Cohomologies and deformations of coassociative coderivations

TL;DR

This paper develops a deformation theory for coderivations of coassociative coalgebras, introducing Coder pairs and their cohomology to analyze rigidity and extendability of deformations.

Contribution

It introduces the concept of Coder pairs, defines their cohomology, and establishes criteria for rigidity and extension of deformations in coassociative coalgebras.

Findings

Coder pairs are rigid if second cohomology is trivial.

Deformations can be extended if the third cohomology class vanishes.

A cohomology framework for deformation analysis of coderivations is constructed.

Abstract

The motivation of this paper is to construct a deformation theory of coderivations of coassociative coalgebras. We introduce a notion of a Coder pair, that is, a coassociative coalgebra with a coderivation. Then we define a proper corepresentation of a Coder pair and study the corresponding cohomology. Finally, we show that a Coder pair is rigid, provided that the second cohomology group is trivial and point out that a deformation of finite order is extensible to higher order deformations if the obstruction class, which is defined to be in the third cohomology group, is trivial.