Yetter–Drinfeld category

In mathematics a Yetter–Drinfeld category is a special type of braided monoidal category. It consists of modules over a Hopf algebra which satisfy some additional axioms.

Definition

Let H be a Hopf algebra over a field k. Let denote the coproduct and S the antipode of H. Let V be a vector space over k. Then V is called a Yetter–Drinfeld module over H if

is a left H-module, where denotes the left action of H on V,
is a left H-comodule, where denotes the left coaction of H on V,
the maps and satisfy the compatibility condition

Examples

Any left H-module over a cocommutative Hopf algebra H is a Yetter–Drinfeld module with the trivial left coaction.
The trivial module with,, is a Yetter–Drinfeld module for all Hopf algebras H.
If H is the group algebra kG of an abelian group G, then Yetter–Drinfeld modules over H are precisely the G-graded G-modules. This means that
More generally, if the group G is not abelian, then Yetter–Drinfeld modules over H=kG are G-modules with a G-gradation
Over the base field all finite-dimensional, irreducible/simple Yetter–Drinfeld modules over a group H=kG are uniquely given through a conjugacy class together with an irreducible group representation of the centralizer of some representing :
:
* As G-module take to be the induced module of :
::
:
* To define the G-graduation assign any element to the graduation layer:
::
* It is very custom to directly construct as direct sum of X´s and write down the G-action by choice of a specific set of representatives for the -cosets. From this approach, one often writes
::
:

Braiding

Let H be a Hopf algebra with invertible antipode S, and let V, W be Yetter–Drinfeld modules over H. Then the map,
A monoidal category consisting of Yetter–Drinfeld modules over a Hopf algebra H with bijective antipode is called a Yetter–Drinfeld category. It is a braided monoidal category with the braiding c above. The category of Yetter–Drinfeld modules over a Hopf algebra H with bijective antipode is denoted by.