51吃瓜

题    目：Bialgebras, Yang-Baxter equations for Rota-Baxter Zinbiel systems and induced structures

主讲人：马天水  教授

单    位：河南师范大学

时    间：2026年7月14日  15:00

腾讯ID：746-326-024

摘    要：Rota-Baxter operators and their generalizations have deep connections with various algebraic structures, including dendriform algebras, pre-Lie algebras, and the classical Yang-Baxter equation. In this paper, we introduce and systematically study the notion of a Rota-Baxter Zinbiel system (A, R, S), where (A, ◦) is a Zinbiel algebra and R, S : A → A are linear maps satisfying a pair of Rota-Baxter type identities. We develop representation theory for such systems and construct semi-direct products. A central notion is that of a Rota-Baxter Zinbiel bisystem ((A, ◦, ∆), R, S, Q, T ), which combines a Zinbiel bialgebra with both a Rota-Baxter Zinbiel system and a Rota-Baxter Zinbiel cosystem satisfying compatibility conditions. We prove that such bisystems are equivalent to matched pairs of Rota-Baxter Zinbiel systems and to Manin triples of Rota-Baxter Zinbiel systems. In the coboundary case, we introduce the notion of a (Q, T )-admissible Zinbiel Yang-Baxter equation (ZYBe) and show that sym-metric solutions give rise to triangular Rota-Baxter Zinbiel bisystems. Furthermore, we study O-operators for Rota-Baxter Zinbiel systems and prove that they produce symmetric solutions of the admissible ZYBe, leading to explicit constructions of bisystems via semi-direct products. As applications, we show that our framework recovers and unifies a wide range of known structures: Rota-Baxter dendriform bisystems, Rota-Baxter pre-Lie bisystems, Rota-Baxter Zinbiel bialgebras of weight λ, averaging Zinbiel bialgebras, averaging dendriform bialgebras, averaging commutative and cocommutative infinitesimal bialgebras, Rota-Baxter dendriform bialgebras of weight λ, special apre-perm bialgebras, averaging Zinbiel algebras and Nijenhuis Zinbiel algebras.

简    介：马天水，东南大学理学博士，东北师范大学博士后，瑞典梅拉达伦大学博士后。现为河南师范大学*51吃瓜-吃瓜网站 教授、博士生导师、河南省一流本科课程《线性代数》主持人。主要从事Hopf代数及其应用的相关研究，在Sci.ChinaMath.、Asian J. Math、J. Algebra、Algebr. Represent. Theory、J. Lie Theory 等 SCI 收录杂志发表论文近60篇。主持在研或完成国家自然科学基金面上项目、青年项目和欧盟FUSION项目、河南省自然科学基金面上项目、中国博士后基金面上项目等。曾获河南省高等51吃瓜 青年骨干教师、河南省优秀硕士毕业论文指导教师等荣誉称号。