nLab free Lie algebra

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Context

Lie theory

Idea
Construction
Properties
Literature

Idea

The free Lie algebra functor is the left adjoint functor $FreeLieAlg$ to the forgetful functor $LieAlg\to Set$ that send Lie algebras to their underlying sets.

Construction

Here is a concrete concrete construction. For $X$ a set, define recursively for $n\geq 1$ sets $X_n = \coprod_{p=1}^{n-1} X_p\times X_{n-p}$ with the basis of recursion $X_1 = X$ . If $x\in X_p$ and $y\in X_q$ then denote $x.y$ the element $(x,y)\in X_{p+q}$ ; this defines a binary operation on $\coprod_{n=0}^\infty X_n$ , which is therefore the free magma on the set $X$ .

Let $k$ be the ground ring (commutative and unital). As a $k$ -module define $Lib_k(X) = k[\coprod_{n=0}^\infty X_n]$ , the free $k$ -module with basis $\coprod_{n=0}^\infty X_n$ . It is a nonassociative $k$ -algebra with product $(\sum_i a_i x_i).(\sum_j b_j y_j) = \sum_{i,j} a_i b_j (x_i.y_j)$ where both sums are finite and $a_i, b_j\in k$ , $x_i,y_i\in X_i$ . Define a two-sided ideal $I$ in this nonassociative $k$ -algebra, generated by all elements of $Lib_k(X)$ of the form $a.a$ and all elements of the form $a.(b.c)+b.(c.a)+c.(a.b)$ , where $a,b,c\in Lib_k(X)$ .

Then $FreeLieAlg(X) \simeq Lib_k(X)/I$ .

Properties

The subject of free Lie algebras is combinatorially rich with lots of open problems.

By a 1953 theorem of A. I. Širšov (Shirshov) every Lie subalgebra of a free Lie subalgebra is free (an analogue of the Nielsen-Schreier theorem in combinatorial group theory).

The study of bases of a free Lie algebra considered as a vector space is very nontrivial; special attention has been paid to so-called Hall bases.

Literature

Jean-Pierre Serre: Free Lie Algebra, Chapter IV of: Lie Algebras and Lie Groups – 1964 Lectures given at Harvard University, Lecture Notes in Mathematics 1500, Springer (1992) [doi:10.1007/978-3-540-70634-2]
Nicolas Bourbaki: Free Lie Algebras, Chap. II of: Lie groups and Lie algebras – Chapters 1-3, Springer (1975, 1989) [ISBN:9783540642428]
Christophe Reutenauer, Free Lie algebras, Oxford University Press (1993) [ISBN:9780198536796]
Christophe Reutenauer, Free Lie algebras, Handbook of Algebra 3 (2003) 887-903 [doi:10.1016/S1570-7954(03)80075]
Shlomo Sternberg: Free Lie Algebras, Section 1.11 of: Lie Algebras (2004) [pdf, pdf]
Wikipedia: Free Lie algebra