# Polish-German seminar on quantum groups, graphs and symmetries via representation theory

## Talks

- Mini workshop on quantum groups, graphs and symmetries via representation theory

Tuesday, 15 November - Thursday, 17 November

- Tuesday
- Time: 10:00

Location: lecture hall 3, E 1.3

Speaker: Hua Wang

Title: Matched pair of length functions and the construction of some bicrossed products with rapid decay

Abstract: In this talk, I shall present some results regarding the rapid decay (RD) property of (the dual of) the classical bicrossed products viewed as compact/discrete quantum groups. I shall present a theoretical characterization of (RD) in this context, with the key notion being the matched pair of length functions coming from the representation theory of such bicrossed products. If time permits, I shall talk about how to deal with some main technical difficulties when applying the above theoretical results in order to construct some interesting examples. - Time: 11:00

Location: lecture hall 3, E 1.3

Speaker: Daniel Gromada

Title: Quantum symmetries of Hadamard matrices

Abstract: We will introduce a diagrammatic category NCBipartEven extending the category of partitions. It turns out that this category describes the quantum symmetries of Hadamard matrices and Hadamard graphs. As a consequence, this means that all Hadamard matrices/graphs of a fixed size are quantum isomorphic. The talk is based on the preprint arXiv:2210.02047. - Time: 12:00

Location: seminar room 6, E 2.4

Speaker: Michael Brannan

Title: Property (RD) and Free Wreath Products

Abstract: A discrete group G is said to have "the property of rapid decay'' (property (RD) for short) if, roughly speaking, one can control the convolution operator norms of elements of the group algebra in terms of their much easier to compute l^2(G)-norms. For discrete quantum groups, property (RD) was introduced and studied by Roland Vergnioux. In this talk I'll review property (RD) in the quantum setting, and I'll explain a result which shows that property (RD) is stable with respect to the operation of taking free wreath products of (duals of) discrete groups by quantum automorphism groups of finite dimensioanal tracial C*-algebras. This result in particular allows us to see that (the discrete duals of) the hyperoctahedral free quantum groups have property (RD). (This is joint work with Li Gao and John Weeks).

- Time: 10:00
- Wednesday
- Time: 10:00

Location: seminar room 9, E 2.4

Speaker: Mateusz Wasilewski

Title: Quantum Cayley graphs

Abstract: I will propose a definition of a quantum Cayley graph of a discrete quantum group. I will first focus on the unimodular case and discuss how to extend the notion of quantum graphs to the (highly restricted) infinite dimensional setting. It turns out that quantum Cayley graphs to a large extent do not depend on the "generating set", just like for classical groups. In the last part of the talk I will address the case of non-unimodular discrete quantum groups and present some examples. This is very much still work in progress. - Time: 11:00

Location: seminar room 9, E 2.4

Speaker: Moritz Weber

Title: Hypergraph C*-algebras

Abstract: Graph C*-algebras form an important and well studied class of C*-algebras: they generalize Cuntz algebras, Cuntz-Krieger algebras and they contain many important examples of C*-algebras. Also, they are classifiable by K-theoretical means. We propose a definition of hypergraph C*-algebras, extending this class. It is strictly larger as it contains non-nuclear C*-algebras. We are wondering whether there is any connection to quantum Cuntz-Krieger algebras, yet another generalization of graph C*-algebras… This is joint work with Dean Zenner and Mirjam Trieb. - Time: 12:00

Location: lecture hall 1, E 2.5

Speaker: Malte Gerhold

Titel: Classification of two-faced independences

Abstract: Two-faced independences are independence relations for pairs of noncommutative random variables, such as bifree independence, which models the relation between left and right regular representation of the free group in the canonical tracial state. Around 2000, in works of Speicher, Ben Ghorbal & Schürmann, and Muraki, a complete classification of "single-faced" independences was obtained: the only independences in this case are boolean, tensor, free, monotone and anti-monotone independence. I report on the current status of the classification program for two (or multi-faced) independences.

Based on joint work with Takahiro Hasebe & Michaël Ulrich (arXiv:2111.07649) and Philipp

- Time: 10:00
- Thursday
- Time: 10:00

Location: lecture hall 2, E 2.5

Speaker: Adam Skalski

Title: C*-algebraic coupling capacities

Abstract: Suppose we are given two probability measures μ and ν defined respectively on (finite) sets X and Y. Classical Strassen Theorem describes conditions for a set E ⊂ X × Y to support a probability measure with marginals μ and ν. We will discuss at the same time quantitative and noncommutative extensions of this result, introducing the notion of coupling capacity for a projection (or a positive contraction) in a tensor product of C*-algebras. We will show how this can be defined in two different ways via a certain variational formula related to operator systems, and then exploit several consequences.

Based on the joint work with Ivan Todorov and Lyudmyla Turowska - Time: 11:00

Location: lecture hall 2, E 2.5

Speaker: Nicolas Faroß

Title: Spatial Partition Quantum Groups and Symmetries of Finite Quantum Spaces

Abstract: Spatial partition quantum groups were introduced by Cébron and Weber and generalize easy quantum groups by replacing partitions with three-dimensional ones. We will take a look at this kind of quantum groups and see how these can be used to describe symmetries of some finite quantum spaces, i.e. finite-dimensional C*-algebras. - Time: 12:00

Location: lecture hall 2, E 2.5

Speaker: Roberto Palomares**Title:**Quantum symmetries for operator algebras**Abstract:**We will review some general facts about subfactors and their classification program by Jones index. Unitary tensor categories and their actions on von Neumann factors turn out to be at the heart of this problem, as these constitute important invariants associated with subfactors. Examples include --but are not limited to-- groups and their representation categories, and discrete quantum groups. After the recent developments in the classification program for simple C*-algebras, it has become increasingly important to study C*-algebras from the perspective of their quantum symmetries, and to adapt subfactor techniques to this framework. At the end of the talk, we will discuss the current status of this program.

- Time: 10:00

- Tuesday
Date: Friday, 9 September 2022, 11:00

Speaker: Daniel Gromada, Czech Technical University Prague (CVUT)

Title: Constructing simple and non-simple quantum graphs

Abstract: In this talk, we summarize different ways of constructing quantum graphs. We explain the recently obtained classification of simple quantum graphs over 2x2 matrices. We also introduce more general concepts such as weighted quantum graphs.*Das ganze tschechische Volk ist eine Simulantenbande. --*military doctor Bautze (from Hašek's novel Švejk)

(Freely translated by Moritz:*The Czech people are a bunch of simulants/malingerers (Bande=Gang)*)- Date: Wednesday, 22 June 2022, 9:00 (!) Polish-German time / 16:00 Korean time

Speaker: Hun Hee Lee (Seoul National University, Republic of Korea)

Title: Quantum channels with quantum group symmetry

Abstract: In this talk we will focus on the quantum channels satisfying compact quantum group symmetry, i.e. covariant channels. We first examine the known cases using irreducible representations of compact groups for the possible symmetry. Our main contribution is that we could provide a rather complete description of covariant channels when the symmetry satisfies the "multiplicity-free" condition. It turns out that this scenario is closely related to the "measure-and-prepare" programmability of quantum channels due to a recent result by Winter, et al.

Secondly, we will demonstrate that any compact quantum groups can also be used as the symmetry for quantum channels. Through the case of SU_q(2) we will explain the necessity of the Heisenberg picture for non-Kac type quantum groups.

Finally, we will visit a few examples including the case of Weyl covariant channels using projective representations and the cases of the symmetric group of order 4 and its quantum version.

- Date: Wednesday, 27 April 2022, 14:00

Speaker: Alexander Wendel (Saarland U)

Title: Quantum Channels and Entangled States associated to Easy Quantum Groups

Abstract: In 2017 Brannan and Collins constructed highly-entangled spaces and states from the representation theory of the orthogonal quantum groups. A key role in their construction is played by the famous Jones-Wenzl projections. These projections are usually defined via a certain commutation property and are projections onto irreducible representations of the orthogonal quantum group.Shortly before (2014/2016) Freslon and Weber gave a combinatorial characterisation of the irreducible representations and fusion rules of easy quantum groups.

We are going to combine these two papers to investigate howfar one can carry over the results of Brannan and Collins to other easy quantum groups. Especially we are going to show that the projections onto irreducibles satisfy a similar characterisation as the Jones-Wenzl projections and are going to characterise by this the image of these projections. We are then going to transfer some results on entangled spaces to the symmetric quantum group and indicate how to carry out these constructions for other easy quantum groups. - Date: Wednesday, 30 March 2022, 14:00

Speaker: Adam Skalski (IMPAN)

Title: Quantum channels related to representations of compact quantum groups.

Abstract: I will introduce basic definitions and certain qualitative notions related to quantum channels, explaining how via the Stinespring theorem one can relate them to properties of isometries between finite-dimensional Hilbert spaces. Then I will describe how such objects arise from representations of compact quantum groups, and how the quantum group structure gives tools to study resulting channels. The talk will be mainly based on the articles of Mike Brannan, Benoit Collins, Hun Hee Lee and Sang Gyoun Youn.*These are the days of miracle and wonder*

(...)

Staccato signals of constant information

- Date: Wednesday, 16 February 2022, 14:00

Speaker: Mateusz Wasilewski (IMPAN)

Title: Introduction to quantum graphs

Abstract: I will present three different perspectives on quantum graphs. I will then try to convince you that each of these three approaches has its merits, by discussing random models of quantum graphs, quantum Cuntz-Krieger algebras and quantum automorphisms. Along the way I will mention several future research directions.*I’m trying to learn something new. I’m trying to surround myself with people that inspire me, or at least inquire similar desires.*

(Kendrick Lamar)

**Date**: Wednesday, 26 January 2022, 12:00**Speaker**: Moritz Weber (Saarland University)**Title**: Partitions, „easy" quantum groups and representation theory**Abstract**: In this first talk of our new seminar, I will outline the general frame and aims of our joint project. I will then briefly review aspects of the representation theory of compact matrix quantum groups, before discussing some diagrammatical calculus, partitions of sets and so called „easy" quantum groups. Some open questions we are interested in will be mentioned.*It is easy to leave the home of reality and get lost in the woods of mathematics, but only a few know how to return.*

(Hugo Steinhaus)

## Postal address

Saarland University

Department of Mathematics

Postfach 15 11 50

66041 Saarbrücken

Germany

## Physical address

Saarland University

Campus building E 2 4

66123 Saarbrücken

Germany