nLab free Lie algebra

Idea

The free Lie algebra functor is the left adjoint functor $FreeLieAlg$ to the forgetful functor $LieAlg\to Set$. The proof that a left adjoint exists relies on a concrete construction. Let $X$ be a set, then 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 $FreeLieAlg(X) = Lib_k(X)/I$.

The free Lie algebra on the set $X$ is the result $FreeLieAlg(X)$ of evaluating the free Lie algebra functor on object $X$.

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

See also:

• C. Reutenauer, Applications of a noncommutative jacobian matrix, Journal of Pure and Applied Algebra 77, n. 2, 1992, p. 169-181, doi

• Mikhail Kapranov, Free Lie algebroids and space of paths, math,DG/0702584

• sbseminar blog: Tannakian construction of the fundamental group and Kapranov’s fundamental Lie algebra

• Nantel Bergeron, Muriel Livernet, A combinatorial basis for the free Lie algebra of the labelled rooted trees, Journal of Lie Theory 20 (2010) 3–15, pdf

• Leila Schneps, On the Poisson bracket on the free Lie algebra in two generators, pdf

• A. Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series, pdf

• F. Chapoton, Free pre-Lie algebras are free as Lie algebras, math.RA/0704.2153, Bulletin Canadien de Mathe’matiques

category: algebra