Cohomology and deformation theory of Reynolds--Nijenhuis associative algebras

TL;DR

This paper introduces Reynolds--Nijenhuis operators on associative algebras, explores their properties, cohomology, and deformation theory, providing a new framework for understanding hybrid algebraic structures.

Contribution

It defines Reynolds--Nijenhuis operators, develops their cohomology, and establishes a formal deformation theory for these associative algebras.

Findings

Reynolds--Nijenhuis operators unify Reynolds and Nijenhuis structures.

A cohomology theory for Reynolds--Nijenhuis algebras is constructed.

Deformation theory reveals conditions for rigidity and equivalence.

Abstract

In this paper, we introduce and study Reynolds--Nijenhuis operators on associative algebras a novel hybrid structure that simultaneously satisfies the defining identities of both Reynolds and Nijenhuis operators. We investigate their connections with Rota-Baxter and modified Rota-Baxter operators. We develop a representation theory for Reynolds--Nijenhuis associative algebras and introduce a corresponding cohomology theory. Furthermore, we establish a one-parameter formal deformation theory for these algebras, examining the role of infinitesimals, rigidity, and equivalence in the context of deformations.