The free Lie algebra functor is the left adjoint functor $FreeLieAlg$ to the forgetful functor $LieAlg\to Set$.
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.
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wikipedia: free Lie algebra
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sbseminar blog: Tannakian construction of the fundamental group and Kapranov’s fundamental Lie algebra
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