Twisted 풪-operator families on Leibniz algebras and NS-Leibniz family algebras.

In this paper, we first define twisted 풪-operator families on Leibniz algebras indexed by a semigroup Ω. Then we introduce and study NS-Leibniz family algebras as the underlying structures of twisted 풪-operator families. We show that an NS-Leibniz family algebra induces an ordinary NS-Leibniz algebra on the tensor product with the semigroup algebra. Finally, we investigate the cohomology of a twisted 풪-operator family. This cohomology can also be seen as the cohomology of a certain Ω-Leibniz algebra with coefficients in a suitable representation. As an application, we study the formal deformations of twisted 풪-operator families on Leibniz algebras. [ABSTRACT FROM AUTHOR]