The Schur multiplier of an n-Lie superalgebra

Abstract

In the present paper, we study the notion of the Schur multiplier $\mathcal{M}\left(L\right)$ of an n-Lie superalgebra $L=L_0\oplus L_1$ and prove that $\mathrm{dim\hspace{0.17em}}\mathcal{M}\left(L\right)\le {\sum }_{i=0}^n\left(\begin{array}{c}m\\ i\end{array}\right)\mathcal{L}(n-i,k),$ where $\mathrm{dim\hspace{0.17em}}{L}_0=m,\mathrm{dim\hspace{0.17em}}{L}_1=k,\mathcal{L}(0,k)=1$ and $\mathcal{L}(t,k)={\sum }_{j=1}^t\left(\begin{array}{c}t-1\\ j-1\end{array}\right)\left(\begin{array}{c}k\\ j\end{array}\right),$ for $1\le t\le n.$ Moreover, we obtain an upper bound for the dimension of $\mathcal{M}\left(L\right)$ in which L is a nilpotent n-Lie superalgebra with one-dimensional derived superalgebra. It is also provided several inequalities on $\mathrm{dim\hspace{0.17em}}\mathcal{M}\left(L\right)$ as well as an n-Lie superalgebra analogue of the converse of Schur’s theorem.

Keywords:

• n-Lie superalgebra
• Schur multiplier
• Schur’s theorem

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B30
• 17B55

Additional information

Funding

This research was in part supported by the University of Bojnord (Grant No. 99/367/11967).