From homotopy Rota-Baxter algebras to Pre-Calabi-Yau and homotopy double Poisson algebras

TL;DR

This paper explores the connections between homotopy Rota-Baxter algebras, pre-Calabi-Yau, and homotopy double Poisson algebras, introducing cyclic structures and demonstrating inheritance of algebraic properties through interactive pairs.

Contribution

It introduces cyclic homotopy Rota-Baxter algebras and shows how they induce pre-Calabi-Yau and homotopy double Poisson structures on related algebras.

Findings

Cyclic homotopy Rota-Baxter structures lead to pre-Calabi-Yau structures.

Modules over cyclic homotopy Rota-Baxter algebras admit homotopy double Lie algebra structures.

Construction of cyclic completion provides a method to generate cyclic homotopy Rota-Baxter algebras.

Abstract

In this paper, we investigate pre-Calabi-Yau algebras and homotopy double Poisson algebras arising from homotopy Rota-Baxter structures. We introduce the notion of cyclic homotopy Rota-Baxter algebras, a class of homotopy Rota-Baxter algebras endowed with additional cyclic symmetry, and present a construction of such structures via a process called cyclic completion. We further introduce the concept of interactive pairs, consisting of two differential graded algebras-designated as the acting algebra and the base algebra-interacting through compatible module structures. We prove that if the acting algebra carries a suitable cyclic homotopy Rota-Baxter structure, then the base algebra inherits a natural pre-Calabi-Yau structure. Using the correspondence established by Fernandez and Herscovich between pre-Calabi-Yau algebras and homotopy double Poisson algebras, we describe the resulting homotopy Poisson structure on the base algebra in terms of homotopy Rota-Baxter algebra structure. In particular, we show that a module over an ultracyclic (resp. cyclic) homotopy Rota-Baxter algebra admits a (resp. cyclic) homotopy double Lie algebra structure.