Editing Hopf algebra (section)

==Representation theory==
{{Main|Representation theory of Hopf algebras}}

Let ''A'' be a Hopf algebra, and let ''M'' and ''N'' be ''A''-modules. Then, ''M'' ⊗ ''N'' is also an ''A''-module, with
:<math>a(m\otimes n):=\Delta(a)(m \otimes n)=(a_1\otimes a_2)(m\otimes n)=(a_1 m \otimes a_2 n)</math>
for ''m'' ∈ ''M'', ''n'' ∈ ''N'' and Δ(''a'') = (''a''<sub>1</sub>, ''a''<sub>2</sub>). Furthermore, we can define the trivial representation as the base field ''K'' with
:<math>a(m):=\epsilon(a)m</math>
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
:<math>(af)(m):=f(S(a)m)</math>
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.