Tag - Quantum groups

Slaven Kožić: Representations of the quantum affine vertex algebra associated with the trigonometric R-matrix of A

14th October, 2024

One important problem in the vertex algebra theory is to associate certain vertex algebra-like objects, the quantum vertex algebras, to various classes of quantum groups, such as quantum affine algebras or double Yangians. In this talk, I will discuss this problem in the context of Etingof-Kazhdan's quantum affine vertex algebra Vc(𝔤𝔩N) associated with the trigonometric R-matrix of type A. The main focus will be on the explicit description of the centre of Vc(𝔤𝔩N) at the critical level c = -N and, furthermore, on the connection between certain classes of Vc(𝔤𝔩N)-modules and representation theories of the quantum affine algebra of type A and the orthogonal twisted h-Yangian.

Hadi Salmasian: Lie Groups and Quantization

4th September, 2024

This is a 23-lecture course, with each lecture being around 80 minutes long, given by Hadi Salmasian.

The goal of the course is to first cover the foundational theory of Lie groups and then move on to more advanced topics that expose the audience to areas of active research. The following is the list of topics that are intended to be covered:

  • Foundational theory of Lie groups: Lie groups, the exponential map, Lie correspondence. Homomorphisms and coverings. Closed subgroups. Classical groups: Cartan subgroups, fundamental groups. Manifolds. Homogeneous spaces. General Lie groups.
  • Introduction to quantization: Symplectic manifolds, pre-quantization, the orbit method. Poisson manifolds, Manin triples. Universal enveloping algebras, quantum sl(2) and its representations, quantum symmetric spaces.

Jon Brundan: Isomeric Heisenberg and Kac–Moody categorification

29th May, 2024

The isomeric Heisenberg category acts naturally on a number of abelian categories appearing in the representation theory of the isomeric supergroup Q(n), and also on representations of Sergeev’s algebra which is related to the double covers of symmetric groups. I will explain an efficient way to convert an action of the isomeric Heisenberg category on these and other abelian categories into an action of a corresponding super Kac–Moody 2-category. To properly understand the odd simple root indexed by the element zero of the ground field requires the theory of odd symmetric functions developed by Ellis, Khovanov and Lauda, the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka, and the covering quantum groups defined and studied by Clark and Wang.

Jie Du: The quantum queer supergroup via their q-Schur superalgebras

28th May, 2024

Using a geometric setting of q-Schur algebras, Beilinson-Lusztig-MacPherson discovered a new basis for quantum 𝔤𝔩n (i.e., the quantum enveloping algebra Uq(𝔤𝔩n) of the Lie algebra 𝔤𝔩n) and its associated matrix representation of the regular module of Uq(𝔤𝔩n). This beautiful work has been generalized (either geometrically or algebraically) to quantum affine 𝔤𝔩n, quantum super 𝔤𝔩m|n, and recently, to some i-quantum groups of type AIII.

In this talk, I will report on a completion of the work for a new construction of the quantum queer supergroup using their q-Schur superalgebras. This work was initiated 10 years ago, and almost failed immediately after a few months’ effort, due to the complication in computing the multiplication formulas by odd generators. Then, we moved on testing special cases or other methods for some years and regained confidence to continue. Thus, it resulted in a preliminary version which was posted on arXiv in August 2022.

The main unsatisfaction in the preliminary version was the order relation used in a triangular relation and the absence of a normalized standard basis. It took almost two more years for us to tune the preliminary version up to a satisfactory version, where the so-called SDP condition, involving further combinatorics related to symmetric groups and Clifford generators, and an extra exponent involving the odd part of a labelling matrix play decisive roles to fix the problems.

Kang Lu: A Drinfeld presentation of twisted Yangians via degeneration

2nd March, 2024

We formulate a new family of algebras, twisted Yangians (of split type) in current generators and relations, via degeneration of Drinfeld presentations of affine iquantum groups (associated with split Satake diagrams). These new algebras admit PBW type bases and are shown to be a deformation of twisted current algebras. For type AI, it matches with the Drinfeld presentation of twisted Yangian obtained via Gauss decomposition. We conjecture that our twisted Yangians are isomorphic to twisted Yangians constructed in RTT presentation.

Oleksandr Tsymbaliuk: Lyndon words and fused currents in shuffle algebra

2nd March, 2024

Classical q-shuffle algebras provide combinatorial models for the positive half Uq(𝔫) of a finite quantum group. We define a loop version of this construction, yielding a combinatorial model for the positive half Uq(L𝔫) of a quantum loop group. In particular, we construct a PBW basis of Uq(L𝔫) indexed by standard Lyndon words, generalizing the work of Lalonde-Ram, Leclerc and Rosso in the Uq(𝔫) case. We also connect this to Enriquez's degeneration A of the elliptic algebras of Feigin-Odesskii, proving a conjecture that describes the image of the embedding Uq(L𝔫)→A in terms of pole and wheel conditions. The talk shall conclude with the shuffle interpretations of fused currents proposed by Ding-Khoroshkin.

Ivan Losev: Quantum category 𝒪

1st March, 2024

The representation theory of quantum groups including at roots of unity is an important part of Lie representation theory. In this talk, we will study one of categories of representations: the quantum category 𝒪, which is a suitable analogue of the classical Bernstein-Gelfand category 𝒪. We will relate it to a model representation category, the affine Hecke category, more precisely to the heart of the new t-structure on that category (all these terms will be defined in the lectures).

Travis Scrimshaw: An Overview of Kirillov-Reshtikhin Modules and Crystals

19th January, 2024

Kirillov-Reshetikhin (KR) modules are an important class of finite-dimensional representations associated to an affine Lie algebra and the associated Yangian and quantum group. KR modules are known to appear in many integrable systems and govern the dynamics. In this talk, we will give an overview of the role KR modules play in the category of finite-dimensional representations, R-matrices and the fusion construction, their (conjectural) crystal bases, and how they relate to Demazure modules. In particular, we will focus on how to construct their crystal bases combinatorially and the different types of character theories. As time permits, we will discuss some of the relations with (quantum) integrable systems.

Dan Nakano: Realizing Rings of Regular Functions via the Cohomology of Quantum Groups

2nd November, 2023

Let G be a complex reductive group and be a parabolic subgroup of G. In this talk, the presenter will address questions involving the realization of the G-module of the global sections of the (twisted) cotangent bundle over the flag variety G/P via the cohomology of the small quantum group.

Our main results generalize the important computation of the cohomology ring for the small quantum group by Ginzburg and Kumar and provide a generalization of well-known calculations by Kumar, Lauritzen, and Thomsen to the quantum case and the parabolic setting. As an application, we answer the question (first posed by Friedlander and Parshall for Frobenius kernels) about the realization of coordinate rings of Richardson orbit closures for complex semisimple groups via quantum group cohomology. Formulas will be provided that relate the multiplicities of simple G-modules in the global sections with the dimensions of extension groups over the large quantum group.

Jethro van Ekeren: Modular tensor categories from exceptional W-algebras

31st August, 2023

I will present results of joint work with T. Arakawa, on representation theory of simple affine W-algebras. For so-called exceptional W-algebras, the category of representations acquires the structure of a modular tensor category, and in this talk I will describe the modular data and fusion rules for some cases. In many cases the modular data matches that of quantum groups at roots of unity, but in other cases, the results are quite mysterious.