Deformations and homotopy theory of Nijenhuis associative algebras

TL;DR

This paper develops an operadic and homotopical framework for Nijenhuis associative algebras, introducing new concepts, models, and cohomology theories to understand their deformations and structures.

Contribution

It introduces homotopy Nijenhuis associative algebras, constructs their minimal models, and links their deformations to an $L_$-algebra controlling Nijenhuis structures.

Findings

Constructed the minimal model of Nijenhuis associative operad.

Established a cohomology theory for Nijenhuis associative algebras.

Connected homotopy Nijenhuis algebras with Rota-Baxter structures.

Abstract

This paper is the first in a series of works devoted to an operadic study of Nijenhuis structures, focusing on Nijenhuis associative algebras. We introduce the concept of homotopy Nijenhuis associative algebras and demonstrate that the differential graded (=dg) operad $\NjAoperad_{\infty}$ governing these structures serves as the minimal model of the operad $\NjAoperad$ for Nijenhuis associative algebras. Additionally, we determine the Koszul dual homotopy cooperad of $\NjAoperad$. We construct an $L_\infty$-algebra that controls the simultaneous deformations of associative products and Nijenhuis operators. The Maurer-Cartan elements of this $L_\infty$-algebra correspond bijectively to Nijenhuis associative algebra structures. From this, we derive a cochain complex (deformation complex) and an associated cohomology theory of Nijenhuis associative algebras. Finally, we explore the connection between homotopy relative Rota-Baxter associative algebras of weight $0$ and homotopy Nijenhuis associative algebras. A sequel to this work will extend the study to Nijenhuis Lie algebras, with applications to Nijenhuis geometry.