Crossed Homomorphisms on associative superalgebras

Abstract

In this paper, we introduce the notion of crossed homomorphisms on associative superalgebras. We show that crossed homomorphisms are precisely the Maurer--Cartan elements of a naturally associated graded Lie algebra. This characterization enables us to construct a cohomology theory of crossed homomorphisms. We then investigate formal deformations of crossed homomorphisms and derive the corresponding deformation equations. It is proved that the infinitesimal part of a formal deformation is a 1-cocycle in the associated cohomology, while the obstruction to extending a deformation of finite order is a 2-cocycle.