Yetter–Drinfeld category

Definition

• is a left H-module, where denotes the left action of H on V and ⊗ denotes a tensor product,
• 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