If H is a Hopf algebra with bijective antipode and α, β ∈ Aut Hopf (H), we introduce a category H YD H (α, β), generalizing both Yetter-Drinfeld modules and anti-Yetter-Drinfeld modules. We construct a braided T-category YD(H) having all the categories H YD H (α, β) 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 YD H (α, β) is isomorphic to the category of usual YetterDrinfeld modules H YD H .

In this paper we study the simplicial structure of the complex C • ((A, B, ε); M ), associated to the secondary Hochschild cohomology. The main ingredient is the simplicial object B (A, B, ε), which plays a role equivalent to that of the bar resolution associated to an algebra. We also introduce the secondary cyclic (co)homology and establish some of its properties (Theorems 3.9 and 4.11).

We introduce -groups and show how they fit in the context of lattice field theory. To a topological space M we associate a -group Γ (M). We define the symmetric cohomology HS n (G, A) of a group G with coefficients in a G-module A. The -group Γ (M)is determined by the action of π 1 (M) on π 2 (M) and an element of HS 3 (π 1 (M), π 2 (M)).

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