Hopf Algebra - Representation Theory

Representation Theory

Let A be a Hopf algebra, and let M and N be A-modules. Then, M ⊗ N is also an A-module, with

for m ∈ M, n ∈ N and . Furthermore, we can define the trivial representation as the base field K with

for m ∈ K. Finally, the dual representation of A can be defined: if M is an A-module and M* is its dual space, then

where f ∈ M* and m ∈ M.

The relationship between Δ, ε, and S ensure that certain natural homomorphisms of vector spaces are indeed homomorphisms of A-modules. For instance, the natural isomorphisms of vector spaces M → M ⊗ K and M → K ⊗ M are also isomorphisms of A-modules. Also, the map of vector spaces M* ⊗ M → K with f ⊗ m → f(m) is also a homomorphism of A-modules. However, the map M ⊗ M* → K is not necessarily a homomorphism of A-modules.