Detecting capable Lie superalgebras

In this article, we show that distributive law holds for non-abelian tensor product of Lie superalgebras under certain direct sums. Thereby we obtain a rule for non-abelian exterior square of a Lie superalgebra. We define capable Lie superalgebra and then give some characterizations. Specifically, we prove that epicenter of a Lie superalgebra is equal to exterior center. Finally, we classify all capable Lie superalgebras whose derived subalgebra dimension is at most one. As an application to those results, we have shown that there exists at least one non-abelian nilpotent capable Lie superalgebra each of dimension (m/n) when m + n >= 3.