The Schur multiplier of an $n$-Lie superalgebra

Abstract

In the present paper, we study the notion of the Schur multiplier $\mathcal{M}(L)$ of an $n$-Lie superalgebra $L$, and prove that $\dim \mathcal{M}(L) \leq \sum_{i=0}^{n} {m\choose{i}} \mathcal{L}(n-i,k)$, where $\dim L=(m|k)$, $\mathcal{L}(0,k)=1$ and $\mathcal{L}(t,k) = \sum_{j=1}^{t}{{t-1}\choose{j-1}} {k\choose j}$, for $1\leq t\leq n$. Moreover, we obtain an upper bound for the dimension of $\mathcal{M}(L)$ in which $L$ is a nilpotent $n$-Lie superalgebra with one-dimensional derived superalgebra. It is also provided several inequalities on $\dim\mathcal{M}(L)$ as well as an $n$-Lie superalgebra analogue of the converse of Schur's theorem.