The free Lie algebra functor is the left adjoint functor to the forgetful functor . The proof that a left adjoint exists relies on a concrete construction. Let be a set, then define recursively for sets with the basis of recursion . If and then denote the element ; this defines a binary operation on , which is therefore the free magma on the set . Let be the ground ring (commutative and unital). As a -module define , the free -module with basis . It is a nonassociative -algebra with product where both sums are finite and , . Define a two-sided ideal in this nonassociative -algebra, generated by all elements of of the form and all elements of the form , where . Then .
The free Lie algebra on the set is the result of evaluating the free Lie algebra functor on object .
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.
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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
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