Yetter–Drinfeld category
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 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 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