Representation theory of Hopf algebras

In abstract algebra, a representation of a Hopf algebra is a representation of its underlying associative algebra. That is, a representation of a Hopf algebra "H" over a field "K" is a "K"-vector space "V" with an action "H" × "V" → "V" usually denoted by juxtaposition ( that is, the image of ("h","v") is written "hv" ). The vector space "V" is called an "H"-module.
Properties.
The module structure of a representation of a Hopf algebra "H" is simply its structure as a module for the underlying associative algebra. The main use of considering the additional structure of a Hopf algebra is when considering all "H"-modules as a category. The additional structure is also used to define invariant elements of an "H"-module "V". An element "v" in "V" is invariant under "H" if for all "h" in "H", "hv" = ε("h")"v", where ε is the counit of "H". The subset of all invariant elements of "V" forms a submodule of "V".
Categories of representations as a motivation for Hopf algebras.
For an associative algebra "H", the tensor product "V"1 ⊗ "V"2 of two "H"-modules "V"1 and "V"2 is a vector space, but not necessarily an "H"-module. For the tensor product to be a functorial product operation on "H"-modules, there must be a linear binary operation Δ : "H" → "H" ⊗ "H" such that for any "v" in "V"1 ⊗ "V"2 and any "h" in "H",
and for any "v" in "V"1 ⊗ "V"2 and "a" and "b" in "H",
using sumless Sweedler's notation, which is somewhat like an index free form of Einstein's summation convention. This is satisfied if there is a Δ such that Δ("ab") = Δ("a")Δ("b") for all "a", "b" in "H".
For the category of "H"-modules to be a strict monoidal category with respect to ⊗, formula_3 and formula_4 must be equivalent and there must be unit object ε"H", called the trivial module, such that ε"H" ⊗ "V", "V" and "V" ⊗ ε"H" are equivalent. 
This means that for any "v" in 
and for "h" in "H",
This will hold for any three "H"-modules if Δ satisfies 
The trivial module must be one-dimensional, and so an algebra homomorphism ε : "H" → "F" may be defined such that "hv" = ε("h")"v" for all "v" in ε"H". The trivial module may be identified with "F", with 1 being the element such that 1 ⊗ "v" = "v" = "v" ⊗ 1 for all "v". It follows that for any "v" in any "H"-module "V", any "c" in ε"H" 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 formula_11 is the usual pairing of dual vector spaces. If the map formula_12 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.