, x_m] $$ for $x_i \in X$. Friedrichs' theorem says for the case of finite $X$ that $a \in A(X)$ belongs to $L(X)$ if and only if $\delta(a) = 1 \otimes a + a \otimes 1$, where $\delta : A(X) \rightarrow A(X) \otimes A(X)$ is the homomorphism defined by $\delta : x \mapsto 1 \otimes x + x \otimes 1$ for $x \in X$.



How to Cite This Entry:
Lie algebra, free. Encyclopedia of Mathematics. URL:,_free&oldid=38623
This article was adapted from an original article by Yu.A. Bakhturin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article