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

TL;DR

This paper establishes an upper bound for the dimension of the non-abelian tensor product of finite-dimensional Lie superalgebras, providing explicit conditions for equality in certain cases.

Contribution

It introduces a new upper bound for the non-abelian tensor product dimension of Lie superalgebras and characterizes when equality occurs.

Findings

Upper bound for tensor product dimension derived

Explicit conditions for equality when r=1, s=0

Results applicable to finite-dimensional nilpotent Lie superalgebras

Abstract

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) \leq (k+l-(r+s))(k+l-1)+2$. We discuss the conditions when the equality holds for $r=1, s=0$ explicitly.