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Cohomologies, deformations and abelian extensions of Lie super triple systems with superderivations

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  01 October 2025

Xinyue Wang ,
Yao Ma Open the ORCID record for Yao Ma [Opens in a new window]  and
Liangyun Chen Open the ORCID record for Liangyun Chen [Opens in a new window]
Xinyue Wang
Affiliation:
Department of Mathematics, Soochow University, Suzhou, China
Yao Ma
Affiliation:
School of Mathematics and Statistics, Northeast Normal University, Changchun, China
Liangyun Chen*
Affiliation:
School of Mathematics and Statistics, Northeast Normal University, Changchun, China
*
Corresponding author: Liangyun Chen; Email: chenly640@nenu.edu.cn
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Abstract

In this paper, we first describe the cohomology theory of Lie supertriple systems by using the cohomology theory of the associated Leibniz superalgebras. Then we focus on Lie supertriple systems with superderivations, called LSTSDer pairs. We introduce the notion of representations of LSTSDer pairs and investigate their corresponding cohomology theory. We also construct a differential graded Lie algebra whose Maurer–Cartan elements are LSTSDer pairs. Moreover, we consider the relationship between a LSTSDer pair and the associated LeibSDer pair. Furthermore, we develop the 1-parameter formal deformation theory of LSTSDer pairs and prove that it is governed by the cohomology groups. At last, we study abelian extensions of LSTSDer pairs and show that equivalent abelian extensions of LSTSDer pairs are classified by the third cohomology groups.

Keywords

Lie supertriple systemcohomologysuperderivationdeformationabelian extension

MSC classification

Primary: 17B10: Representations, algebraic theory (weights) 17B40: Automorphisms, derivations, other operators 17B56: Cohomology of Lie (super)algebras

Information

Type
Research Article
Information
Glasgow Mathematical Journal , First View , pp. 1 - 22
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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