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

If H is a Hopf algebra with bijective antipode and α, β ∈ AutHopf (H), we introduce a category HYD (α, β), generalizing both Yetter-Drinfeld modules and anti-Yetter-Drinfeld modules. We construct a braided T-category YD(H) having all the categories HYD (α, β) as components, which ifH 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 HYD (α, β) is isomorphic to the category of usual YetterDrinfeld modules HYD H . Introduction Let H be a Hopf algebra with bijective antipode S and α, β ∈ AutHopf (H). We introduce the concept of an (α, β)-Yetter-Drinfeld module, as being a left H-module right H-comodule M with the following compatibility condition: (h ·m)(0) ⊗ (h ·m)(1) = h2 ·m(0) ⊗ β(h3)m(1)α(S (h1)). This concept is a generalization of three kinds of objects appeared in the literature. Namely, for α = β = idH , we obtain the usual Yetter-Drinfeld modules; for α = S 2, β = idH , we obtain the so-called anti-Yetter-Drinfeld modules, introduced in [7], [8], [10] as coefficients for the cyclic cohomology of Hopf algebras defined by Connes and Moscovici in [5], [6]; finally, an (idH , β)Yetter-Drinfeld module is a generalization of the object Hβ defined in [4], which has the property that, if H is finite dimensional, then the map β 7→ End(Hβ) gives a group anti-homomorphism from AutHopf (H) to the Brauer group of H. It is natural to expect that (α, β)-Yetter-Drinfeld modules have some properties resembling the Research partially supported by the programme CERES of the Romanian Ministry of Education and Research, contract no. 4-147/2004. Permanent address: Institute of Mathematics of the Romanian Academy, PO-Box 1-764, RO-014700 Bucharest, Romania.