Multipliers, covers, and stem extensions for Lie superalgebras

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.

Keywords

Multiplier / cover / stem extension / Heisenberg superalgebra / filiform Lie supleralgebra

References

[1] Batten P. Multipliers and Covers of Lie Algebras. Ph D Thesis, North Carolina State University, 1993

[2] Batten P, Moneyhun K, Stitzinger E. On characterizing nilpotent Lie algebras by their multipliers. Comm Algebra, 1996, 24(14): 4319–4330

[3] Batten P, Stitzinger E. On covers of Lie algebras. Comm Algebra, 1996, 24(14): 4301–4317

[4] Eshrati D, Saeedi F, Darabi H. On the multiplier of nilpotent n-Lie algebras. J Algebra, 2016, 450: 162–172

[5] Fialowski A, Millionschikov D. Cohomology of graded Lie algebras of maximal class. J Algebra, 2006, 296(1): 157–176

[6] Gilg M. Low-dimensional filiform Lie superalgebras. Rev Mat Complut, 2001, 14: 463–478

[7] Hardy P. On characterizing nilpotent Lie algebras by their multipliers, III. Comm Algebra, 2005, 33: 4205–4210

[8] 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

[9] Liu W D, Yang Y. Cohomology of model filiform Lie superalgebra. J Algebra Appl, 2018, 17: 1850074

[10] Liu W D, Zhang Y L. Classification of nilpotent Lie superalgebras of multiplier-rank≤2. J Lie Theory, 2020, 30: 1047–1060

[11] Moneyhun K. Isoclinisms in Lie algebras. Algebras, Groups and Geometries, 1994, 11: 9–22

[12] Nayak S. Multipliers of nilpotent Lie superalgebra. Comm Algebra, 2019, 47: 689–705

[13] Niroomand P. On dimension of the Schur multiplier of nilpotent Lie algebras. Cent Eur J Math, 2001, 9(1): 57–64

[14] Niroomand P, Russo F G. A note on the Schur multiplier of a nilpotent Lie algebra. Comm Algebra, 2011, 39(4): 1293–1297

[15] Rodíguez-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

[16] 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

RIGHTS & PERMISSIONS

Higher Education Press