Multipliers of nilpotent Lie superalgebras

Abstract

In this article, first we prove that all finite dimensional special Heisenberg Lie superalgebras with even center have dimension $(2m+1|n)$ for some non-negative integers m, n and are isomorphic to H(m, n). Further, for a nilpotent Lie superalgebra L of dimension $(m|n)$ and $\dim L\prime =(r|s)$ with $r\ge 1$, we find the upper bound $\dim \mathcal{M}(L)\le \frac{1}{2}\left[(m+n+r+s-2)(m+n-r-s-1)\right]+n+1$, where $\mathcal{M}(L)$ denotes the Schur multiplier of L. Moreover, if $\dim L\prime =1$, then the equality holds if and only if $L\cong H(1,0)+A_1,$ where A₁ is an abelian Lie superalgebra with $\dim A_1=(m-3|n)$ and $H(1,0),$ is a special Heisenberg Lie superalgebra of dimension $(3|0)$.

Keywords:

• Nilpotent Lie superalgebra
• multiplier
• stem extension
• special Heisenberg Lie superalgebra

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 17B30
• Secondary 17B05

Acknowledgements

The author would like to thank the referee for the useful suggestions. The author would also like to thank Dr. Sudhansu Sekhar Rout for all the fruitful discussions during preparation of the manuscript.

Additional information

Funding

Department of Atomic Energy, Government of India.