On multipliers of pairs of Lie superalgebras

TL;DR

This paper investigates the Schur multiplier of pairs of Lie superalgebras, providing upper bounds on their dimensions and characterizing specific finite-dimensional nilpotent pairs with precise multiplier dimensions.

Contribution

It introduces new bounds on the Schur multiplier of Lie superalgebra pairs and characterizes cases where the dimension attains a specific formula.

Findings

Derived upper bounds for the dimension of the Schur multiplier.

Characterized pairs of finite-dimensional nilpotent Lie superalgebras with specific multiplier dimensions.

Provided explicit formulas for special cases with t=0,1.

Abstract

In this article, we study the notion of the Schur multiplier $\mathcal{M}(N,L)$ of a pair $(N,L)$ of Lie superalgebras and obtain some upper bounds concerning dimensions. Moreover, we characterize the pairs of finite dimensional (nilpotent) Lie superalgebras for which $\dim \mathcal{M}(N,L)= \frac{1}{2}\big{(}(m+n)^2+(n-m)\big{)}+\dim N\dim(L/N)-t$, for $t=0,1$, where $\dim N=(m|n)$.