Yetter–Drinfeld category

Definition

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

( V , . ) is a left H-module, where . : H ⊗ V → V denotes the left action of H on V and ⊗ denotes a tensor product,

( V , δ ) is a left H-comodule, where δ : V → H ⊗ V denotes the left coaction of H on V,

the maps . and δ satisfy the compatibility condition

where, using Sweedler notation, ( Δ ⊗ i d ) Δ ( h ) = h ( 1 ) ⊗ h ( 2 ) ⊗ h ( 3 ) ∈ H ⊗ H ⊗ H denotes the twofold coproduct of h ∈ H , and δ ( v ) = v ( − 1 ) ⊗ v ( 0 ) .

Examples

Any left H-module over a cocommutative Hopf algebra H is a Yetter–Drinfeld module with the trivial left coaction δ ( v ) = 1 ⊗ v .

The trivial module V = k { v } with h . v = ϵ ( h ) v , δ ( v ) = 1 ⊗ v , 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

where each V g is a G-submodule of V.

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 k = C all finite-dimensional, irreducible/simple Yetter–Drinfeld modules over a (nonabelian) group H=kG are uniquely given through a conjugacy class [ g ] ⊂ G together with χ , X (character of) an irreducible group representation of the centralizer C e n t ( g ) of some representing g ∈ [ g ] : V = O [ g ] χ = O [ g ] X V = ⨁ h ∈ [ g ] V h = ⨁ h ∈ [ g ] X

As G-module take O [ g ] χ to be the induced module of χ , X :

(this can be proven easily not to depend on the choice of g)

To define the G-graduation (comodule) assign any element t ⊗ v ∈ k G ⊗ k C e n t ( g ) X = V to the graduation layer:

It is very custom to directly construct V as direct sum of X´s and write down the G-action by choice of a specific set of representatives t i for the C e n t ( g ) -cosets. From this approach, one often writes

(this notation emphasizes the graduation h ⊗ v ∈ V h , rather than the module structure)

Braiding

Let H be a Hopf algebra with invertible antipode S, and let V, W be Yetter–Drinfeld modules over H. Then the map c V , W : V ⊗ W → W ⊗ V ,

is invertible with inverseFurther, for any three Yetter–Drinfeld modules U, V, W the map c satisfies the braid relation ( c V , W ⊗ i d U ) ( i d V ⊗ c U , W ) ( c U , V ⊗ i d W ) = ( i d W ⊗ c U , V ) ( c U , W ⊗ i d V ) ( i d U ⊗ c V , W ) : U ⊗ V ⊗ W → W ⊗ V ⊗ U .

A monoidal category C 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 H H Y D .