
Journal of Lie Theory 30 (2020), No. 1, 179199 Copyright Heldermann Verlag 2020 Wells Exact Sequence and Automorphisms of Extensions of Lie Superalgebras Samir Kumar Hazra Dept. of Mathematics, School of Natural Sciences, Shiv Nadar University, Uttar Pradesh 201314, India sk896@snu.edu.in Amber Habib Dept. of Mathematics, School of Natural Sciences, Shiv Nadar University, Uttar Pradesh 201314, India amber.habib@snu.edu.in [Abstractpdf] Let $0\rightarrow \mathfrak{a} \rightarrow \mathfrak{e} \rightarrow \mathfrak{g}\rightarrow 0$ be an abelian extension of Lie superalgebras. In this article, corresponding to this extension we construct two exact sequences connecting the various automorphism groups and the 0th homogeneous part of the second cohomology group, $H^2(\mathfrak{g},\mathfrak{a})_0$. These exact sequences constitute an analogue of the wellknown Wells exact sequence for group extensions. It follows that the obstruction for a pair of automorphism $(\phi,\psi) \in Aut(\mathfrak{a}) \times Aut( \mathfrak{g})$ to be induced from an automorphism in $Aut_\mathfrak{a}(\mathfrak{e})$ lies in $H^2(\mathfrak{g},\mathfrak{a})_0$. Then we consider the family of Heisenberg Lie superalgebras and show that not all pairs are inducible in this family. We also give some necessary and sufficient conditions for inducibility of pairs arising in this family. Keywords: Lie superalgebras, extensions, cohomology, Heisenberg Lie superalgebras. MSC: 17B40, 17B56. [ Fulltextpdf (172 KB)] for subscribers only. 