Mini-courses

    Schedule:

    Fujita's courses will be given online via ZOOM. The ZOOM ID is 841 1245 6417, and the password will be provided elsewhere.

    Abstract:

  "Quantum affine algebras and quantum Grothendieck rings",  by Ryo Fujita (RIMS, Kyoto University).

  In this mini-course, we consider the (untwisted) quantum affine algebra associated with a complex simple Lie algebra, and discuss its monoidal category of finite-dimensional modules. Analogous to the classical highest weight theory, simple modules in this category are classified by the so-called highest l-weights. However, determining their characters (or dimensions) from general highest l-weights is an open problem at this moment.
  The quantum Grothendieck ring and its canonical basis are introduced to formulate the Kazhdan-Lusztig type approach to this character problem. For the simply-laced types (ADE), the canonical basis is known to enjoy some nice properties thanks to the geometric theory of Nakajima quiver varieties. Although such a geometric theory is not applicable for non-simply-laced types (BCFG), we can still verify the same properties of the canonical basis via an isomorphism of cluster theoretical nature between the quantum Grothendieck ring of non-simply-laced type and that of "unfolded" simply-laced type.
  In this mini-course, I will explain these stories, based on my collaboration with David Hernandez, Se-jin Oh, and Hironori Oya.

  Lecture: Lecture 1 (Fujita), Lecture 2 (Fujita), Lecture 3 (Fujita).

  "Representations of quiver Hecke algebras and generalized quantum affine Schur-Weyl duality", by Myungho Kim (Kyung Hee University).

  Quiver Hecke algebra is a family of graded associative algebras which categorifies the half of quantum groups. More precisely, the category of finite-dimensional (graded) modules over a quiver Hecke algebra is a monoidal category whose Grothendieck ring is isomorphic to the (dual) integral form of a half of a quantum group.
  Interestingly, the category of finite-dimensional modules on a quiver Hecke algebra is very similar to the category of finite-dimensional modules on a quantum affine algebra. Indeed, there are functors between these categories, called generalized quantum affine Schur-Weyl dualities, which exhibit these similarities.
  In this course, we will review the categories of finite-dimensional modules over quiver Hecke algebras and those over quantum affine algebras. It will be explained how to construct functors from the category of a quiver Hecke algebra modules to that of quantum affine algebra modules. Some related topics, such as the monoidal categorification of cluster algebras will also be presented.

  Lecture: Lecture 1 (Kim), Lecture 2 (Kim), Lecture 3 (Kim).

  "Hall algebras and i-quantum groups",  by Ming Lu (Sichuan University).

   The i-quantum groups arising from quantum symmetric pairs can be viewed as a natural generalization of Drinfeld-Jimbo quantum groups. As suggested by Bao-Wang, most of the fundamental constructions in the theory of quantum groups should admit generalizations in the setting of i-quantum groups.
  This mini-course addresses the Hall algebra realization of i-quantum groups. As quantum groups can be viewed as i-quantum groups, this construction includes a reformulation of Bridgeland's Hall algebra realization for the whole quantum group, and Bridgeland's construction is in turn built on Ringel's Hall algebra realization for halves of quantum groups.  
  To that end, we introduce a class of finite-dimensional algebras called i-quiver algebras, and develop its representation theory. We shall also explain the i-divided powers, Serre presentation, braid group symmetries for i-quantum groups in the setting of Hall algebras.
  This mini-course should be accessible to graduate students with background in Lie theory and representation theory. Some of its preliminaries are given by Weiqiang Wang's mini-course.

  Lecture: Lecture 1-3 (Lu).

  "Introduction to i-quantum groups",  by Weiqiang Wang (University of Virginia).

   In this mini-course, we introduce i-quantum groups arising from quantum symmetric pairs and explain why it is natural to view it as a generalization of Drinfeld-Jimbo quantum groups. We introduce i-divided powers and show how they lead to iSerre relations and Serre presentations for (quasi-)split i-quantum groups. 
  We then establish an iSchur duality between i-quantum group of type AIII and Hecke algebra of type B acting on some q-tensor space. This is a generalization of Schur-Jimbo duality between the quantum group and Hecke algebra of type A. We explain how to study Kazhdan-Lusztig (KL) bases of classical types in such settings. 
  We construct a new bar involution and i-canonical basis for any integrable module over quantum group (viewed as modules over i-quantum groups); i-divided powers are  i-canonical basis elements on the i-quantum group of split rank one. In the setting of iSchur duality, the i-canonical basis on the q-tensor space coincides with KL basis of type B/D. (A generalization of this i-canonical basis leads to KL theory for super BGG category.)
  This mini-course should be accessible to graduate students with background in Lie theory.

  Lecture: Lecture 1 (Wang), Lecture 2 (Wang),  Lecture 3 (Wang).