On the dimension of non-abelian tensor square of Lie superalgebras

In this paper, we determine upper bound for the non-abelian tensor product of finite dimensional Lie superalgebra. More precisely, if $L$ is a non-abelian nilpotent Lie superalgebra of dimension $(k\mid l)$ and its derived subalgebra has dimension $(r\mid s)$, then $\dim (L\otimes L)\le (k+l-(r+s))(k+l-1)+2$. We discuss the conditions when the equality holds for $r=1,s=0$ explicitly.