Cocentral Split Abelian Hopf Algebra Extensions from Crossed Cocycles

Abstract

We study cocentral algebra-split abelian Hopf algebra extensions over an algebraically closed field of characteristic zero for a fixed action of a group on a finite abelian group. We describe the extension classes in terms of crossed families of normalized 2-cocycles and construct the obstruction to representing crossed families of cohomology classes by cocycles satisfying the crossed identity. A bicharacter obstruction maps to the strict lifting obstruction and may remain nonzero even when its image vanishes. For permutation modules, we obtain an explicit description of the corresponding cocentral extension groups. We also study finite reductions of geometric representations of Coxeter groups and compute the first cohomology of the associated linear coefficient modules for finite dihedral groups and the infinite dihedral group.

27 pages. Revised and expanded version. We clarify the distinction between the strict and bicharacter lifting obstructions, compute the full extension group for permutation modules, and extend the infinite-dihedral cohomology calculation. Several proofs and the exposition have also been improved