Mikhail Bershtein
Title : Free Field Realizations of Deformed W-Algebras and Cluster Algebras
Abstract : There is an interesting analogy between cluster algebras and deformed W-algebras. Many important notions from the cluster and q-W worlds, such as mutations, global functions, screening operators, and R-matrices, emerge naturally in this context. I will illustrate this analogy with examples of regular and subregular W-algebras in type A. Based on joint work with J. E. Bourgine and E. Fursman.
Benjamin Enriquez
Title : Shuffle algebras and multiple zeta values
Abstract : We will review the construction, due to Chen, of homotopy-invariant iterated integrals on manifolds based on the cohomology of the "bar-complex" of differential forms. As a consequence, we will derive a set of “shuffle” relations between the MZVs. We will explain two additionally known sets of relations: (a) the associator relations, which rely on a compatible collection of differential systems on the configuration spaces of points on the plane (Le-Murakami, Drinfeld) ; (b) the stuffle relations, which rely on the expression of MZVs as iterated sums (Racinet, Ihara-Kaneko-Zagier). By Furusho's theorem (2009), (a) implies (b). We will give indications on the original proof and on more recent variants.
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.
Nicolle Gonzalez
Title : The Elliptic Hall and double Dyck path algebra
Abstract : The double Dyck path algebra Bqt was originally introduced by Carlsson, Gorsky, Mellit in search of a geometric realization of the similarly named Aqt algebra arising in the proof of the Shuffle Theorem. In this talk I will explain how the elliptic Hall algebra arises as the spherical subalgebra of Bqt. If time permits, I’ll discuss how an algebra Bqt(Q) may be defined for arbitrary quivers Q whose spherical subalgebras, conjecturally, recover the K-theoretic Hall algebras of Q.
Iva Halacheva
Title : Categorical braid group actions and perverse equivalences
Abstract : Two classical braid group actions on the representations of a quantum group are Lusztig’s ‘internal’ action, where the braid group matches the type of the underlying Lie algebra, and the action of the n-strand braid group on n-fold tensor products, via the R-matrix. These actions have been categorified, in the first instance by Chuang and Rouquier, through so-called Rickard complexes, and in the second by Webster, via KLRW algebras. I will discuss these two constructions in the setting of perverse equivalences – prior work on the former, and work in progress on the latter joint with Erika Beserra.
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.
Duncan Laurie
Title : Representations of quantum toroidal algebras
Abstract : Quantum toroidal algebras are the double affine objects within the quantum setting. They possess two primary module categories Ô and Ô_int, with the latter a natural toroidal analogue of the finite-dimensional modules for quantum affine algebras.
After briefly introducing these algebras and discussing their structure, we shall outline some recent results on their representation theory. These include well-defined tensor product and monoidal structures on Ô and Ô_int, and meromorphic braidings by R-matrices.
I’ll also mention work in progress with Théo Pinet (McGill) which seeks to exhibit special subcategories of Ô_int as monoidal categorifications of cluster algebras, as well as a variety of questions that have arisen in our project thus far.
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.
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.
Yegor Zenkevich
Title : Wall-crossing, K-theoretic Coulomb branches and coproducts on shifted quantum affine algebras
Abstract : We investigate a family of coproducts on shifted quantum affine algebra U_q^b(^sl_2). The coproducts are related to each other by Drinfeld twists or by twisting with automorphisms of the algebra. We show that the Drinfeld twists evaluated in vector representations of the shifted algebra reproduce quantum dilogarithms associated with mutations of the BPS quiver of a 4d N=2 SU(k) gauge theory. Particular distinguished coproducts lead to open Toda quantum integrable system and its dual. We analyze in a similar way quantum toroidal algebra U_{q,t}(^^gl_1) obtaining trigonometric Ruijsenaars-Schneider integrable system and show that in the Inozemtsev type limit t -> 0 it reduces to the shifted quantum affine case.