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 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
- for all ,
- where, using Sweedler notation, denotes the twofold coproduct of , and .
- 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 basfield all finite-dimensional, irreducible/simple Yetter–Drinfeld modules over a (nonabelian) group H=kG are uniquely given 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)
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 .
- S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Lecture Notes vol 82, American Mathematical Society, Providence, RI, 1993. ISBN 0-8218-0738-2
- N. Andruskiewitsch and M.Grana: Braided Hopf algebras over non abelian groups, Bol. Acad. Ciencias (Cordoba) 63(1999), 658-691