Deformations theory and minimal model of operads for Nijenhuis algebras morphisms

TL;DR

This paper develops a new cohomology framework for Nijenhuis algebra morphisms, linking deformation theory with operad minimal models, and provides a comparison theorem for their cohomologies.

Contribution

It introduces a cohomology theory for Nijenhuis algebra morphisms, including a comparison theorem and a minimal operad model, advancing deformation theory in this area.

Findings

Defined a cohomology theory for Nijenhuis algebra morphisms.

Proved a cohomology comparison theorem for these morphisms.

Constructed a minimal model for the operad of Nijenhuis algebra morphisms.

Abstract

Nijenhuis operators are very useful in the deformation theory of algebras. In this paper, we introduce a new cohomology theory related to deformation of Nijenhuis algebra morphisms, this notion involves simultaneous deformation of two Nijenhuis algebras and a morphism between them. As a consequence, we define a cohomology theory of Nijenhuis algebra morphisms to interpret the lower degree cohomology groups as formal deformation. We also prove a cohomology comparison Theorem of Nijenhuis algebra morphisms, i.e. the cohomology of a morphism of Nijenhuis algebras is isomorphic to the cohomology of an auxiliary Nijenhuis algebra. Finally, we construct a minimal model for the operad governing Nijenhuis algebras morphisms.