# Workshop talks

Workshop talks for “Quantum groups and their analysis”,  August 5–9, 2019

## Schedule

 08.05 (Mon) 08.06 (Tue) 08.07 (Wed) 08.08 (Thu) 08.09 (Fri) 09:15–10:05 Loïc Foissy Gabriella Böhm Alexander Stottmeister Ehud Meir Jonas Wahl 10:05–10:40 Coffee break Coffee break Coffee break Coffee break Coffee break 10:40–11:30 Martina Balagović Alexander Shapiro Arnaud Brothier Jason Crann Hun Hee Lee 11:35–12:10 Simon Schmidt Hua Wang Arkadiusz Bochniak Tobias Ohrmann Yuri Manin 11:40–12:30 Paweł Joziak Damien Rivet End of workshop 12:10–13:40 Lunch Lunch Lunch 13:40–14:30 Francesco Fidaleo Alexandru Chirvasitu Free afternoon Adam Skalski 14:30–15:00 Coffee break Coffee break Coffee break 15:00–15:50 Wenqing Tao Christian Voigt Ami Viselter evening Conference dinner

## Talks

Martina Balagović (Newcastle University)

Universal K-matrix for quantum symmetric pairs

I will review some recent progress on quantum symmetric pair coideal subalgebras of quantum groups, in particular the construction of the universal K matrix, which bears significant resemblance to the construction of the universal R-matrix for the quantum group.

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Connes and Riemannian Differential Geometry on Finite Groups

The two points of view on noncommutative geometry, one based on spectral triple approach, and the second one known as quantum Riemannian geometry, are trying to describe the generalized, noncommutative geometries. Both approaches were successfully used in a broad range of physical models. There is a natural question how to connect them. We start with the discussion of Riemannian geometry on finite groups and prepare for further analysis of its relations with spectral triples. This is work in progress, joint with Andrzej Sitarz and Paweł Zalecki.

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Gabriella Böhm (Wigner Research Centre for Physics)

Crossed modules of Hopf algebras: an approach via monoids

Whitehead’s crossed modules received a lot of attention since their appearance in 1941 because of their occurrence in many different contexts. They admit several equivalent descriptions: a ‘simplicial group whose Moore complex is concentrated in degrees 1 and 2’ turns out to be the internal nerve of a ‘strict 2-group’ and the Moore complex yields a ‘crossed module’. These constructions establish, in fact, equivalences between these three notions. A highly elegant proof of these equivalences, using the theory of semi-Abelian categories, was initiated by George Janelidze.

Recently the definition of, and some results on crossed modules have been extended from groups to Hopf algebras. However, in contrast to groups, Hopf algebras do not constitute a semi-Abelian category. So Janelidze’s methods can not be applied to them. In the talk we report about a program which aims at a (categorical) theory of crossed modules over monoids in quite general monoidal categories. Applying it to the category of coalgebras — whose monoids are bialgebras — it covers also the crossed modules of Hopf algebras; in particular we obtain a proof of the equivalences between crossed modules, certain internal category objects, and certain simplicial Hopf algebras.

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Arnaud Brothier (University of New South Wales)

Jones' actions of the Thompson's groups: applications to group theory and mathematical physics

Motivating in constructing conformal field theories Jones recently discovered a very general process that produces actions of the Thompson groups FT and V such as unitary representations or actions on C*-algebras. I will explain this constructions via examples and present various applications regarding analytical properties of groups and, if time permits, in lattice theory (e.g. quantum field theory).

Alexandru Chirvasitu (University at Buffalo)

Residually finite discrete quantum groups

The quantum symmetric (or more generally, reflection) groups provide a good testing ground for all manner of problems, conjectures and properties borrowed from the theory of plain (non-quantum) discrete groups.

The main result exemplifying this will be that the discrete duals of the quantum reflection groups are residually finite and hence in particular hyperlinear. In turn, this has consequences for the free entropy dimension of the (von Neumann algebras attached to) these discrete quantum groups. A particularly satisfying result is that two possible flavors of free entropy dimension are equal to 1 for the discrete duals of the quantum permutation groups. (joint with Michael Brannan and Amaury Freslon)

Jason Crann (Carleton University)

Mapping ideals of quantum group multipliers

A well-known result of Gilbert establishes a fundamental correspondence between completely bounded multipliers of the Fourier algebra McbA(G) and convolution maps L1(G) → L(G) which factor through a Hilbert space. In this talk we present a quantum version of Gilbert's result for a large class of quantum groups, including group von Neumann algebras VN(G) for quasi-SIN groups G. Within this class, we also characterize the mapping ideals of completely integral, completely nuclear, and completely compact quantum group convolution maps. The latter two characterizations yield new connections with the quantum Bohr compactification. This is joint work with Mahmood Alaghmandan and Matthias Neufang.

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Francesco Fidaleo (University of Rome Tor Vergata)

On the uniform convergence of Cesaro averages for C*-dynamical systems

Let (𝔄, Φ) be a C*-dynamical system based on an identity-preserving *-endomorphism Φ of the unital C*-algebra 𝔄. We study the uniform convergence of Cesaro averages

$$M_{a,\lambda}(n):=\frac1{n}\sum_{k=0}^{n-1}\lambda^{-k}\Phi^k(a),\quad a\in\mathfrak A,$$

uniformly for all values λ in the unit circle.

For such a purpose, we define a spectral set $$\sigma^{(\mathop{\rm ph, f})}_{\mathop{\rm pp}}(\Phi)\subset\mathbb{T}$$ canonically associated to the given dynamical system, and show that

$$\lim_{n\to+\infty}M_{a,\lambda}(n)=0$$

whenever $$\lambda\in\mathbb{T}\smallsetminus\sigma^{(\mathop{\rm ph, f})}_{\mathop{\rm pp}}(\Phi)$$.

If in addition, if (𝔄, Φ) is uniquely ergodic with respect to the fixed-point algebra, then we can provide conditions for which the sequence (Ma,λ(n))n uniformly converges for $$\lambda\in\sigma^{\mathop{\rm ph}}_{\mathop{\rm pp}}(\Phi)$$, providing the formula of such a limit.

To end, we also discuss some simple examples arising from quantum probability, the first one not enjoying the property to be uniquely ergodic with respect to the fixed point subalgebra, and the second one satisfying that ergodic property, to which our results apply. Other examples based on more involved examples arising from noncommutative geometry (i.e. the noncommutative 2-torus) can be also exhibited.

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Loïc Foissy (University of the Littoral Opal Coast)

Twisted Hopf algebras

A twisted Hopf algebra is a Hopf algebra in the category of linear species. The Fock functors allow to recover “classical” Hopf algebras from twisted ones. Numerous constructions and results can be lifted to the level of twisted bialgebras, such that cofreeness, shuffle and quasi-shuffles products, etc.

Using two tensor products in the category of species, we define the notion of cointeraction of twisted bialgebras. Examples on finite topologies and graphs allow to reconstruct Ehrahrt polynomials and chromatic polynomials in a canonical way from cointeraction of twisted bialgebras.

Paweł Joziak (Institute of Mathematics of the Polish Academy of Sciences)

On Quantum Increasing Sequences

Quantum increasing sequences were introduced by S. Curran to characterize free amalgamated independence of an infinite sequence of random variables (over the tail algebra) by means of the so-called quantum spreadability. This is a de Finetti type theorem that in the classical case was first established by C. Ryll-Nardzewski, but the assumptions are formally weaker. The plan of the talk is to discuss the connection of this structure to quantum permutation groups and describe in full generality the quantum subgroups generated by them. This is done by establishing a certain inductive-type framework for generation in quantum groups and relies on recent topological generation result of Brannan–Chirvasitu–Freslon. Based on arXiv:1904.07721.

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Hun Hee Lee (Seoul National Univelsity)

Spectral theory of weighted Fourier algebras of (locally) compact quantum groups

To a locally compact group G one can associate a commutative Banach algebra A(G) called the Fourier algebra, whose Gelfand spectrum recovers G as a topological space. Recently, weighted versions of Fourier algebra have been introduced, whose spectrum is realized in the complexification GC of the Lie group G. In this talk we discuss about its quantum extension. We first consider weighted versions of Fourier algebras of compact quantum groups focusing on the spectral aspects of the resulting Banach algebras in two different ways. The first option would be investigating their Gelfand spectrum, which shows a connection to the maximal classical closed subgroup and its complexification. Second, we study specific finite dimensional representations coming from the complexification of the underlying quantum group. We demonstrate that the weighted Fourier algebras can detect the complexification structure in the special case of SUq(2), whose complexification is the quantum Lorentz group SLq(2, C). We finally discuss about some possibility of non-compact cases by reviewing the case of Lie groups.

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Yuri I. Manin (Max Planck Institute for Mathematics)

Quadratic algebras as quantum linear spaces: monoidal structures, dualities, and enrichmets

In my Montreal lectures of 1988, I developed the approach to quantum group putting in the foreground non-commutative versions of their group rings rather than universal enveloping algebras. In this approach, the classical category of vector spaces is replaced by the category of quadratic algebras.

In this talk, I make a short survey of basic properties of these “quantum linear spaces”, and then extend the relevant definitions and results to the category of operads whose components are quadratic algebras.

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Ehud Meir (University of Aberdeen)

Geometric invariant theory and Hopf algebras

Geometric invariant theory (GIT) has applications in several areas of the theory of Hopf algebras. It can be used to find a complete set of scalar invariants for semisimple Hopf algebras and for Hopf cocycles, and also to study non-semisimple Hopf algebras in braided monoidal categories. In this talk I will explain how scalar invariants can be used to prove the finite number of Hopf orders of a given finite dimensional semisimple Hopf algebra, and an ongoing work about the rigidity of Nichols algebras.

Tobias Ohrmann (Leibniz University Hannover)

Non-semisimple modular tensor categories from small quantum groups

One of the main reasons to study modular tensor categories is their importance in the description of 2d conformal field theories (CFTs). More precisely, it is believed that the chiral part of a 2d CFT is encoded in the representation category of a vertex operator algebra (VOA). At least in the rational (i.e. finite and non-semisimple) case, this is known to be a modular tensor category.

In the talk, I first introduce a family of finite-dimensional quasi-triangular quasi-Hopf algebras, generalizing Lusztig's small quantum groups. By modularizing their representation categories, we obtain a large class of finite non-semisimple modular tensor categories. Conjecturally, these are equivalent to the representation categories of
the underlying VOAs (known as W-algebras) of a certain family of logarithmic (i.e. non-semisimple) 2d CFTs (known as logarithmic extensions of minimal models). We will then see how to generalize this approach to a larger class of quasi-triangular quasi-Hopf algebras.

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Damien Rivet (University Clermont Auvergne)

Explicit Rieffel induction module for quantum groups

For G an algebraic (or more generally, a bornological) quantum group, and B a closed quantum subgroup one can build an induction module by explicitly defining an inner product which takes its value in the convolution algebra of B, as in the original approach of Rieffel. In this context, I will discuss the link with the induction functor defined by Vaes for von Neumann algebraic quantum groups.
I will also illustrate this construction with the example of parabolic induction for complex semi-simple quantum groups.

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Simon Schmidt (Saarland University)

On the quantum symmetry of distance-transitive graphs

To capture the symmetry of a graph one studies its automorphism group. A generalization of this concept is the quantum automorphism group of a graph. An important task now is to see whether or not a graph has quantum symmetry, i.e. whether or not its automorphism group and its quantum automorphism group coincide. We will discuss this problem for distance-transitive graphs and show that certain families of graphs have no quantum symmetry.

Alexander Shapiro (The University of Edinburgh)

Positive Peter–Weyl theorem

The classical Peter–Weyl theorem asserts that the regular representation of a compact Lie group G on the space of square-integrable functions L2(G) decomposes as the direct sum of all irreducible unitary representations of G. In the talk, I will use positive representations of cluster varieties, to obtain a “non-compact” quantum analogue of the Peter–Weyl theorem. This is a joint work with Ivan Ip and Gus Schrader.

slides

On C*-completions of discrete quantum group rings

Recently Grigorchuk, Musat and Rordam introduced and studied the class of just infinite C*-algebras and noted that for group C*-algebras this property is related to the existence of various C*-completions of a group ring. They asked in particular whether such a completion is unique if and only if the group in question is locally finite. We will discuss corresponding results for discrete quantum groups and show that in the quantum world, i.e. when we look at C*-completions of Hopf *-algebras associated to compact quantum groups, the answer is negative. (Joint work with Martijn Caspers)

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Alexander Stottmeister (University of Münster)

Quantum field theory on the lattice: renormalization via operator algebras

We discuss Hamiltonian lattice quantum field theory and the implementation of Wilson’s approach to the renormalization group from an operator-algebraic perspective. We indicate how the problem of identifying suitable continuum limits can be used to construct interesting examples of von Neumann algebras. As working examples, we present the constructions of scalar field theory in a box and Yang–Mills theory on a space-time cylinder and the latter's potential relations with quantum stochastic processes. Moreover, we point out connections with recently proposed actions of Thompson’s groups by Vaughan Jones (see the talk by Arnaud Brothier).

Wenqing Tao (University of Hasselt)

Noncommutative Riemannian geometry on finite groups and Hopf quivers

We explore the differential geometry of finite groups where the differential structure is given by a Hopf quiver (in the sense of C. Cibils and M. Rosso) rather than as more usual by a Cayley graph. In this setting, the duality between the geometries on the function algebra and group algebra emerges and is realised as an extension of duality between path algebra and path coalgebra on the same quiver. Interestingly, linear (left) connections are given by quiver representations. We show how quiver geometries arise naturally in the context of quantum principal bundles. If time allows I will illustrate these on the symmetric group S3. The talk is based on the joint works with Shahn Majid.

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Ami Viselter (The University of Haifa)

Generating functionals on quantum groups

We will discuss generating functionals on locally compact quantum groups. One type of examples comes from probability: the family of distributions of a L\'evy process form a convolution semigroup, which in turn admits a natural generating functional. Another type of examples comes from (locally compact) group theory, involving semigroups of positive-definite functions and conditionally negative-definite functions, which provide important information about the group's geometry. We will explain how these notions are related and how they extend to locally compact quantum groups; see how generating functionals may be (re)constructed and study their domains; and indicate several applications, in particular to cocycles. Based on joint work with Adam Skalski.

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Christian Voigt (The University of Glasgow)

Categorical representations of quantum groups

In this talk I’ll discuss representations of (locally compact) quantum groups on C*-categories and some aspects of their duality theory.

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Jonas Wahl (The University of Bonn)

Representation theory for compact quantum groups at the limit

This talk is intended as a survey on recent developments regarding the q-deformation of the representation theory of the infinite unitary group. We will give a detailed overview of Gorin's classification of q-central measures on the boundary of the Gelfand–Tsetlin graph and recent results by Sato on quantized characters on the Stratila–Voiculescu algebra. We will define the tensor product operation in this context and we will give explicit decomposition rules for tensor products of ‘discrete’ quantized characters. This is part of ongoing work joint with Alexey Bufetov.

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Hua Wang (The Mathematics Institute of Jussieu–Paris Rive Gauche)

Bicrossed products whose dual has property (RD) but not polynomial growth

From the work of Haargerup in the classical case, to the work of Vergnioux in the quantum case, among many others, property (RD) becomes an interesting approximation property of discrete quantum groups due to its intimate connection to K-theory. It is weaker than the better known notion of polynomial growth. To study property (RD) of a discrete quantum group requires a deep understanding of the representation theory of its dual on the one hand, and a careful treatment of the relevant length function on the other. Thus, in the quantum setting, it is nontrivial to come up with interesting examples with property (RD). In my talk, based on a previous work of Pierre Fima and myself, using the classic bicross products, I will describe a concrete construction of compact quantum groups whose dual has property (RD) but not polynomial growth.

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Published Apr. 2, 2019 11:42 PM - Last modified Aug. 19, 2019 1:34 PM