The minimal model of Rota-Baxter operad with arbitrary weight

TL;DR

This paper constructs a minimal model for the Rota-Baxter operad with arbitrary weights, introducing homotopy Rota-Baxter algebras and analyzing their deformation theory from an operadic perspective.

Contribution

It explicitly constructs the minimal model of the Rota-Baxter operad with arbitrary weights and defines homotopy Rota-Baxter algebras, advancing the operadic understanding of these structures.

Findings

Constructed a homotopy cooperad as the Koszul dual of the Rota-Baxter operad.

Proved the cobar construction yields the minimal model of the Rota-Baxter operad.

Described the deformation complex and $L_$-algebra structure of Rota-Baxter algebras.

Abstract

This paper investigates Rota-Baxter associative algebras of of arbitrary weights, that is, associative algebras endowed with Rota-Baxter operators of arbitrary weights from an operadic viewpoint. Denote by $\RB$ the operad of Rota-Baxter associative algebras. A homotopy cooperad is explicitly constructed, which can be seen as the Koszul dual of $\RB$ as it is proven that the cobar construction of this homotopy cooperad is exactly the minimal model of $\RB$. This enables us to give the notion of homotopy Rota-Baxter associative algebras. The deformation complex of a Rota-Baxter associative algebra and the underlying $L_\infty$-algebra structure over it are exhibited as well.