We develop a mechanism for classication of isomorphism types of non-trivial semisimple Hopf algebras whose group of grouplikes G(H) is abelian of prime index p which is the smallest prime divisor of |G(H)|. We describe structure of the second cohomology group of extensions of kCp by k G where Cp is a cyclic group of order p and G a finite abelian group. We carry out an explicit classification f...

Given a group π, we use involutary Hopf π-coalgebras to define a scalar invariant of flat π-bundles over 3-manifolds. When π = 1, this invariant equals to the one of 3-manifolds constructed by Kuperberg from involutary Hopf algebras. We give examples which show that this invariant is not trivial.

We describe the quasitriangular structure (universal R-matrix) on the non-standard quantum group U q (H 1 , H 2 , X ±) associated to the Alexander-Conway matrix solution of the Yang-Baxter equation. We show that this Hopf algebra is connected with the super-Hopf algebra U q gl (1|1) by a general process of superization.

We obtain deformations of a crossed product of a polynomial algebra with a group, under some conditions, from universal deformation formulas. These formulas arise from actions of Hopf algebras generated by automorphisms and skew derivations. They are universal in the sense that they apply to deform all algebras with such Hopf algebra actions, and we give one additional example.

We compute the Hochschild and Gerstenhaber-Schack cohomological dimensions of the universal cosovereign Hopf algebras, when the matrix of parameters is a generic asymmetry. Our main tools are considerations on the cohomologies of free product of Hopf algebras, and on the invariance of the cohomological dimensions under graded twisting by a finite abelian group.

In 1988, Brown and Ellis published [3] a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that condition. The main aim of this paper is, however, to generalise this corrected result to derive formulae of Hopf type for the n-fold Čech derived functors of the...

Every finite dimensional Hopf algebra is a Frobenius algebra, with Frobenius homomorphism given by an integral. The Nakayama automorphism determined by it yields a decomposition with degrees in a cyclic group. For a family of pointed Hopf algebras, we determine necessary and sufficient conditions for this decomposition to be strongly graded.