action of comultiplication in this algebra. See tensor algebra for a detailed exposition of the inter-relation between the shuffle product and comultiplication. Hall sets An explicit basis of the free Lie algebra can be given in terms of a Hall set, which is a particular kind of subset inside the free magma on X. Elements of the free magma are binary trees, with their leaves labelled by elements of X. Hall sets were introduced by based on work of Philip Hall on groups. Subsequently, Wilhelm Magnus showed that they arise as the graded Lie algebra associated with the filtration on a free group given by the lower central series. This correspondence was motivated by commutator identities in group theory due to Philip Hall and Witt. Lyndon basis The Lyndon words are a special case of the Hall words, and so in particular there is a basis of the free Lie algebra corresponding to Lyndon words. This is called the Lyndon basis, named after Roger Lyndon. (This is also called the Chen–Fox–Lyndon basis or the Lyndon–Shirshov basis, and is essentially the same as the Shirshov basis.) There is a bijection γ from the Lyndon words in an ordered alphabet to a basis of the free Lie algebra on this alphabet defined as follows: If a word w has length 1 then (considered as a generator of the free Lie algebra). If w has length at least 2, then write for Lyndon words u, v with v as long as possible (the "standard factorization"). Then . Shirshov–Witt theorem and showed that any Lie subalgebra of a free Lie algebra is itself a free Lie algebra. Applications Serre's theorem on a semisimple Lie algebra uses a free Lie algebra to construct a semisimple algebra out of generators and relations. The Milnor invariants of a link group are related to the free Lie algebra on the components of the link, as discussed in that article. See also Lie operad for the use of a free Lie algebra in the construction of the operad. See also Free object Free algebra Free group References Properties of Lie algebras Free algebraic structures