On capability and the Schur multiplier of some nilpotent Lie superalgebras

Abstract

A Lie superalgebra H satisfying $H^2=Z\left(H\right)$ is called generalized Heisenberg Lie superalgebra. If $d\left(H\right):=(m\mid n)$ is the minimum number of generators required to describe H, then in this article we intend to find the structure of H when precisely $\dim \left(H^2\right)=\frac{1}{2}\left[\right(m+n{)}^2+(n-m)]$. Further, we give some results about the capability, and the Schur multiplier of H. Moreover, we find multiplier $\mathcal{M}\left(L\right)$ for any nilpotent Lie superalgebra L of nilpotency class 2 with $d\left(L\right)=(r\mid s)$ when its derived subalgebra is of maximum possible dimension and finally show that such an algebra is capable.

Keywords:

• Generalized Heisenberg Lie superalgebra
• multiplier
• capable
• nilpotent

2010 Mathematics subject classifications:

• Primary 17B55
• Secondary 17B05

Acknowledgments

We would like to thank K. C. Pati for his encouragement. We are also grateful to the referee for his/her helpful corrections which improved the presentation of this paper. S. Nayak is supported by NBHM Post Doctoral Fellowship, Govt. of India.

Disclosure statement

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

Additional information

Funding

S. Nayak is supported by NBHM Post Doctoral Fellowship, Govt. of India.