# Hopf Algebra - Representation Theory

Representation Theory

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

for mM, nN and . Furthermore, we can define the trivial representation as the base field K with

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

where fM* and mM.

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 MMK and MKM are also isomorphisms of A-modules. Also, the map of vector spaces M* ⊗ MK with fmf(m) is also a homomorphism of A-modules. However, the map MM* → K is not necessarily a homomorphism of A-modules.

Read more about this topic:  Hopf Algebra

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