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

Let Uε(g) be the simply connected quantized enveloping algebra at roots of one associated to a finite dimensional complex simple Lie algebra g. The De ConciniKac-Procesi conjecture on the dimension of the irreducible representations of Uε(g) is proved for the representations corresponding to the spherical conjugacy classes of the simply connected algebraic group G with Lie algebra g. We achieve this result by means of a new characterization of the spherical conjugacy classes of G in terms of elements of the Weyl group.

We classify the orbits of coquasi-triangular structures for the Hopf algebra E(n) under the action of lazy cocycles and the Hopf automorphism group. This is applied to detect subgroups of the Brauer group BQ(k, E(n)) of E(n) that are isomorphic. For a triangular structure R on E(n) we prove that the subgroup BM (k, E(n), R) of BQ(k, E(n)) arising from R is isomorphic to a direct product of BW (k), the BrauerWall group of the ground field k, and Sym n (k), the group of n × n symmetric matrices under addition. For a general quasi-triangular structure R ′ on E(n) we construct a split short exact sequence having BM (k, E(n), R ′ ) as a middle term and as kernel a central extension of the group of symmetric matrices of order r < n (r depending on R ′ ). We finally describe how the image of the Hopf automorphism group inside BQ(k, E(n)) acts on Sym n (k).

Let G be a simple algebraic group over an algebraically closed field of characteristic zero or positive odd, good characteristic. Let B be a Borel subgroup of G. We show that the spherical conjugacy classes of G intersect only the double cosets of B in G corresponding to involutions in the Weyl group of G. This result is used in order to prove that for a spherical conjugacy class O with dense B-orbit v(0). BwB there holds l(w) + rk(1-w) = dim O extending to the case of groups over fields of odd, good characteristic a characterization of spherical conjugacy classes obtained by Cantarini, Costantini and the author

We show that the sheets for a connected reductive algebraic group G over an algebraically closed field in good characteristic acting on itself by conjugation are in bijection with G-conjugacy classes of tripleswhere M is the connected centralizer of a semisimple element in G, Z(M ) • t is a suitable coset in Z(M )/Z(M ) • and O is a rigid unipotent conjugacy class in M . Any semisimple element is contained in a unique sheet S and S corresponds to a triple with O = {1}. The sheets in G containing a unipotent conjugacy class are precisely those corresponding to triples for which M is a Levi subgroup of a parabolic subgroup of G and such a class is unique.