Braided Sweedler Cohomology

We introduced a braided Sweedler cohomology, which is adequate to work with the H-braided cleft extensions studied in [G-G1]. Introduction In [Sw] a cohomology theory H(H,A) for a commutative module algebra A over a cocommutative Hopf algebra H was introduced. This cohomology is related to those of groups an Lie algebras in the following sense. When H is a group algebra k[G], then H(H,A) is canonically isomorphic to the group cohomology of G in the multiplicative group of invertible elements of A, and when H is the enveloping algebra U(L) of a Lie algebra L, then H(H,A) is canonically isomorphic to the cohomology of L in the underlying vector space of A, for all n ≥ 2. One of the man properties of the Sweedler cohomology is that there is a bijective correspondence between H(H,A) and the equivalences classes of H-cleft extensions of A. This result it was extended in [D1], where it was shown that the hypothesis of commutativity of A can be removed. Let H be a braided bialgebra. In [G-G1] a notion of clef extension of an Hbraided module algebra (A, s) was presented (for the definitions see Section 1). This concept is more general than the one defined in [B-C-M] still when H is a standard Hopf algebra. Assume that H is a braided cocommutative Hopf algebra. In this paper we present a braided version of the Sweedler cohomology in order to classify the cleft extensions introduced in [G-G1]. The paper is organized as follows: Section 1 is devoted to review some notions from [G-G1] and to introduced some concepts that we will need later. In Section 2, we define, by means of a explicit complex, the braided Sweedler cohomology of a braided cocommutative Hopf algebra H with coefficients in an H-braided module algebra A. When H is a cocommutative standard Hopf algebra and H is an usual module algebra, our complex reduced to the classical one of Sweedler. In Section 3 we show that the second cohomology group of our complex classify the cleft extensions of an H-braided module algebra (A, s). In Section 4 we prove that when H is a group algebra k[G], the braided Sweedler cohomology of H with coefficients in an H-braided module algebra (A, s) coincides with a variant of the group homology of G with coefficients in the multiplicative group of invertible elements of A and in Section 5 we prove a similar result for the cohomology groups of degree greater than 1, when H is the enveloping algebra of a Lie algebra L. In Section 6 we show that in order to compute the cohomology mentioned in the previous section, a Chevalley-Eilenberg type complex can be used. Finally, in Section 7, we calculate all the cleft extensions in a particular case. 2000 Mathematics Subject Classification. Primary 18G60; Secondary 16W30. Supported by PICT 12330, UBACYT X294 and CONICET. 1 2 SERGIO D. CORTI, JORGE A. GUCCIONE, AND JUAN J. GUCCIONE 1. Preliminaries In this article we work in the category of vector spaces over a field k. Then we assume implicitly that all the maps are k-linear and all the algebras and coalgebras are over k. The tensor product over k is denoted by ⊗, without any subscript, and the category of k-vector spaces is denoted by Vect. Given a vector space V and n ≥ 1, we let V n denote the n-fold tensor power V ⊗ · · · ⊗ V . Given vector spaces U, V,W and a map f : V → W we write U ⊗ f for idU ⊗f and f ⊗ U for f ⊗ idU . We assume that the algebras are associative unitary and the coalgebras are coassociative counitary. Given an algebra A and a coalgebra C, we let μ : A ⊗ A → A, η : k → A, ∆: C → C ⊗ C and ǫ : C → k denote the multiplication, the unit, the comultiplication and the counit, respectively, specified with a subscript if necessary. Some of the results of this paper are valid in the context of monoidal categories. In fact we use the nowadays well known graphic calculus for monoidal and braided categories. As usual, morphisms will be composed from up to down and tensor products will be represented by horizontal concatenation in the corresponding order. The identity map of a vector space will be represented by a vertical line. Given an algebra A, the diagrams , ◦ and stand for the multiplication map, the unit and the action of A on a left A-module, respectively, and for a coalgebra C, the comultiplication and the counit will be represented by the diagrams