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 Interactions of quantum affine algebras with cluster algebras, current algebras and categorification

Schedule

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June 2 - June 4: Summer School

June 5 - June 8: Conference

June 6: Reception and Banquet dinner

Summer School

Alistair Savage (University of Ottawa, Canada)

String diagrams and categorification

Diagrammatic methods have come to play a vital role in the modern theory of categorification. The goal of this mini-course is to introduce students to these methods. We will discuss the string diagram formalism for monoidal categories, where morphisms are represented by planar diagrams, and will see how the concept of adjunction is extremely natural from this point of view. We will then discuss applications to categorification, emphasizing the importance of pivotal categories, where one has a natural topological notion of duals.

Bernard Leclerc (Université de Caen, France)

Quantum affine algebras and cluster algebras

Ten years ago, it was observed that cluster algebras play a role in the representation theory of quantum affine algebras. Certain monoidal categories of finite-dimensional representations were shown to have a Grothendieck ring with a natural structure of a cluster algebra of finite type $A$, $D$, $E$. Since then, cluster algebras have developed into a new combinatorial tool for studying finite-dimensional simple modules over quantum affine algebras. For instance the special class of real simple modules, which conjecturally correspond to cluster monomials, have been shown to enjoy interesting properties. This minicourse will provide an introduction to these ideas, following the works of many people: Hernandez-Leclerc, Nakajima, Kimura-Qin, Kang-Kashiwara-Kim-Oh, Qin, Brito-Chari-Moura, ...

Abstracts of talks

Aaron Lauda (University of Southern California, USA)

Current Algebras and Categorification

Many important algebraic objects such as quantum groups, Hecke algebras, and Heisenberg algebras admit diagrammatically defined categorifications. These categorifications are certain monoidal categories defined by planar diagrams modulo local relations. Typically, decategorification is the operation of reducing a monoidal category to an algebra by taking the Grothendieck ring of the category. However, the diagrammatic nature of these categorifications suggests an alternative decategorification, the 'trace', which can be thought of as the result of restricting to diagrams that live on an annulus. In this talk we will explain how the theory of categorified quantum groups recovers the current algebra and some of its representations via its trace decategorification. A basic consequence of this theory is that any 2-representation of the categorified quantum group automatically gives a representation of the current algebra on a part of the 2-representation.

Yiqiang Li (University of Buffalo, USA)

Quiver varieties and symmetric pairs

To an $ADE$ Dynkin diagram, one can attach a simply-laced complex simple Lie algebra, say $\mathfrak g$, and a class of Nakajima's quiver varieties. The latter provides a natural home for a geometric representation theory of the former. If the algebra $\mathfrak g$ is further equipped with an involution, it leads to a so-called symmetric pair $(\mathfrak g,\mathfrak k)$ where $\mathfrak k$ is the fixed-point subalgebra under the involution. In this talk, I’ll present bridges at several levels between symmetric pairs and Nakajima varieties.

 Funded by the National Science Foundation Partially supported by the European Research Council with the ERC Grant "Qaffine"