Yetter–Drinfeld module

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.

Let H be a Hopf algebra over a field k. Let ${\displaystyle \Delta }$ 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

Let H be a Hopf algebra with invertible antipode S, and let V, W be Yetter–Drinfeld modules over H. Then the map ${\displaystyle c_{V,W}:V\otimes W\to W\otimes V}$,

A monoidal category ${\displaystyle {\mathcal {C}}}$ 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 ${\displaystyle {}_{H}^{H}{\mathcal {YD}}}$.

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