In
mathematics, a free Lie algebra over a
fieldK is a
Lie algebra generated by a
setX, without any imposed relations other than the defining relations of alternating K-bilinearity and the
Jacobi identity.
Definition
The definition of the free Lie algebra generated by a set X is as follows:
Let X be a set and a
morphism of sets (
function) from X into a Lie algebra L. The Lie algebra L is called free on X if is the
universal morphism; that is, if for any Lie algebra A with a morphism of sets , there is a unique Lie algebra morphism such that .
Given a set X, one can show that there exists a unique free Lie algebra generated by X.
The free Lie algebra on a set X is naturally
graded. The 1-graded component of the free Lie algebra is just the
free vector space on that set.
One can alternatively define a free Lie algebra on a
vector spaceV as left adjoint to the forgetful functor from Lie algebras over a field K to vector spaces over the field K – forgetting the Lie algebra structure, but remembering the vector space structure.
Universal enveloping algebra
The
universal enveloping algebra of a free Lie algebra on a set X is the
free associative algebra generated by X. By the
Poincaré–Birkhoff–Witt theorem it is the "same size" as the symmetric algebra of the free Lie algebra (meaning that if both sides are graded by giving elements of X degree 1 then they are
isomorphic as graded vector spaces). This can be used to describe the dimension of the piece of the free Lie algebra of any given degree.
The graded dual of the universal enveloping algebra of a free Lie algebra on a finite set is the
shuffle algebra. This essentially follows because universal enveloping algebras have the structure of a
Hopf algebra, and the
shuffle product describes the action of comultiplication in this algebra. See
tensor algebra for a detailed exposition of the inter-relation between the shuffle product and comultiplication.
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
Marshall Hall (
1950) 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"[1]). Then .