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 H × V → V usually denoted by juxtaposition ( that is, the image of (h,v) is written hv ). The vector space V is called an H-module. Properties 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, hv = ε(h)v, where ε is the counit of H. The subset of all invariant elements of V forms a submodule of V. Categories of representations as a motivation for Hopf algebras 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 Einstein's summation convention. This is satisfied if there is a Δ such that Δ(ab) = Δ(a)Δ(b) for all a, b in H. For the category of H-modules to be a strict monoidal category with respect to ⊗, and must be equivalent and there must be unit object εH, called the trivial module, such that εH ⊗ V, V and V ⊗ εH are equivalent. 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 =