Abstract
Suppose that the underlying field is of characteristic different from 2 and 3. We first prove that the so-called stem deformations of a free presentation of a finite-dimensional Lie superalgebra L exhaust all the maximal stem extensions of L, up to equivalence of extensions. Then we prove that multipliers and covers always exist for a Lie superalgebra and they are unique up to superalgebra isomorphisms. Finally, we describe the multipliers, covers, and maximal stem extensions of Heisenberg superalgebras and model filiform Lie superalgebras.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Batten P. Multipliers and Covers of Lie Algebras. Ph D Thesis, North Carolina State University, 1993
Batten P, Moneyhun K, Stitzinger E. On characterizing nilpotent Lie algebras by their multipliers. Comm Algebra, 1996, 24(14): 4319–4330
Batten P, Stitzinger E. On covers of Lie algebras. Comm Algebra, 1996, 24(14): 4301–4317
Eshrati D, Saeedi F, Darabi H. On the multiplier of nilpotent n-Lie algebras. J Algebra, 2016, 450: 162–172
Fialowski A, Millionschikov D. Cohomology of graded Lie algebras of maximal class. J Algebra, 2006, 296(1): 157–176
Gilg M. Low-dimensional filiform Lie superalgebras. Rev Mat Complut, 2001, 14: 463–478
Hardy P. On characterizing nilpotent Lie algebras by their multipliers, III. Comm Algebra, 2005, 33: 4205–4210
Hardy P, Stitzinger E. On characterizing nilpotent Lie algebras by their multipliers t(L) = 3, 4, 5, 6. Comm Algebra, 1998, 26(11): 3527–3539
Liu W D, Yang Y. Cohomology of model filiform Lie superalgebra. J Algebra Appl, 2018, 17: 1850074
Liu W D, Zhang Y L. Classification of nilpotent Lie superalgebras of multiplier-rank ≼ 2. J Lie Theory, 2020, 30: 1047–1060
Moneyhun K. Isoclinisms in Lie algebras. Algebras, Groups and Geometries, 1994, 11: 9–22
Nayak S. Multipliers of nilpotent Lie superalgebra. Comm Algebra, 2019, 47: 689–705
Niroomand P. On dimension of the Schur multiplier of nilpotent Lie algebras. Cent Eur J Math, 2001, 9(1): 57–64
Niroomand P, Russo F G. A note on the Schur multiplier of a nilpotent Lie algebra. Comm Algebra, 2011, 39(4): 1293–1297
Rodríguez-Vallarte M C, Salgado G, Sánchez-Valenzuela O A. Heisenberg Lie super-algebras and their invariant superorthogonal and supersymplectic forms. J Algebra, 2011, 332(1): 71–86
Saeedi F, Arabyani H, Niroomand P. On dimension of the Schur multiplier of nilpotent Lie algebras II. Asian-Eur J Math, 2017, 10(4): 1750076
Acknowledgements
The authors thank the reviewers for their valuable suggestions. This work was supported in part by the National Natural Science Foundation of China (Grant No. 12061029), the Natural Science Foundation of Hainan Province (No. 120RC587), and the Natural Science Foundation of Heilongjiang Province (No. YQ2020A005).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, W., Miao, X. Multipliers, covers, and stem extensions for Lie superalgebras. Front. Math. China 16, 979–995 (2021). https://doi.org/10.1007/s11464-021-0907-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-021-0907-8