Lazy cohomology: An analogue of the Schur multiplier for arbitrary Hopf algebras

Abstract

We propose a detailed systematic study of a group H 2 L (A) associated, by elementary means of lazy 2-cocycles, to any Hopf algebra A. This group was introduced by Schauenburg in order to generalize Kac's exact sequence. We study the various interplays of lazy cohomology in Hopf algebra theory: Galois and biGalois objects, Brauer groups and projective representations. We obtain a Kac-Schauenburg-type sequence for double crossed products of possibly infinite-dimensional Hopf algebras. Finally, the explicit computation of H 2 L (A) for monomial Hopf algebras and for a class of cotriangular Hopf algebras is performed.In 1968, Sweedler defined in [32] the cohomology for a cocommutative Hopf algebra H with coefficients in a module algebra A, relating it to the Brauer group of a field and to well-known cohomology theories such as Lie algebra cohomology, group cohomology, Galois cohomology. For instance, he showed that when the algebra A is commutative, there

Extending lazy 2-cocycles on Hopf algebras and lifting projective representations afforded by them

“…In general, the results in [2] suggest that, for an arbitrary Hopf algebra, lazy cocycles are much closer to the cocommutative case than general left cocycles. Hence, a sort of general principle is suggested: results that hold for an arbitrary 2-cocycle on a cocommutative Hopf algebra are likely to hold also for a lazy 2-cocycle on an arbitrary Hopf algebra.…”

“…Lazy cocycles and lazy cohomology were also used in [20] to give a generalized version of Kac's exact sequence. A general theory of lazy cocycles and lazy cohomology started to be developed recently in [2]. The remarkable fact is that the set Z 2 L (H ) of normalized and convolution invertible lazy 2-cocycles on H form a group (this was noted in [11]), and that one can also define lazy 2-coboundaries B 2 L (H ) and the second lazy cohomology group H 2 L (H ) = Z 2 L (H )/B 2 L (H ), generalizing the second Sweedler cohomology group of a cocommutative Hopf algebra (note that for cocommutative Hopf algebras any 2-cocycle is lazy).…”

“…The remarkable fact is that the set Z 2 L (H ) of normalized and convolution invertible lazy 2-cocycles on H form a group (this was noted in [11]), and that one can also define lazy 2-coboundaries B 2 L (H ) and the second lazy cohomology group H 2 L (H ) = Z 2 L (H )/B 2 L (H ), generalizing the second Sweedler cohomology group of a cocommutative Hopf algebra (note that for cocommutative Hopf algebras any 2-cocycle is lazy). The group H 2 L (H ) can be regarded as a subgroup of Bigal(H ), the group of Bigalois objects of H , and the examples in [2] show that it is much easier to compute H 2 L (H ) than Bigal(H ).…”

A Sequence to Compute the Brauer Group of Certain Quasi-Triangular Hopf Algebras

“…After the group of lazy 2-cocycles was studied in [7], and knowing the computations of Brauer groups done in [45], [12], [13] and [11], it was suspected that the group of lazy 2-cocycles would embed in the Brauer group, as mentioned in the introduction of [7]. In view of the following result, there is an embeding of the second braided cohomology group of the braided Hopf algebra into the Brauer group of the corresponding Radford biproduct.…”

On the subgroup structure of the full Brauer group of Sweedler Hopf algebra

“…We recall from [3] that a lazy cocycle on H is a left 2-cocycle σ such that twisting H by σ does not modify the product in H. In other words: for every h, l, m ∈ H, (2) ).…”