题    目：Vertex algebras and coalgebras over the representation category of a diagonalizable algebraic group

主讲人：林宗柱 教授

单    位：美国堪萨斯州立大学

时    间：2025年6月12日 11：00

地    点：学院二楼会议室

摘    要：As part of our effort to define vertex algebras over symmetric tensor categories, in this talk we will define vertex coalgebras. The test category is the category of rational representations of a diagonalizable algebraic group. In this category, every object is graded by the character group. This is a symmetric monoidal category in which dualizable objects and compact objects are different. We will define the vertex algebras and vertex coalgebras over this category and discuss the duality between them as well as their representations. We also define and establish the corresponding $C_2$-algebras for vertex algebras and $C_2$-coalgebras which turn out to be co-Poisson co-algebra. The representation theory is very much parallel to the comodules for the coordinate algebras of an algebraic group and the modules for the algebra of distributions of an affine algebraic group. Although not considered in the paper, it is natural to define the corresponding vertex algebras and vertex coalgebras in the category of differential complexes, which will be part of our effort to define derived vertex algebras and vertex co-algebras in the frame of Lurie using homotopy theory. Using the category of representation category of the diagonalizable group also serves as a test case for defining vertex algebras over an algebraic stack. This paper is a joint work with Antoine Caradot.

简    介：美国Kansas州立大学教授，博士生导师，主要从事代数表示论的研究，在美国麻省大学师从李代数名师Humphrey教授。曾在美国国家自然科学基金会担任职务，是活跃在代数群、量子群表示、顶点算子代数理论研究领域的知名专家学者。同时也是Ameri.Math Soc（美国数学学会会员）、美国《数学期刊》评论家、德国《数学文摘》期刊评论家。Invent. Math.，Adv. Math., Trans. Amer. Math. Soc., CMP 和J. Algebra 等重要学术期刊上发表论文数十篇，标志性成果包括林-Nakano定理，出版学术著作五部，并组织过多次国际会议。