Tag - Quantum groups

Sergei Gukov: Surprising VOA Structures from Quantum Topology

6th June, 2022

In quantum topology, one usually constructs invariants of knots and 3-manifolds starting with an algebraic structure with suitable properties that can encode braiding and surgery operations in three dimensions. ln this talk, 1 review recent work on q-series invariants of 3-manifolds, associated with quantum groups at generic q, that provide a connection between quantum topology and algebra going in the opposite direction: starting with a 3-manifold and a choice of Spin-C structure, the q-series invariant turns out to be a character of a (logarithmic) vertex algebra that depends on the 3-manifold.

Yuri Bahturin: Group Gradings and Actions of Pointed Hopf Algebras

12th May, 2022

Pointed Hopf algebras are a wide class of Hopf algebras, including group algebras and enveloping algebras of Lie algebras. In this talk, based on a recent work with Susan Montgomery, we study actions of pointed Hopf algebras on simple algebras. These actions are known to be inner, as in the case of Skolem-Noether theorem. We try to give explicit descriptions, whenever possible, and consider Taft algebras, their Drinfeld doubles and some quantum groups.

Kent Vashaw: Can a Zhang twist be a cocycle twist?

26th October, 2021

The Zhang twist of a graded algebra was defined by J. Zhang in 1996, and has provend an important tool in non-commutative algebra and non-commutative algebraic geometry. On the other hand, in the world of Hopf algebras and quantum groups, the 2-cocycle twist of a Hopf algebra gives a new Hopf algebra which is Morita-Takeuchi equivalent to the original Hopf algebra. We provide sufficient conditions for a Zhang twist of a graded Hopf algebra H to be again a Hopf algebra, to be an H-cleft object, or a 2-cocycle twist of H. In particular, we introduce the notion of a twisting pair for H such that the Zhang twist of H by such a pair is a 2-cocycle twist. This new notion is investigated in the context of various examples of Hopf algebras including Manin's universal quantum groups, and the quantized coordinate rings of general linear groups.

Weiqiang Wang: From Quantum Groups to iQuantum Groups

16th October, 2021

Drinfeld-Jimbo quantum groups have made major impacts on representation theory and other areas. i-Quantum groups arise from quantum symmetric pairs. We shall explain why it is natural to view i-quantum groups as a generalization of quantum groups, and then discuss some of the many new developments and applications of i-quantum groups as initiated in Huanchen Bao’s UVA dissertation.

Daniel Nakano: Non-commutative Tensor Triangular Geometry and Applications

27th September, 2021

In this talk, I will show how to develop a general non-commutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (MΔC). Insights from non-commutative ring theory are used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an MΔC, K, and then to associate to K a topological space: the Balmer spectrum Spc(K). We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that Spc(K) is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an MΔC. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of K, which in turn can be applied to classify the thick two-sided ideals and Spc(K). Applications will be given for quantum groups and non-cocommutative finite-dimensional Hopf algebras studied by Benson and Witherspoon.

Geordie Williamson: Spectra in representation theory

18th June, 2021

In geometric representation theory cohomology, intersection cohomology and constructible sheaves show up everywhere. This might seem strange to an algebraic topologist, who might ask: why this emphasis on cohomology, when there are so many other interesting cohomology theories (like K-theory, elliptic cohomology, complex cobordism, ...) out there? They might also ask: is there something like "intersection K-theory", or "intersection complex cobordism"? This is something I've often wondered about. I will describe work in progress with Ben Elias, where we use Soergel bimodules to investigate what KU-modules look like on the affine Grassmannian. We have checked by hand that in types A1, A2 and B2, one gets something roughly resembling the quantum group. Speaking very roughly, the intersection K-theory of Schubert varieties in the affine Grassmannian should recover the irreducible representations of the quantum group. Inspirations for this work include a strange Cartan matrix discovered by Ben Elias, and work of Cautis-Kamnitzer.

Chris Chung: iQuantum Covering Groups: Serre presentation and canonical basis

26th January, 2021

In 2016, Bao and Wang developed a general theory of canonical basis for quantum symmetric pairs (U,Ui), generalizing the canonical basis of Lusztig and Kashiwara for quantum groups and earning them the 2020 Chevalley Prize in Lie Theory. The i-divided powers are polynomials in a single generator that generalize Lusztig's divided powers, which are monomials. They can be similarly perceived as canonical basis in rank one, and have closed form expansion formulas, established by Berman and Wang, that were used by Chen, Lu and Wang to give a Serre presentation for coideal subalgebras Ui, featuring novel i-Serre relations when τ(i)=i. Quantum covering groups, developed by Clark, Hill and Wang, are a generalization that `covers' both the Lusztig quantum group and quantum supergroups of anisotropic type. In this talk, I will talk about how the results for i-divided powers and the Serre presentation can be extended to the quantum covering algebra setting, and subsequently applications to canonical basis for Uiπ, the quantum covering analogue of Ui, and quantum covering groups at roots of 1.

Rob Muth: Specht modules and cuspidal ribbon tableaux

27th October, 2020

Representation theory of Khovanov-Lauda-Rouquier (KLR) algebras in affine type A can be studied through the lens of Specht modules, associated with the cellular structure of cyclotomic KLR algebras, or through the lens of cuspidal modules, associated with categorified PBW bases for the quantum group of affine type A. Cuspidal ribbons provide a sort of combinatorial bridge between these approaches. I will describe some recent results on cuspidal ribbon tableaux, and some implications in the world of KLR representation theory, such as bounds on labels of simple factors of Specht modules, and the presentation of cuspidal modules.

Milen Yakimov: Quantum cluster algebras at roots of unity and discriminants

3rd October, 2020

Cluster algebras were invented by Fomin and Zelevinsky twenty years ago. Since then they have played an important role in a number of settings in combinatorics, geometry, representation theory and topology. We will introduce a notion of root of unity quantum cluster algebras which are PI algebras, and will show that they have large canonical central subalgebras isomorphic to the original cluster algebras. These are far reaching generalizations of the De Concini-Kac-Procesi central subalgebras that appear in the study of the irreducible representations of big quantum groups. We will describe a general theorem computing the discriminants of these algebras. In a special situation it yields a formula for the discriminants of the quantum unipotent cells at roots of unity associated to all symmetrizable Kac-Moody algebras.