Future directions for quantum groups


Y. Bazlov: Noncommutative reflections

In the first part of my talk, I will reflect on two constructions, both due to Drinfeld, and the role they play in the ongoing research in noncommutative algebra and representation theory. Quasitriangular structures on quantum groups provide an algebraic formalism for understanding deformation quantisation; degenerate affine Hecke algebras (more specifically, rational Cherednik algebras) have become a favourite of representation theorists. I will then talk about my recent joint work with Berenstein where both constructions come together.

Recall that every symmetric polynomial in n variables can be uniquely expressed in terms of elementary symmetric polynomials. In the last century, this result was extended by Chevalley, Shephard, Todd and Serre to invariants of finite groups generated by reflections. We are interested in invariants of groups acting on a noncommutative space (algebra) A and introduce the notion of a noncommutative reflection of A. It turns out that there is a better behaved version of invariants if the group is replaced by a certain quantum group. The case of an n-dimensional quantum plane yields a family of "twisted" Coxeter groups and explains two recent results: one due to Berenstein and Bazlov and independently due to Kirkman, Kuzmanovich and Zhang, the other due to Ellis, Khovanov and Lauda.

Slides: FDQG-Bazlov.pdf

T. Brzezinski: Bundles over quantum weighted projective spaces

Quantum weighted projective spaces are defined as quotients of weighted circle actions on quantum spheres and other quantum spaces. They are examples of deformations of orbifolds. We describe in detail algebraic structure of the lower dimensional cases: quantum weighted projective complex lines and quantum weighted projective real planes. In addition we construct principal U(1)-bundles and associated line bundles over these spaces. Joint work with Simon A. Fairfax.

Slides: FDQG-Brzezinski.pdf

D. Jordan: Cyclically quantized Weyl algebras and their semiclassical limits

I will discuss some aspects of the passage between families of noncommutative algebras and their semiclassical limits in the context of a particular class of examples. The Poisson algebras concerned arise from work of Fordy and Marsh on periodic cluster mutation and the noncommutative algebras are akin to polynomial algebras in one variable over Weyl algebras. I plan to look at three aspects where the Poisson situation is more easily understood and tells us what to expect on the noncommutative side but where a theorem would be much better. These are complete integrability (the search for commutative subalgebras of the right size), the determination of finite-dimensional simple modules and the analysis of the prime spectrum.

Slides: FDQG-Jordan.pdf

S. Kolb: Quantum symmetric Kac-Moody pairs

Quantum group analogs of Lie subalgebras appear as coideal subalgebras of quantized enveloping algebras. This talk will give an introduction to the theory of quantum symmetric pairs as developed by Gail Letzter. It will be explained how her theory can be formulated consistently for involutions of the second kind of symmetrisable Kac-Moody algebras. The resulting quantum symmetric Kac-Moody pairs allow triangular decompositions, can be written in terms of generators and relations, and behave well under specialization. Time permitting, it will be explained how the twisted q-Yangians introduced by Molev, Ragoucy, and Sorba appear as examples.

Slides: FDQG-Kolb.pdf

U. Kraehmer: New remarks on the Dirac operators on quantised Hermitian symmetric spaces

In this joint work with Matthew Tucker-Simmons (U Berkeley) the ̄-complex of the quantised compact Hermitian symmetric spaces is identified with the Koszul complexes of the quantised symmetric algebras of Berenstein and Zwicknagl, which then leads to an explicit construction of the relevant quantised Clifford algebras.

K. Kremnizer: Geometry of quantum groups at roots of unity

I will discuss the relationship between quantum groups at roots of unity and Springer fibers. This relationship has many applications. I will describe these applications. I will also speculate about possible extensions of this work to affine quantum groups.

T. Lenagan: Totally nonnegative matrices

A real matrix is totally nonnegative if each of its minors is nonnegative. Totally nonnegative matrices arise in a variety of contexts, and have a history going back about 100 years.

Recent work with Goodearl and Launois has shown a deep connection between the theory of totally nonnegative matrices and the invariant prime spectrum of quantum matrices. This talk will explain the connection, give an introduction to the theory of totally nonnegative matrices and prove one or more results in this theory that were motivated by results in the quantum world.

J.M. Lindsay: Lévy processes on compact quantum groups and noncommutative manifolds

The aim of this talk is to indicate some directions in which research in compact quantum groups, and their applications, is currently headed.

A general structure theory of Lévy processes on compact quantum groups will be described. This extends the bounded-generator case developed with my former student, Adam Skalski, and the original algebraic theory of Lévy processes on Hopf *-algebras due to Schürmann and coworkers. By restriction to classical compact groups, all the classical Lévy processes are realised. If time permits, a characterisation will be given of the class of quantum stochastic flows on an "admissible" spectral triple of finite compact type.

The key ingredients are: the representation theory of compact quantum groups, in particular the Peter-Weyl theory; quantum stochastic cocycles, and their generation via quantum stochastic differential equations in the sense of Hudson, (Evans) and Parthasarathy; and quantum isometry groups of noncommutative manifolds in the sense of Goswami.

This is joint work with Biswarup Das. It was supported by the UKIERI Research Collaboration Network Quantum Probability, Noncommutative Geometry & Quantum Information.

S. Majid: Noncommutative cohomology and coverings of finite groups

We transfer ideas from Lie theory to finite group theory using quantum group methods and noncommutative geometry. Specifically, we introduce the notion of an "inverse property" (IP) quandle C which we propose as the right notion of "Lie algebra" in the category of sets and for which we construct an associated group GC. In a certain "locally skew" case and when GC is finite we show that the noncommutative de Rham cohomology H1(GC) with respect to the calculus corresponding to C is trivial aside from a single generator that has no classical analogue. This computes the noncommutative cohomology or a large class of finite groups equipped with differential structure, including all Weyl groups with C the conjugacy class of reflections and dihedral groups D6m. As another application we we start with a finite group G and C a generating ad-stable inversion-stable subset as "Lie algebra" then we show that GC→G provides a natural "covering group" in analogy with the similar result for the simply-connected covering group of a Lie group. As an example we consider C=ZP1ZP1 as a locally skew IP-quandle `Lie algebra' of SL2(Z) and show that its covering group GC is isomorphic to B3, the braid group on 3 strands, in accordance with the universal covering map of SL2(R) restricted to the inverse image of SL2(Z). The talk will be based on my recent paper with K. Rietsch and also picks up on earlier work with her and J. López Peña.

Slides: FDQG-Majid.pdf