Title: Deformation theory and Koszul duality for Rota-Baxter systems Show Chinese title

Title: 变形理论与Rota-Baxter系统之间的Koszul对偶性

Abstract: This paper investigates Rota-Baxter systems in the sense of Brzezi\'nski 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. Show Chinese abstract

Abstract: 本文从操作理论的角度研究了Brzeziński意义上的Rota-Baxter系统。 构造了Rota-Baxter系统操作的最小模型，等价地给出了其Koszul对偶同伦协操作的显式构造。 从Koszul对偶同伦协操作中推导出了同伦Rota-Baxter系统的概念以及控制Rota-Baxter系统变形的$L_\infty$-代数。 引入了infinity-Yang-Baxter对的概念，这是传统Yang-Baxter对的高阶推广。 表明在分次空间的自同态代数上的同伦Rota-Baxter系统结构等价于该分次代数上的结合infinity-Yang-Baxter对，从而推广了经典Yang-Baxter对与Rota-Baxter系统之间的对应关系。