Cohomology and deformations of dendriform coalgebras

TL;DR

This paper develops a cohomology theory for dendriform coalgebras, explores their deformations, relates it to existing cohomologies, and introduces homotopy analogues to deepen understanding of their algebraic structure.

Contribution

It introduces a new cohomology framework for dendriform coalgebras, links it with related cohomologies, and defines homotopy versions, advancing the theoretical understanding of these structures.

Findings

Cohomology governs formal deformations of dendriform coalgebras.

Established relations between different cohomology theories.

Introduced homotopy dendriform coalgebras and studied their properties.

Abstract

Dendriform coalgebras are the dual notion of dendriform algebras and are splitting of associative coalgebras. In this paper, we define a cohomology theory for dendriform coalgebras based on some combinatorial maps. We show that the cohomology with self coefficients governs the formal deformation of the structure. We also relate this cohomology with the cohomology of dendriform algebras, coHochschild (Cartier) cohomology of associative coalgebras and cohomology of Rota-Baxter coalgebras which we introduce in this paper. Finally, using those combinatorial maps, we introduce homotopy analogue of dendriform coalgebras and study some of their properties.