Exact sequences in the cohomology of a Lie superalgebra extension

2019/04/14 by Samir Kumar Hazra, Hazra, Samir Kumar, Amber Habib +1 · 2 citations
Mathematics · #17B40 #17B56 #18G40 #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT #msc:17B40 #msc:17B56 #msc:18G40

paper · pdf · doi:10.48550/arxiv.1904.06650

17 pages

arxiv created 2020/11/08 · arxiv updated 2020/11/10

Abstract

Let 0→ \mathfraka → \mathfrake → \mathfrakg → 0 be an abelian extension of the Lie superalgebra \mathfrakg. In this article we consider the problems of extending endomorphisms of \mathfraka and lifting endomorphisms of \mathfrakg to certain endomorphisms of \mathfrake. We connect these problems to the cohomology of \mathfrakg with coefficients in \mathfraka through construction of two exact sequences, which is our main result, involving various endomorphism groups and the second cohomology. The first exact sequence is obtained using the Hochschild-Serre spectral sequence corresponding to the above extension while to prove the second we rather take a direct approach. As an application of our results we obtain descriptions of certain automorphism groups of semidirect product Lie superalgebras.

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• Non-abelian extensions and Wells exact sequences of Bol algebras
• Cohomology and automorphisms for a matched pair of 3-Lie algebras

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