On the triple tensor product of generalized Heisenberg Lie superalgebra of rank $\le 2$

In this article, we compute the Schur multiplier of all generalized Heisenberg Lie superalgebras of rank $2$. We discuss the structure of ${\otimes }^{3}H$ and ${\wedge }^{3}H$ where $H$ is a generalized Heisenberg Lie superalgebra of rank $\le 2$. 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 $\dim {\otimes }^{3}L\le (m+n)(m+n-(r+s){)}^2$. In particular, for $r=1,s=0$ the equality holds if and only if $L\cong H(1\mid 0)$.