Yetter-Drinfeld category
From Wikipedia, the free encyclopedia
In mathematics a Yetter-Drinfel'd category is a special type of braided monoidal category. It consists of modules over a Hopf algebra which satisfy some additional axioms.
Contents |
[edit] 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 Yetter-Drinfel'd module over H if
- is a left H-module, where denotes the left action of H on V and ⊗ denotes a tensor product,
- (V,δ) is a left H-comodule, where denotes the left coaction of H on V,
- the maps and δ satisfy the compatibility condition
-
- for all ,
- where, using Sweedler notation, denotes the twofold coproduct of , and .
[edit] Examples
- Any left H-module over a cocommutative Hopf algebra H is a Yetter-Drinfel'd module with the trivial left coaction .
- The trivial module V = k{v} with , , is a Yetter-Drinfel'd module for all Hopf algebras H.
- If H is the group algebra kG of an abelian group G, then Yetter-Drinfel'd modules over H are precisely the G-graded G-modules. This means that
-
- ,
- where each Vg is a G-submodule of V.
- More generally, if the group G is not abelian, then Yetter-Drinfel'd modules over H=kG are G-modules with a G-gradation
-
- , such that .
[edit] Braiding
Let H be a Hopf algebra with invertible antipode S, and let V, W be Yetter-Drinfel'd modules over H. Then the map ,
-
- ,
- is invertible with inverse
- .
- Further, for any three Yetter-Drinfel'd modules U, V, W the map c satisfies the braid relation
[edit] Yetter-Drinfel'd category
A monoidal category consisting of Yetter-Drinfel'd modules over a Hopf algebra H with bijective antipode is called a Yetter-Drinfel'd category. It is a braided monoidal category with the braiding c above. The category of Yetter-Drinfel'd modules over a Hopf algebra H with bijective antipode is denoted by .
[edit] References
- S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Lecture Notes vol 82, American Math Society, Providence, RI, 1993. ISBN-10: 0821807382