**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.

Read more about this topic: Hopf Algebra

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