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We provide an intrinsic definition of the fundamental group of a linear category over a ring as the automorphism group of the fibre functor on Galois coverings. If the universal covering exists, we prove that this group is isomorphic to the Galois group of the universal covering. The grading deduced from a Galois covering enables us to describe the canonical monomorphism from its automorphism group to the first Hochschild-Mitchell cohomology vector space. [ABSTRACT FROM AUTHOR]

We consider $${\mathsf{N}}$$ -complexes as functors over an appropriate linear category in order to show first that the Krull-Schmidt Theorem holds, then to prove that amplitude cohomology (called generalized cohomology by M. Dubois-Violette) only vanishes on injective functors providing a well defined functor on the stable category. For left truncated $${\mathsf{N}}$$ -complexes, we show that amplitude cohomology discriminates the isomorphism class up to a projective functor summand. Moreover amplitude cohomology of positive $${\mathsf{N}}$$ -complexes is proved to be isomorphic to an Ext functor of an indecomposable $${\mathsf{N}}$$ -complex inside the abelian functor category. Finally we show that for the monoidal structure of $${\mathsf{N}}$$ -complexes a Clebsch-Gordan formula holds, in other words the fusion rules for $${\mathsf{N}}$$ -complexes can be determined. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic p. Let H be a normal subgroup of odd index, prime to p, in a finite group G. We prove that an indecomposable kH-module G-stable and selfdual can always be extended to G. If the kH-module is irreducible, only the odd index hypothesis is required. There exists an exact sequence which is the obstruction to extending the action in such a way that the resulting kG-module is still selfdual. This enables us to make our results more precise. [ABSTRACT FROM AUTHOR]

We construct quantum groups at a root of unity and we describe their monoidal module category using techniques from the representation theory of finite dimensional associative algebras. [ABSTRACT FROM AUTHOR]

In this paper we present a direct proof of what is suggested by Holm’s results (T. Holm, The Hochschild cohomology ring of a modular group algebra: the commutative case, Comm. Algebra 24, 1957–1969 (1996)): there is an isomorphism of algebras HH*(kG,kG) → kG ⊗ H*(G,k) where G is a finite abelian group, k a ring, HH*(kG,kG) is the Hochschild cohomology algebra and H*(G,k) the usual cohomology algebra. This result agrees with the well-known additive structure result in force for any group G; we remark that the multiplicative structure result we have obtained is quite similar to the description of the monoidal category of Hopf bimodules over kG given in “C. Cibils, Tensor product of Hopf bimodules, to appear in Proc. Amer. Math. Soc”. This similarity leads to conjecture the structure of HH*(kG,kG) for G a finite group. [ABSTRACT FROM AUTHOR]