On the triple tensor product of generalized Heisenberg Lie superalgebra of rank $\leq2$

TL;DR

This paper investigates the structure and properties of the triple tensor and exterior powers of generalized Heisenberg Lie superalgebras of rank at most 2, providing explicit calculations and bounds for their dimensions.

Contribution

It computes the Schur multiplier and describes the structure of tensor and exterior powers for these superalgebras, establishing a dimension bound for the tensor power of certain nilpotent Lie superalgebras.

Findings

Explicit structure of $ ensor^3H$ and $igwedge^3H$ for rank $ extless= 2$

Dimension bounds for tensor powers of nilpotent Lie superalgebras

Characterization of when equality holds for the dimension bound

Abstract

In this article, we compute the Schur multiplier of all generalized Heisenberg Lie superalgebras of rank $2$. We discuss the structure of $\otimes^3H$ and $\wedge^3H$ where $H$ is a generalized Heisenberg Lie superalgebra of rank $\leq2$. Moreover, we prove that if $L$ is an $(m\mid n)$-dimensional non-abelian nilpotent Lie superalgebra with derived subalgebra of dimension $(r\mid s)$, then $\dim\otimes^3L \leq (m+n)(m+n - (r+s))^2$. In particular, for $r=1,s=0$ the equality holds if and only if $L \cong H(1\mid 0)$.