Cohomologies and deformations of coassociative coderivations

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.

Keywords:

• Coassociative coalgebras
• coderivations
• cohomologies
• defomations

2020 Mathematics Subject Classification:

• 13D10
• 16T15

Acknowledgments

The authors would like to thank the anonymous referee for his/her helpful comments. In particular, the interesting question in Remark 3.1 were raised by him/her.

Additional information

Funding

Y.-H. Bao was supported by NSFC (Grant No. 11871071).