Deformations of Courant pairs and Poisson algebras

TL;DR

This paper investigates the deformation theory of Courant pairs with a focus on cohomology, universal infinitesimal deformations, and obstructions, culminating in explicit computations for Poisson algebra structures.

Contribution

It introduces a bi-complex for deformation cohomology, describes universal infinitesimal deformations, and extends deformation theory to include versal deformations for Courant pairs.

Findings

Defined deformation cohomology bi-complex for Courant pairs

Constructed universal infinitesimal deformation explicitly

Computed universal infinitesimal deformation for Poisson algebras

Abstract

We study deformation of Courant pairs with a commutative algebra base. We consider the deformation cohomology bi-complex and describe a universal infinitesimal deformation. In a sequel, we formulate an extension of a given deformation of a Courant pair to another with extended base. This leads to describe the obstruction in extending a given deformation. We also discuss about the construction of versal deformation of Courant pairs. As an application, we explicitly compute universal infinitesimal deformation of Poisson algebra structures