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. 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 Δ and ε satisfying these conditions. This is the motivation for the definition of a bialgebra, where Δ is called the comultiplication and ε 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 S : H → H 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. Representations on an 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 μ : A ⊗ A → A, and ρ : H ⊗ A → A is a representation of H on A, then ρ is said to be a representation of H on an algebra if μ is H-equivariant. As special cases, Lie algebras, Lie superalgebras and groups can also have representations on an algebra. See also Tannaka–Krein reconstruction theorem Hopf algebras Representation theory