Characterization and Cohomology of Crossed Homomorphisms on Lie Superalgebras
RB Yadav, Arpan Sharma
TL;DR
The paper develops a Maurer-Cartan framework for crossed homomorphisms on Lie superalgebras by embedding LieSupAct triples into a $ obreak ext{Z} imes ext{Z}_2$-graded Lie algebra and identifying the MC element $igl( ext{pi}+ ho+ extmuigr)$. It constructs a cohomology theory for LieSupAct triples via a differential graded Lie algebra and defines a corresponding cohomology $H^*( rak g, rak h, ho)$. For crossed homomorphisms, it provides a parallel MC-based characterization within a dedicated cochain complex $(C^*( rak g, rak h),oxed{ rbracketullet,ullet rbracket},oxed{ ext{partial}_{ ext{pi}+ ho}})$ and develops a twisted differential $d_D$ giving the cohomology $H^*(D)$. The deformation theory is then developed: infinitesimals of deformations are 1-cocycles, linear deformations are controlled by these cocycles, and analogous results hold for crossed homomorphisms via the $d_D$-cohomology, enabling systematic study of obstructions and deformations in the Lie superalgebra context.
Abstract
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.