Yetter–Drinfeld category

Definition

Let H be a Hopf algebra over a field k. Let \Delta 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,\boldsymbol{.}) is a left H-module, where \boldsymbol{.}: H\otimes V\to V denotes the left action of H on V,
• (V,\delta;) is a left H-comodule, where \delta : V\to H\otimes V denotes the left coaction of H on V,
• the maps \boldsymbol{.} and \delta satisfy the compatibility condition :: \delta (h\boldsymbol{.}v)=h_{(1)}v_{(-1)}S(h_{(3)}) \otimes h_{(2)}\boldsymbol{.}v_{(0)} for all h\in H,v\in V, :where, using Sweedler notation, (\Delta \otimes \mathrm{id})\Delta (h)=h_{(1)}\otimes h_{(2)} \otimes h_{(3)} \in H\otimes H\otimes H denotes the twofold coproduct of h\in H , and \delta (v)=v_{(-1)}\otimes v_{(0)} .

Examples

• Any left H-module over a cocommutative Hopf algebra H is a Yetter–Drinfeld module with the trivial left coaction \delta (v)=1\otimes v.
• The trivial module V=k{v} with h\boldsymbol{.}v=\epsilon (h)v, \delta (v)=1\otimes 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 :: V=\bigoplus _{g\in G}V_g, :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 :: V=\bigoplus {g\in G}V_g, such that g.V_h\subset V{ghg^{-1}}.
• Over the base field k=\mathbb{C}; all finite-dimensional, irreducible/simple Yetter–Drinfeld modules over a (nonabelian) group H=kG are uniquely given through a conjugacy class [g]\subset G; together with \chi,X; (character of) an irreducible group representation of the centralizer Cent(g); of some representing g\in[g]:
• :V=\mathcal{O}{[g]}^\chi=\mathcal{O}{[g]}^{X}\qquad V=\bigoplus_{h\in[g]}V_{h}=\bigoplus_{h\in[g]}X
  • As G-module take \mathcal{O}_{[g]}^\chi to be the induced module of \chi,X;:
• ::Ind_{Cent(g)}^G(\chi)=kG\otimes_{kCent(g)}X
• :(this can be proven easily not to depend on the choice of g)
  • To define the G-graduation (comodule) assign any element t\otimes v\in kG\otimes_{kCent(g)}X=V to the graduation layer:
• ::t\otimes v\in V_{tgt^{-1}}
  • 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 Cent(g);-cosets. From this approach, one often writes
• ::h\otimes v\subset[g]\times X ;; \leftrightarrow ;; t_i\otimes v\in kG\otimes_{kCent(g)}X \qquad\text{with uniquely};;h=t_igt_i^{-1}
• :(this notation emphasizes the graduation h\otimes v\in 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\otimes W\to W\otimes V,

:is invertible with inverse

:Further, for any three Yetter–Drinfeld modules U, V, W the map c satisfies the braid relation ::(c_{V,W}\otimes \mathrm{id}U)(\mathrm{id}V\otimes c{U,W})(c{U,V}\otimes \mathrm{id}W)=(\mathrm{id}W\otimes c{U,V}) (c{U,W}\otimes \mathrm{id}_V) (\mathrm{id}U\otimes c{V,W}):U\otimes V\otimes W\to W\otimes V\otimes U.

A monoidal category \mathcal{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\mathcal{YD}.

References

References

1. (1999). "Braided Hopf algebras over non abelian groups". Bol. Acad. Ciencias (Cordoba).