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

If H is a Hopf algebra with bijective antipode and \alpha, \beta \in Aut_{Hopf}(H), we introduce a category_H{\cal YD}^H(\alpha, \beta), generalizing both Yetter-Drinfeld and anti-Yetter-Drinfeld modules. We construct a braided T-category {\cal YD}(H) having all these categories 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 (\alpha, \beta) admits a so-called pair in involution, then_H{\cal YD}^H(\alpha, \beta) is isomorphic to the category of usual Yetter-Drinfeld modules_H{\cal YD}^H.