Tag - Quantum algebra

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.

Thomas Creutzig: Tensor categories of modules of W-algebras

28th August, 2023

Let V be an affine vertex algebra of some simple Lie algebra 𝔤 and some level. Let KL be the category of V-modules whose conformal weight spaces are integrable 𝔤-modules. A famous result of Kazhdan and Lusztig tells us that for almost all levels KL is a braided tensor category and as such equivalent to a category of weight modules of the quantum group Uq(𝔤) of 𝔤 for suitable q.

It is desired to have similar results for suitable categories of W-algebras and superalgebras. In particular one wants to understand tensor structure and equivalences to quantum supergroups.

I will outline how to prove such statements and illustrate this in some examples.

Eric Vasserot: Non-symmetric quantum groups and critical cohomology of quiver varieties

28th August, 2023

We realize the quantum loop groups and shifted quantum loop groups of arbitrary types, possibly non-symmetric, using critical K-theory. This gives a generalization of Nakajima’s construction of symmetric quantum loop groups via quiver varieties to non-symmetric types. This also yields a geometric realization of some simple modules, in particular the Kirillov-Reshethikin modules, and the tensor product of prefundamental modules.

Ziqing Xiang: Quantum wreath product

28th July, 2023

The classical wreath product G ≀ Σd is a semidirect product Gd ⋊ Σd with Σd acting on Gd by permutations. We deform this classical wreath product by deforming G into an associative algebra B, deforming Σd into a Hecke algebra, and deforming the action. The result is called a quantum wreath product B ≀ H(d). Many variants of Hecke algebras can be viewed as quantum wreath products, hence could be treated in a unified manner.

In this talk, we will discuss necessary and sufficient conditions for quantum wreath products to have a basis of suitable size. We will also discuss some other structural results, the Schur algebras of these quantum wreath products, and their representations.

Nathan Brownlowe: Self-similar quantum groups

23rd June, 2023

In this talk I will introduce the notion of self-similarity for compact quantum groups. I will start by looking at the quantum automorphism group of an infinite homogeneous rooted tree. Self-similar quantum groups are then certain quantum subgroups of these quantum automorphisms. I will then look at a class of examples called finitely-constrained self-similar quantum groups, and I will describe a subclass as quantum wreath products by subgroups of the quantum permutation group.

Fabienne Chouraqui: Connections between the Yang-Baxter equation and Thompson’s group F

22nd June, 2023

The quantum Yang-Baxter equation is an equation in mathematical physics and it lies in the foundation of the theory of quantum groups. One of the fundamental problems is to find all the solutions of this equation. Drinfeld suggested the study of a particular class of solutions, derived from the so-called set-theoretic solutions. A set-theoretic solution of the Yang-Baxter equation is a pair (X,r), where X is a set and

r : X ⨯ X → X ⨯ X     r(x,y)=(σx(y),γy(x))

is a bijective map satisfying r12r23r12 = r23r12r23, where r12 = r ⨯ IdX and r23 = IdX ⨯ r. We define non-degenerate involutive partial solutions as a generalization of non-degenerate involutive set-theoretical solutions of the quantum Yang-Baxter equation (QYBE). The induced operator is not a classical solution of the QYBE, but a braiding operator as in conformal field theory. We define the structure inverse monoid of a non-degenerate involutive partial solution and prove that if the partial solution is square-free, then it embeds into the restricted product of a commutative inverse monoid and an inverse symmetric monoid. Furthermore, we show that there is a connection between partial solutions and the Thompson's group F. This raises the question of whether there are further connections between partial solutions and Thompson's groups in general.

Kang Lu: A Drinfeld presentation of twisted Yangians

12th May, 2023

In this talk, I will discuss the Drinfeld’s new presentation for (Olshanski’s) twisted Yangians of type AI that is open for 30 years. This presentation comes from the Gauss decomposition and turns out to be compatible with the one obtained from degeneration of affine iQuantum Groups. If time permits, I will also discuss our work on twisted Yangians of other types.

Kent Vashaw: Twisting of universal quantum groups and comodule algebras

1st May, 2023

We consider 2-cocycle twists (and more generally, Morita-Takeuchi equivalences between) Manin's universal quantum groups and their comodule algebras. We show when Zhang twists of connected graded algebras can be realized as cocycle twists, thus concretely connected the (graded) representation theory of an algebra A to the corepresentation theory of its universal quantum group. We also prove that fundamental properties of non-commutative associative algebras, such as Artin-Schelter regularity and Koszuality are preserved under 2-cocycle twist.

Vladimir Dotsenko: Operad filtrations and quantization

24th April, 2023

The celebrated problem of deformation quantization discusses deformations of Poisson algebras into associative algebras, a question that is, in the end, motivated by quantum mechanics. I shall discuss this question and some of its generalisations from the purely algebraic point of view using the theory of operads. In particular, I shall show how to prove that there are, in a strict mathematical sense, only two meaningful deformation problems for Poisson algebras, namely deforming them in the class of all Poisson algebras or all associative algebras, and there is only one meaningful deformation problem for the so called almost Poisson algebras (also sometimes known as generic Poisson algebras), namely deforming them in the class of all almost Poisson algebras. For instance, this explains the existing body of work in the mathematical physics literature asserting that some classes of non-associative star products cannot be alternative, are always flexible etc.