Vol. 13, No. 5, 2019

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A five-term exact sequence for Kac cohomology

César Galindo and Yiby Morales

Vol. 13 (2019), No. 5, 1121–1144
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