Generalized (anti) Yetter-Drinfeld modules as components of a braided T-category

AbstractIf H is a Hopf algebra with bijective antipode and α, β ∈ AutHopf(H), we introduce a category $$_H \mathcal{Y}\mathcal{D}^H (\alpha ,\beta )$$ , generalizing both Yetter-Drinfeld modules and anti-Yetter-Drinfeld modules. We construct a braided T-category $$\mathcal{Y}\mathcal{D}(H)$$ having all the categories $$_H \mathcal{Y}\mathcal{D}^H (\alpha ,\beta )$$ as components, which, if H is finite dimensional, coincides with the representations of a certain quasitriangular T-coalgebra DT(H) that we construct. We also prove that if (α, β) admits a so-called pair in involution, then $$_H \mathcal{Y}\mathcal{D}^H (\alpha ,\beta )$$ is isomorphic to the category of usual Yetter-Drinfeld modules $$_H \mathcal{Y}\mathcal{D}^H $$ .