Lie algebra, free

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.

References

[1]	A.I. Shirshov, "Subalgebras of free Lie algebras" Mat. Sb. , 33 : 2 (1953) pp. 441–452 (In Russian)
[2]	E. Witt, "Die Unterringe der freien Lieschen Ringe" Math. Z. , 64 (1956) pp. 195–216
[3]	G.P. Kukin, "Intersection of subalgebras of a free Lie algebra" Algebra and Logic , 16 (1977) pp. 387–394 Algebra i Logika , 16 (1977) pp. 577–587
[4]	W. Magnus, "Ueber Beziehungen zwischen höheren Kommutatoren" J. Reine Angew. Math. , 177 (1937) pp. 105–115
[5]	Yu.A. Bakhturin, "Identical relations in Lie algebras" , VNU , Utrecht (1987) (Translated from Russian)

Comments

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 $x \in X$.

Free Lie algebras are the best context for the formulation of the Campbell–Baker–Hausdorff formula in its most general form.

References

[a1]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[a2]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)