TY - JOUR

T1 - A five-term exact sequence for kac cohomology

AU - Galindo, César

AU - Morales, Yiby

N1 - Funding Information:
C.G. is partially supported by Faculty of Science of Universidad de los Andes, Convocatoria 2018–2019 para la Financiación de Programas de Investigación, programa ”Simetría T (inversión temporal) en categorías de fusión y modulares”. MSC2010: 16T05. Keywords: Hopf algebras, relative cohomology, abelian extensions of Hopf algebras.
Publisher Copyright:
© 2019, Mathematical Sciences Publishers. All rights reserved.

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

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U2 - 10.2140/ant.2019.13.1121

DO - 10.2140/ant.2019.13.1121

M3 - Research Article

AN - SCOPUS:85071869015

SN - 1937-0652

VL - 13

SP - 1121

EP - 1144

JO - Algebra and Number Theory

JF - Algebra and Number Theory

IS - 5

ER -