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 $\dim \mathcal{M}(L)\le {\sum }_{i=0}^n\left(\genfrac{}{}{0ex}{}{m}{i}\right)\mathcal{L}(n-i,k)$, where $\dim L=(m|k)$, $\mathcal{L}(0,k)=1$ and $\mathcal{L}(t,k)={\sum }_{j=1}^t\left(\genfrac{}{}{0ex}{}{t-1}{j-1}\right)\left(\genfrac{}{}{0ex}{}{k}{j}\right)$, for $1\le t\le n$. 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.

References in corpus (3)

• Cohomology of Heisenberg Lie Superalgebras
• Classification of nilpotent Lie superalgebras of multiplier-rank $\le 2$
• On multipliers of pairs of Lie superalgebras