That is, a representation of a Hopf algebra H over a field K is a K-vector space V with an action H × V → V usually denoted by juxtaposition (that is, the image of (h, v) is written hv).
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, hv = ε(h)v, where ε 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 V1 ⊗ V2 of two H-modules V1 and V2 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 Δ : H → H ⊗ H such that for any v in V1 ⊗ V2 and any h in H, and for any v in V1 ⊗ V2 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 means that for any v in and for h in H, This will hold for any three H-modules if Δ satisfies The trivial module must be one-dimensional, and so an algebra homomorphism ε : H → F may be defined such that hv = ε(h)v for all v in εH.
The trivial module may be identified with F, with 1 being the element such that 1 ⊗ v = v = v ⊗ 1 for all v. It follows that for any v in any H-module V, any c in εH and any h in H, The existence of an algebra homomorphism ε satisfying is a sufficient condition for the existence of the trivial module.
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 S : H → H such that for any h in H, x in V and y in V*, where
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.