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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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References

Ayala, V., Kizil, E. and Azevedo, I., On an algorithm for finding derivations of Lie algebras, Proyecciones 31(1) (2012), 81–90.Google Scholar
Chtioui, T., Hajjaji, A., Mabrouk, S. and Makhlouf, A., Cohomologies and deformations of $\mathcal{O}$ -operators on Lie triple systems, J. Math. Phys 64(8) (2023), 25.Google Scholar
Das, A., Leibniz algebras with derivations, J. Homotopy Relat. Struct 16(2) (2021), 245–274.10.1007/s40062-021-00280-wCrossRefGoogle Scholar
Das, A. and Mandal, A., Extensions, deformation and categorification of assDer pairs, (2020), arXiv: 2002.11415 Google Scholar
Dzhumadil’daev, A., Cohomologies of colour Leibniz algebras: pre-simplicial approach, Lie theory and its applications in physics, III (1999), 124–136.Google Scholar
Coll, V., Gerstenhaber, M. and Giaquinto, A., An explicit deformation formula with noncommuting derivations, Isr. Math. Conf. Proc., 1 (1989), 396–403.Google Scholar
Feng, J., T*-extension of lie supertriple systems, Commun. Math. Res. 30(1) (2014), 51–59.Google Scholar
Guo, S., Cohomology and Nijenhuis operators of Lie supertriple systems, J. East China Norm. Univ. Natur. Sci. Ed 4 (2020), 1–11, (in Chinese).Google Scholar
Guo, S., Cohomology and deformation of Hom-Lie supertriple systems, J. Cent. China Norm. Univ. Nat. Sci 54(5) (2020), 758–765, (in Chinese).Google Scholar
Guo, S., Central extensions and deformations of Lie triple systems with a derivation, J. Math. Res. Appl 42(2) (2022), 189–198.Google Scholar
Guo, S. and Saha, R., On $3$ -Lie algebras with a derivation, afr, Mat 33(2) (2022), 15.Google Scholar
Kamiya, N. and Okubo, S., On $\delta$ -Lie supertriple systems associated with $(\epsilon ,\delta )$ -Freudenthal-Kantor supertriple systems, Proc. Edinburgh Math. Soc 43(2) (2000), 243–260.10.1017/S0013091500020903CrossRefGoogle Scholar
Li, Q. and Ma, L., 1-parameter formal deformations and abelian extensions of Lie color triple systems, Electron. Res. Arch., 30(7) (2022), 2524–2539.10.3934/era.2022129CrossRefGoogle Scholar
Liu, D. and Hu, N., Leibniz superalgebras and central extensions, J. Algebra Appl 5(6) (2006), 765–780.Google Scholar
Loday, J., On the operad of associative algebras with derivation, Georgian Math. J 17(2) (2010), 347–372.Google Scholar
Loday, J. and Vallette, B., Algebraic operads. (Springer, Heidelberg, 2012).10.1007/978-3-642-30362-3CrossRefGoogle Scholar
Ma, Y., Chen, L. and Liu, D., Generalized derivations of Lie supertriple systems, Acta Math. Sinica (Chinese Ser.), 43 (2013), 961–970.Google Scholar
Magid, A., Lectures on differential galois theory, University Lecture Series, vol. 7, (American Mathematical Society, Providence, RI, 1994).10.1090/ulect/007CrossRefGoogle Scholar
Okubo, S., Para-statistics as Lie-super triple systems, J. Math. Phys 35 (1994), 2785–2803.10.1063/1.530486CrossRefGoogle Scholar
Okubo, S. and Kamiya, N., Quasi-classical Lie superalgebras and Lie supertriple systems, Comm. Algebra 30(8) (2002), 3825–3850.10.1081/AGB-120005822CrossRefGoogle Scholar
Peng, J., Chen, L. and Sun, B., Centroids of Lie supertriple systems, Adv. Math. Phys 949046 (2015), Art. ID 949046, 9.Google Scholar
Tang, R., Frégier, Y. and Sheng, Y., Cohomologies of a Lie algebra with a derivation and applications, J. Algebra 534 (2019), 65–99.10.1016/j.jalgebra.2019.06.007CrossRefGoogle Scholar
Voronov, T., Higher derived brackets for arbitrary derivations, J. Pure Appl. Algebra 202 (2005), 133–153.10.1016/j.jpaa.2005.01.010CrossRefGoogle Scholar
Wu, X., Ma, Y., Sun, B. and Chen, L., Cohomology of Leibniz triple systems with derivations, J. Geom. Phys 179 (2022), 104594.10.1016/j.geomphys.2022.104594CrossRefGoogle Scholar
Xu, S. and Liu, J., Cohomologies of 3-Lie algebras with derivations, (2021), arXiv: 2110.04215.Google Scholar
Yamaguti, K., On the cohomology space of Lie triple system, Kumamoto J. Sci. Ser. A 5 (1960), 44–52.Google Scholar
Yadav, R. B., Cohomology and deformation of Leibniz superalgebras, (2021), arxiv: 2101.07652.Google Scholar
Yadav, R. B., Behera, N. and Bhutia, R., Equivariant one-parameter deformations of Lie triple systems, J. Algebra 568 (2021), 467–479.10.1016/j.jalgebra.2020.10.015CrossRefGoogle Scholar
Zhang, T., Notes on cohomologies of Lie triple systems, J. Lie Theory 24 (2014), 909–929.Google Scholar
Zhao, X. and Chen, L., Cohomologies and deforamtions of Lie superalgebras with derivations, (2021), https://www. researchgate.net/publication/355675478.Google Scholar
Zhang, Z. and Jia, P., The killing forms and decomposition theorems of Lie supertriple systems, Acta Math. Sci. Ser. B (Engl. Ed.) 29(2) (2009), 360–370.Google Scholar