nLab
free Lie algebra

The free Lie algebra functor is the left adjoint functor FreeLieAlgFreeLieAlg to the forgetful functor LieAlgSetLieAlg\to Set.

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

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

Revised on October 17, 2010 07:32:04 by Toby Bartels (98.19.58.126)