In abstract algebra, a representation of a Hopf algebra is a representation of its underlying associative algebra. That is, a representation of a Hopf algebra H over a field K is a K-vector space V with an action usually denoted by juxtaposition (that is, the image of is written hv). The vector space V is called an H-module.
The module structure of a representation of a Hopf algebra H is simply its structure as a module for the underlying associative algebra. The main use of considering the additional structure of a Hopf algebra is when considering all H-modules as a category. The additional structure is also used to define invariant elements of an H-module V. An element v in V is invariant under H if for all h in H, , where õ is the counit of H. The subset of all invariant elements of V forms a submodule of V.
For an associative algebra H, the tensor product of two H-modules V<sub>1</sub> and V<sub>2</sub> is a vector space, but not necessarily an H-module. For the tensor product to be a functorial product operation on H-modules, there must be a linear binary operation such that for any v in and any h in H,
and for any v in V<sub>1</sub> â V<sub>2</sub> and a and b in H,
using sumless Sweedler's notation, which is somewhat like an index free form of the Einstein summation convention. This is satisfied if there is a ÃÂ such that for all a, b in H.
For the category of H-modules to be a monoidal category with respect to âÂÂ, and must be equivalent and there must be unit object õ<sub>H</sub>, called the trivial module, such that , V and are equivalent.
This means that for any v in
and for h in H,
This will hold for any three H-modules if ÃÂ satisfies
The trivial module must be one-dimensional, and so an algebra homomorphism may be defined such that for all v in õ<sub>H</sub>. The trivial module may be identified with F, with 1 being the element such that for all v. It follows that for any v in any H-module V, any c in õ<sub>H</sub> and any h in H,
The existence of an algebra homomorphism õ satisfying
is a sufficient condition for the existence of the trivial module.
It follows that in order for the category of H-modules to be a monoidal category with respect to the tensor product, it is sufficient for H to have maps àand õ satisfying these conditions. This is the motivation for the definition of a bialgebra, where àis called the comultiplication and õ is called the counit.
In order for each H-module V to have a dual representation V such that the underlying vector spaces are dual and the operation * is functorial over the monoidal category of H-modules, there must be a linear map such that for any h in H, x in V and y in V*,
where is the usual pairing of dual vector spaces. If the map induced by the pairing is to be an H-homomorphism, then for any h in H, x in V and y in V*,
which is satisfied if
for all h in H.
If there is such a map S, then it is called an antipode, and H is a Hopf algebra. The desire for a monoidal category of modules with functorial tensor products and dual representations is therefore one motivation for the concept of a Hopf algebra.
A Hopf algebra also has representations which carry additional structure, namely they are algebras.
Let H be a Hopf algebra. If A is an algebra with the product operation , and is a representation of H on A, then àis said to be a representation of H on an algebra if ü is H-equivariant. As special cases, Lie algebras, Lie superalgebras and groups can also have representations on an algebra.