nLab
free Lie algebra

The free Lie algebra functor is the left adjoint functor $\mathrm{FreeLieAlg}$ to the forgetful functor $\mathrm{LieAlg}\to \mathrm{Set}$.

The free Lie algebra on the set $X$ is the result $\mathrm{FreeLieAlg}(X)$ 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.

• N. Bourbaki, Lie groups and Lie algebras, Chap. II: Free Lie Algebras

• Christophe Reutenauer, Free Lie algebras, Oxford Univeristy Press 1993

• C. Reutanauer, Free Lie algebras, Handbook of Algebra, vol. 3, 2003, 887-903, doi

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

• wikipedia: free Lie algebra

• 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