Tag - Quantum algebra

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.

Iana Anguelova: Microlocality and Chiral Algebras

16th October, 2021

In one of my last conversations with Ben Cox, we discussed our mutual desire to work together on the axiomatic approach to multilocal and quantum chiral algebras. We both had worked already on issues related to multilocality; situations where the fields/vertex operators in question have Operator Product Expansions (OPEs) with more than the one singularity at 'z=w'. In particular, we worked together on the theory of N-point local chiral algebras, i.e., algebras that are 'complete' with respect to OPEs, and have singularities at roots of unity. But we were planning to work on the outstanding case where the OPEs have singularities at infinite multiplicative lattices. Such is the example of the Frenkel-Jing quantum vertex operators. In this talk I will discuss some problems arising in the axiomatic approach to multilocal chiral algebras, both N-point local, and quantum.

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.

Marcy Robertson: Expansions, completions and automorphisms of welded tangled foams

22nd April, 2021

Welded tangles are knotted surfaces in ℝ4. Bar-Natan and Dancso described a class of welded tangles which have 'foamed vertices' where one allows surfaces to merge and split. The resulting welded tangled foams carry an algebraic structure, similar to the planar algebras of Jones, called a circuit algebra. In joint work with Dancso and Halacheva we provide a one-to-one correspondence between circuit algebras and a form of rigid tensor category called 'wheeled props'. This is a higher-dimensional version of the well-known algebraic classification of planar algebras as certain pivotal categories.

This classification allows us to connect these 'welded tangled foams' to the Kashiwara-Vergne conjecture in Lie theory. In work in progress, we show that the group of homotopy automorphisms of the (rational completion of) the wheeled prop of welded foams is isomorphic to the group of symmetries KV, which acts on the solutions to the Kashiwara-Vergne conjecture. Moreover, we explain how this approach illuminates the close relationship between the group KV and the pro-unipotent Grothendieck–Teichmueller group.

Catharina Stroppel: Verlinde rings and DAHA actions

16th March, 2021

In this talk we will briefly recall how quantum groups at roots give rise Verlinde algebras which can be realised as Grothendieck rings of certain monoidal categories. The ring structure is quite interesting and was very much studied in type A. I will try to explain how one gets a natural action of certain double affine Hecke algebras and show how known properties of these rings can be deduced from this action and in which sense modularity of the tensor category is encoded.

Alina Vdovina: Buildings, quaternions and Drinfeld-Manin solutions of Yang-Baxter equations

11th February, 2021

We will give a brief introduction to the theory of buildings and present their geometric, algebraic and arithmetic aspects. In particular, we present explicit constructions of infinite families of quaternionic cube complexes, covered by buildings. We will introduce new connections of geometric group theory and theoretical physics by using quaternionic lattices to find new infinite families of Drinfeld-Manin solutions of Yang-Baxter equations.

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.