Deformation theory and Koszul duality for Rota-Baxter systems

TL;DR

Bringing operad theory to Brzeziński's Rota-Baxter systems, the paper constructs the Koszul dual homotopy cooperad $\mathfrak{RBS}^\mathrm{!'}$ and shows that its cobar construction $\Omega(\mathfrak{RBS}^\mathrm{!'})$ is a minimal model for the RBS operad. It then defines homotopy Rota-Baxter systems and associative infinity-Yang-Baxter pairs, proving a precise equivalence between HRBS structures on $\mathrm{End}(V)$ and associative infinity-Yang-Baxter pairs on $\mathrm{End}(V)$. From the minimal model, it derives the controlling $L_\infty$-algebra $\frak{C}_{\mathsf{RBS}}(V)$ and identifies its deformation complex with the suspension of the cochain complex previously studied by Liu, Wang, and Yin. These results furnish a homotopical deformation framework for Rota-Baxter systems and extend the Yang-Baxter correspondence to an infinity-level setting.

Abstract

This paper investigates Rota-Baxter systems in the sense of Brzeziński from the perspective of operad theory. The minimal model of the Rota-Baxter system operad is constructed, equivalently a concrete construction of its Koszul dual homotopy cooperad is given. The concept of homotopy Rota-Baxter systems and the $L_\infty$-algebra that governs deformations of a Rota-Baxter system are derived from the Koszul dual homotopy cooperad. The notion of infinity-Yang-Baxter pairs is introduced, which is a higher-order generalization of the traditional Yang-Baxter pairs. It is shown that a homotopy Rota-Baxter system structure on the endomorphism algebra of a graded space is equivalent to an associative infinity-Yang-Baxter pair on this graded algebra, thereby generalizing the classical correspondence between Yang-Baxter pairs and Rota-Baxter systems.

Figures (1)

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Theorems & Definitions (40)

• Lemma 1.1
• proof
• Definition 1.2
• Definition 1.3
• Proposition 1.4
• proof
• Theorem 1.5
• proof
• Proposition 2.1
• Definition 2.2