Czechoslovak Mathematical Journal, Vol. 71, No. 2, pp. 335-350, 2021


Cohomology and deformations of 3-dimensional Heisenberg Hom-Lie superalgebras

Junxia Zhu, Liangyun Chen

Received July 15, 2019.   Published online February 3, 2021.

Abstract:  We study Hom-Lie superalgebras of Heisenberg type. For 3-dimensional Heisenberg Hom-Lie superalgebras we describe their Hom-Lie super structures, compute the cohomology spaces and characterize their infinitesimal deformations.
Keywords:  Hom-Lie superalgebra; Lie superalgebra; Heisenberg Hom-Lie superalgebra; cohomology; deformation
Classification MSC:  17B56, 17B60, 17B61, 17B99


References:
[1] M. A. Alvarez, F. Cartes: Cohomology and deformations for the Heisenberg Hom-Lie algebras. Linear Multilinear Algebra 67 (2019), 2209-2229. DOI 10.1080/03081087.2018.1487379 | MR 4002345 | Zbl 07116666
[2] F. Ammar, Z. Ejbehi, A. Makhlouf: Cohomology and deformations of Hom-algebras. J. Lie Theory 21 (2011), 813-836. MR 2917693 | Zbl 1237.17003
[3] F. Ammar, A. Makhlouf: Hom-Lie superalgebras and Hom-Lie admissible superalgebras. J. Algebra 324 (2010), 1513-1528. DOI 10.1016/j.jalgebra.2010.06.014 | MR 2673748 | Zbl 1258.17008
[4] F. Ammar, A. Makhlouf, N. Saadaoui: Cohomology of Hom-Lie superalgebras and $q$-deformed Witt superalgebra. Czech. Math. J. 63 (2013), 721-761. DOI 10.1007/s10587-013-0049-6 | MR 3125651 | Zbl 1299.17018
[5] Y. Liu, L. Chen, Y. Ma: Hom-Nijienhuis operators and $T^*$-exentions of Hom-Lie superalgebras. Linear Algebra Appl. 439 (2013), 2131-2144. DOI 10.1016/j.laa.2013.06.006 | MR 3090460 | Zbl 1281.17033
[6] A. Makhlouf, S. Silvestrov: Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras. Forum Math. 22 (2010), 715-739. DOI 10.1515/forum.2010.040 | MR 2661446 | Zbl 1201.17012
[7] R. Peniche, O. A. Sánchez-Valenzuela: On Heisenberg-like super group structures. Ann. Henri Poincaré 10 (2010), 1395-1417. DOI 10.1007/s00023-010-0020-0 | MR 2593111 | Zbl 1206.81066
[8] M. Rodríguez-Vallarte, G. Salgado, O. A. Sánchez-Valenzuela: Heisenberg Lie superalgebras and their invariant superorthogonal and supersymplectic forms. J. Algebra 332 (2011), 71-86. DOI 10.1016/j.jalgebra.2011.02.003 | MR 2774679 | Zbl 1264.17007
[9] Y. Sheng: Representations of Hom-Lie algebras. Algebr. Represent. Theory 15 (2012), 1081-1098. DOI 10.1007/s10468-011-9280-8 | MR 2994017 | Zbl 1294.17001
[10] C. Wang, Q. Zhang, Z. Wei: A classification of low dimensional multiplicative Hom-Lie superalgebras. Open Math. 14 (2016), 613-628. DOI 10.1515/math-2016-0056 | MR 3552510 | Zbl 1345.17020
[11] Z. Wang: A Classification of Low Dimensional Lie Superalgebras: Master Thesis. East China Normal University, Shanghai (2006). (In Chinese.)

Affiliations:   Junxia Zhu, Liangyun Chen (corresponding author), School of Mathematics and Statistics, Northeast Normal University, No. 5268, Renmin Street, Changchun, Jilin Province, Changchun, 130024, P. R. China, e-mail: chenly640@nenu.edu.cn


 
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