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

We propose a detailed systematic study of a group H_L^2(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 and is called second lazy cohomology group of A. 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_L^2(A) for monomial Hopf algebras and for a class of cotriangular Hopf algebras is performed.