nLab free Lie algebra

The free Lie algebra functor is the left adjoint functor FreeLieAlgFreeLieAlg to the forgetful functor LieAlgSetLieAlg\to Set. The proof that a left adjoint exists relies on a concrete construction. Let XX be a set, then define recursively for n1n\geq 1 sets X n= p=1 n1X p×X npX_n = \coprod_{p=1}^{n-1} X_p\times X_{n-p} with the basis of recursion X 1=XX_1 = X. If xX px\in X_p and yX qy\in X_q then denote x.yx.y the element (x,y)X p+q(x,y)\in X_{p+q}; this defines a binary operation on n=0 X n\coprod_{n=0}^\infty X_n, which is therefore the free magma on the set XX. Let kk be the ground ring (commutative and unital). As a kk-module define Lib k(X)=k[ n=0 X n]Lib_k(X) = k[\coprod_{n=0}^\infty X_n], the free kk-module with basis n=0 X n\coprod_{n=0}^\infty X_n. It is a nonassociative kk-algebra with product ( ia ix i).( jb jy j)= i,ja ib j(x i.y j)(\sum_i a_i x_i).(\sum_j b_j y_j) = \sum_{i,j} a_i b_j (x_i.y_j) where both sums are finite and a i,b jka_i, b_j\in k, x i,y iX ix_i,y_i\in X_i. Define a two-sided ideal II in this nonassociative kk-algebra, generated by all elements of Lib k(X)Lib_k(X) of the form a.aa.a and all elements of the form a.(b.c)+b.(c.a)+c.(a.b)a.(b.c)+b.(c.a)+c.(a.b), where a,b,cLib k(X)a,b,c\in Lib_k(X). Then FreeLieAlg(X)=Lib k(X)/IFreeLieAlg(X) = Lib_k(X)/I.

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.

category: algebra

Last revised on March 29, 2023 at 16:36:00. See the history of this page for a list of all contributions to it.