A five-term exact sequence for Kac cohomology

Galindo, César; Morales, Yiby

doi:10.2140/ant.2019.13.1121

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A five-term exact sequence for Kac cohomology

César Galindo and Yiby Morales

Vol. 13 (2019), No. 5, 1121–1144

DOI: 10.2140/ant.2019.13.1121

Abstract

We use relative group cohomologies to compute the Kac cohomology of matched pairs of finite groups. This cohomology naturally appears in the theory of abelian extensions of finite dimensional Hopf algebras. We prove that Kac cohomology can be computed using relative cohomology and relatively projective resolutions. This allows us to use other resolutions, besides the bar resolution, for computations. We compute, in terms of relative cohomology, the first two pages of a spectral sequence which converges to the Kac cohomology and its associated five-term exact sequence. Through several examples, we show the usefulness of the five-term exact sequence in computing groups of abelian extensions.

Keywords

Hopf algebras, relative cohomology, abelian extensions of Hopf algebras

Mathematical Subject Classification 2010

Primary: 16T05

Milestones

Received: 15 June 2018

Revised: 18 September 2018

Accepted: 22 February 2019

Published: 12 July 2019

Authors

César Galindo
Departamento de Matemáticas Universidad de Los Andes Bogota Colombia

Yiby Morales
Departamento de Matemáticas Universidad de los Andes Bogota Colombia

Vol. 13, No. 5, 2019