Characterization and Cohomology of Crossed Homomorphisms on Lie Superalgebras

In this article, we give a characterisation of crossed homomorphisms on Lie superalgebras as a Maurer-Cartan element of a graded Lie algebra. Using this characterisation we study cohomology of these crossed homomorphisms. As an application of this cohomology we study formal deformation of crossed homomorphisms. We show that linear deformations of these homomorphisms are characterised by one cocycles.