数学学科2020系列学术报告十五——十七

报告人：宋林亮(助理教授)，同济大学

题目: Representations of cyclotomic Brauer categories and Kauffman categories

时间：2020/11/4, 8:30-9:20，腾讯会议：会议号：761158941，密码：201104

摘要：Brauer category and Kauffman category are category versions of Brauer algebras and BMW algebras, respectively.  Let $A$ be the locally unital algebra associated to (cyclotomic) Brauer categories or Kauffman categories. We prove that the category $A$-lfdmod of locally finite dimensional left $A$-modules is an upper finite fully stratified category in the sense of Brundan-Stroppel.  It is an upper finite highest weight category if the (degenerate) cyclotomic Hecke algebras are semisimple. Moreover, $A$ is semisimple if and only if its centralizer subalgebras associated to certain idempotent elements are semisimple.  Furthermore, certain endofunctors are defined and give categorical actions of some Lie algebras on the subcategory of $A$-lfdmod consisting of all objects which have a finite standard filtration. This leads to categorifications of certain representations of the classical limits of coideal algebras. This talk is based on joint works with Hebing Rui, Mengmeng Gao.

报告人：黎允楠(副教授)，广州大学

Title: Construction of free differential algebras by extending Grobner-Shirshov bases

时间：2020/11/4, 9:30-10:20，腾讯会议：会议号：761158941，密码：201104

摘要：As a fundamental notion, the free differential algebra on a set is concretely constructed as the polynomial algebra on the differential variables. Such a construction is not known for the more general notion of the free differential algebra on an algebra, from the left adjoint functor of the forgetful functor from differential algebras to algebras, instead of sets. In this talk we show that generator-relation properties of a base algebra can be extended to the free differential algebra on this algebra. More precisely, Gr¨obner-Shirshov basis property of the base algebra can be extended to provide a Poincar´e-BirkhoffWitt type basis for these more general free differential algebras. Examples are given as illustrations. This is joint work with Li Guo.

报告人：胡峻（教授），北京理工大学

题目: Crystal of affine type A and modular branching rules for Hecke algebras of types B_n and D_n

时间：2020/11/4 10:30-11:20   腾讯会议：会议号：761158941，密码：201104

摘要：Crystal bases of integral highest weight modules over symmetrizable Kac-Moody algebras were introduced by Kashiwara and have found many applications in representation theory. Earlier work of Ariki and Grojnowski shows that the crystal of type A_{e-1}^{(1)} controls the modular representation theory of type A affine Hecke algebras and their cyclotomic quotients at a primitive e-th root of unity. In this talk I shall explain a similar connection between the crystal of type A_{/ell-1}^{(1)} with the Hecke algebras of types B_n and D_n at a primitive (2/ell)-th root of unity.