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.
N. Bourbaki, Lie groups and Lie algebras, Chap. II: Free Lie Algebras
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wikipedia: free 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
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F. Chapoton, Free pre-Lie algebras are free as Lie algebras, math.RA/0704.2153, Bulletin Canadien de Mathe’matiques