On degenerations of Lie superalgebras

ABSTRACT

We give necessary conditions for the existence of degenerations between two complex Lie superalgebras of dimension $(m,n)$. As an application, we study the variety ${\mathcal{L}\mathcal{S}}^{(2,2)}$ of complex Lie superalgebras of dimension $(2,2)$. First we give the algebraic classification and then obtain that ${\mathcal{L}\mathcal{S}}^{(2,2)}$ is the union of seven irreducible components, three of which are the Zariski closures of rigid Lie superalgebras. As a byproduct, we obtain an example of a nilpotent rigid Lie superalgebra, in contrast to the classical case where no example is known.

KEYWORDS:

• Lie superalgebras
• geometric classification
• degenerations

AMS CLASSIFICATIONS:

• 17B30
• 17B56
• 17B99

Acknowledgments

Both authors thank Ivan Kaygorodov for useful comments about the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

María Alejandra Alvarez http://orcid.org/0000-0002-2221-2325

Isabel Hernández http://orcid.org/0000-0002-4595-5228

Additional information

Funding

The first author was supported by grant FOMIX-CONACYT YUC-2013-C14 - 221183 and Becas Iberoamérica de Jóvenes Profesores e Investigadores, Santander Universidades. The second author was supported by grants FOMIX-CONACYT YUC-2013-C14 - 221183 and 222870. The first author expresses her gratitude to Universidad Autónoma de Yucatán and Centro de Investigación en Matemáticas – Unidad Mérida for their hospitality during her research stays in both centres.