David Hernandez (mini-course)
Title : Quantum loop algebras and monoidal categorification of cluster algebras
Abstract : The representation theory of quantum loop algebras is remarkably rich and has been explored from a variety of perspectives. In particular, finite-dimensional representations can be studied through the lens of cluster algebras, remarkable commutative algebras endowed with distinguished sets of generators produced by inductive processes. The aim of these lectures is to explain this connection and to present some recent developments, especially in the context of shifted quantum groups.
Sasha Tsymbaliuk (mini-course)
Title : Shuffle algebras and combinatorics of words
Abstract : We shall discuss the shuffle approach to various quantum loop groups and applications. This approach goes back to the works of Feigin-Odesskii and Enriquez more than 2 decades ago, generalizing the classical shuffle approach of Green and Rosso to usual quantum groups. However, the new machinery that allows to establish long-thought results was only recently developed in the works of Negut, highlighting the importance of slope subalgebras and utilizing the discrete combinatorics of words.
Alexandr Garbali
Title : Shuffle algebras and the Yang—Baxter equation
Abstract : In this talk I will describe a mechanism for constructing commuting shuffle algebras from R-matrices which are solutions of the Yang—Baxter equation. I will then discuss several examples in which I will recover shuffle products and commuting elements of Feigin—Odesskii shuffle algebras and some of their generalizations.
Bernard Leclerc
Title : Schemes of bands and categories of modules over (shifted) quantum affine algebras
Abstract : The Grothendieck rings of several categories of modules over (shifted) quantum affine algebras have been shown to carry the structure of a cluster algebra of infinite rank. In this talk I will present some infinite-dimensional affine schemes, whose rational points are called bands, and I will show that their coordinate rings exhibit the same cluster algebra structures. I will explain how this yields a proof of a conjecture of E. Frenkel and N. Reshetikhin describing the q-character homomorphism as a q-difference Miura transformation for formal loop groups. This is joint work with Luca Francone.