Yetter–Drinfeld category

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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

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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 (left left) 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
  for all  ,
where, using Sweedler notation,   denotes the twofold coproduct of  , and  .

Examples

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  • 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
 ,
where each   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
 , such that  .
  • Over the base field   all finite-dimensional, irreducible/simple Yetter–Drinfeld modules over a (nonabelian) group H=kG are uniquely given[1] through a conjugacy class   together with   (character of) an irreducible group representation of the centralizer   of some representing  :
     
    • As G-module take   to be the induced module of  :
     
    (this can be proven easily not to depend on the choice of g)
    • To define the G-graduation (comodule) 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
     
    (this notation emphasizes the graduation  , rather than the module structure)

Braiding

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

 
is invertible with inverse
 
Further, for any three Yetter–Drinfeld modules U, V, W the map c satisfies the braid relation
 

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  .

References

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  1. ^ Andruskiewitsch, N.; Grana, M. (1999). "Braided Hopf algebras over non abelian groups". Bol. Acad. Ciencias (Cordoba). 63: 658–691. arXiv:math/9802074. CiteSeerX 10.1.1.237.5330.