Detecting capable Lie superalgebras

Abstract

In this article, we show that distributive law holds for non-abelian tensor product of Lie superalgebras under certain direct sums. Thereby we obtain a rule for non-abelian exterior square of a Lie superalgebra. We define capable Lie superalgebra and then give some characterizations. Specifically, we prove that epicenter of a Lie superalgebra is equal to exterior center. Finally, we classify all capable Lie superalgebras whose derived subalgebra dimension is at most one. As an application to those results, we have shown that there exists at least one non-abelian nilpotent capable Lie superalgebra each of dimension (m/n) when $m+n\ge 3.$

Keywords:

• Heisenberg Lie superalgebra
• multiplier
• capability
• non-abelian tensor and exterior product

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 17B30
• Secondary: 17B05

Acknowledgements

We are grateful to the referee for his/her helpful corrections which improved the presentation of this paper.

Disclosure statement

No potential conflict of interest is reported by the author(s).

Additional information

Funding

Saudamini Nayak research is supported by National Board for Higher Mathematics (NBHM) Post Doctoral Fellowship, Govt. of India.