## Noncommutative geometry

### Organisers: Piotr M. Hajac and Tomasz Maszczyk

*Venue: Ul. Śniadeckich 8, room 321, Mondays*

*To see the video recording of a talk, click on its title.*

**Academic year 2018/2019:**

### 15 October 2018, 10:30-12:00

**AN EQUIVARIANT PULLBACK STRUCTURE OF TRIMMABLE GRAPH C*-ALGEBRAS**

We prove that the graph C*-algebra C*(*E*) of a trimmable graph *E* is *U*(*1*)-equivariantly isomorphic to a pullback C*-algebra of a subgraph C*-algebra C*(*E"*) and the C*-algebra of functions on a circle tensored with another subgraph C*-algebra *C**(*E'*). This allows us to unravel the K-theory of the fixed-point subalgebra C*(*E*)^{U(1)} through the (typically simpler) K-theory of C*(*E'*), C*(*E"*) and C*(*E"*)^{U(1)}. To obtain interesting examples of trimmable graphs, we consider one-loop extensions of the standard graphs encoding, respectively, the Cuntz algebra *O _{2}* and the Toeplitz algebra

*T*. Then we analyze equivariant pullback structures of trimmable graphs yielding the C*-algebras of the quantum lens spaces

*L*

^{3}_{q}_{(k;1,k)}. Based on joint work with Francesca Arici, Francesco D'Andrea and Piotr M. Hajac.

### 15 October 2018, 14:00-15:30

**COMPLEMENTED PARTITIONS OF UNITY AND THE TOMS-WINTER CONJECTURE**

This talk will be based on the recent result of myself, Castillejos, Tikuisis, White and Winter that simple, separable, unital, nuclear C*-algebras which absorb the Jiang-Su algebra tensorially have nuclear dimension at most 1. I will begin by explaining the broader context for this result and its applications for classifying C*-algebras. I will then outline the proof of result, before discussing the key new idea in the proof, complemented partitions of unity (*CPoU*), in some detail. I also will discuss my recent research with Castillejos on the non-unital version of this result.

SAMUEL EVINGTON (IMPAN)

### 22 October 2018, 10:30-12:00

**ON THE STRUCTURE OF AMPLE GROUPOID C*-ALGEBRAS**

This talk is an introduction to the structure theory of groupoid C*-algebras. I will focus on a recent joint work with Kang Li on the ideal structure and pure infiniteness of these algebras. If time permits, I will also mention some work in progress on the finite setting.

CHRISTIAN BÖNICKE (University of Münster)

### 22 October 2018, 14:00-15:30

**COVARIANT REPRESENTATION OF QUANTUM GROUPS AND ADMISSIBILITY CONJECTURE**

A finite-dimensional unitary representation of a locally compact quantum group is called admissible if it factors through a matrix quantum group. An important and a very old open question in the representation theory of quantum groups is to determine whether all finite-dimensional (unitary) representations are admissible. Of course, one big hurdle in this direction is to have enough examples of locally compact quantum groups with admissible representations. In this talk, we will first introduce a new class of quantum group representations, which are covariant with respect to the scaling automorphism group. We will link this with the aforementioned problem, and then using this connection, we will construct concrete examples of locally compact quantum groups with admissible representations. Based on joint work (and an ongoing project) with M. Daws and P. Salmi.

BISWARUP DAS (Uniwersytet Wrocławski)

### 29 October 2018, 10:30-12:00

**QUANTUM DIRICHLET FORMS AND THEIR APPLICATIONS TO LOCALLY COMPACT QUANTUM GROUPS (PART 1)**

The theory of Dirichlet forms is a crucial tool in the analysis of classical Markov semigroups. Following the work of Goldstein, Lindsay and Cipriani, we will first outline some results on quantum Dirichlet forms in the most general context of von Neumann algebras equipped with weights. Then, we will show how, in the presence of translation invariance, these can be used in the study of locally compact quantum groups and the related convolution semigroups of states. Some examples, applications to approximation properties, and very recent results on generating functionals for convolution semigroups will also be presented. Based on joint work with Ami Viselter.

ADAM SKALSKI (IMPAN)

### 29 October 2018, 14:00-15:30

**QUANTUM DIRICHLET FORMS AND THEIR APPLICATIONS TO LOCALLY COMPACT QUANTUM GROUPS (PART 2)**

(see above)

ADAM SKALSKI (IMPAN)

### 5 November 2018, 10:30-12:00

**GENERALIZED (CO)INTEGRALS ON COIDEAL SUBALGEBRAS**

Given a Hopf algebra *H*, its left-coideal subalgebra *A* and a non-zero multiplicative functional *ω* on *A*, the space of left *ω*-integrals consists of elements *x* in *A* satisfying *ax = ω(a)x* for all *a* in *A*. We observe that, if *A* is a Frobenius algebra, then the space of left *ω*-integrals is one-dimensional. In particular, the above equality holds for finite-dimensional left-coideal subalgebras of a weakly finite Hopf algebra. In general, we prove that, if the space of *ω*-integrals is nontrivial, then *A* is finite dimensional. Given a group-like element *g* in *H*, we define the space of *g*-cointegrals on *A*, and by linking this concept with the theory of *ω*-integrals, we observe the following: (1) Every semisimple left-coideal subalgebra that is preserved by the antipode squared admits a faithful 1-cointegral. (2) Every unimodular finite-dimensional left-coideal subalgebra admitting a faithful 1-cointegral is preserved by the antipode squared. (3) Every non-degenerate right group-like projection in a cosemisimple Hopf algebra is a two-sided group-like projection. Finally, we list all counit-integrals and all *g*-cointegrals for left-coideal subalgebras in Taft algebras. Partially based on joint work with A. Chirvasitu and P. Szulim.

PAWEŁ KASPRZAK (Uniwersytet Warszawski)

### 5 November 2018, 14:00-15:30

**THETA-DEFORMATION OF COMPACT QUANTUM GROUPS AND THEIR ACTIONS**

In his framework, Rieffel showed that compact Lie groups of rank at least 2 admit nontrivial *Θ*-deformations as compact quantum groups. In my recent work, I showed that an action of such a Lie group *G* on a manifold *M* with a toric action can be extended to an action of the *Θ*-deformation *G _{Θ}* on the

*Θ*-deformation

*M*(two different parameter matrices). Of course, an action cannot be extended to an action of the deformed algebras for arbitrary

_{Θ'}*Θ*-parameters. I will explain exactly when this happens and give one example. If time permits, I will also explain how the noncommutative sphere

*S*can be viewed as a quantum homogeneous space.

^{7}_{Θ}MITSURU WILSON (IMPAN)

### 12 November 2018, 10:30-12:00

**LINE BUNDLES AND DRINFEL'D TWISTS IN DEFORMATION QUANTIZATION**

In this talk, I will illustrate the role of line bundles and Drinfel'd twists in the deformation quantization. A condition that guarantees the existence of a non-formal quantization for a compact symplectic manifold is that its symplectic form has an integer class, and the quantization procedure uses a line bundle that is non-trivial. Among homogeneous symplectic manifolds, an important class of examples is given by integral coadjoint orbits of Lie groups. In the framework of deformation quantization, we are particularly interested in deformations coming from a Drinfel'd twist (twisted star products). While the existence of a suitable non-trivial line bundle is a prerequisite for the geometric quantization, I will argue that the existence of a non-trivial equivariant line bundle is an obstruction for twisted star products. In this sense, the geometric quantization and the twist quantization are complementary. In particular, if *M* is a compact integral coadjoint *G*-orbit (with the canonical symplectic structure) and *g* is the Lie algebra of *G*, there is no deformation quantization of *M* coming from a Drinfel'd twist based on the formal deformation quantization *U(g)[[h]]* of the universal enveloping algebra of the Lie algebra *g*.

FRANCESCO D'ANDREA (University of Naples)

### 12 November 2018, 14:00-15:30

**DIAGONAL DIMENSION FOR C*-PAIRS**

We will introduce the notion of diagonal dimension for diagonal pairs of C*-algebras, and will compare it with the usual nuclear dimension for C*-algebras. We show that the diagonal dimension of a uniform Roe algebra with respect to the standard diagonal is equal to the asymptotic dimension of its underlying metric space. Finally, we will discuss its relation to the dynamic asymptotic dimension of groupoids introduced by Guentner, Willett and Yu and the (fine) tower dimension of topological dynamical systems introduced by Kerr. This is joint work with Hung-Chang Liao and Wilhelm Winter.

KANG LI (IMPAN)

### 19 November 2018, 10:30-12:00

**HOPF ALGEBRAS WITH AND WITHOUT MODULAR PAIRS IN INVOLUTION**

A modular pair in involution of a Hopf algebra *H* is a pair consisting of a grouplike element *g* in *H* and a character *f* on *H* such that *f*(*g*)* = *1 and the antipode square is given by the adjoint action of *g* and *f*. The ground field *k* of *H* can be viewed as a module and a comodule via the character and the grouplike, respectively. The module comodule *k* is then a stable anti-Yetter-Drinfeld module playing the role of coefficients in Hopf-cyclic cohomology introduced by Connes and Moscovici. When *H* is finite dimensional, the aforementioned antipode condition for (*g,f*) is a square root of the celebrated Radford formula for the 4th power of the antipode, and is thus related to the work of Radford and Kauffman concerning the existence of a ribbon element in the Drinfeld double of *H*. Using Radford's biproduct (also known as Majid's bosonization) of certain Nichols algebras over cyclic groups, we construct finite-dimensional Hopf algebras that do not admit a modular pair in involution. Based on joint work with U. Krähmer.

SEBASTIAN HALBIG (Technische Universität Dresden)

### 19 November 2018, 14:00-15:30

**C*-ALGEBRAIC QUANTUM GROUPOIDS AND MULTIPLICATIVE PARTIAL ISOMETRIES**

Motivated by the purely algebraic notion of weak multiplier Hopf algebras, we developed a C*-algebraic framework for a subclass of quantum groupoids, which extends the class of locally compact quantum groups. From the axioms, one can construct certain *multiplicative partial isometries* and the antipode map. However, in this talk, I wish to take a different approach: I will explore possible conditions, including the pentagonal equation among others, to be given on a partial isometry *W* in such a way that *W* encodes information about a dual pair of C*-algebraic quantum groupoids. As a special case, when *W* is a unitary, we obtain the Baaj-Skandalis notion of a multiplicative unitary. Finally, I will propose definitions that generalize the "regularity" of a locally compact quantum group and the "manageability" of a multiplicative unitary. Joint work with Van Daele (Leuven).

BYUNG-JAY KAHNG (Canisius College / SUNY, Buffalo, USA)

### 26 November 2018, 10:30-12:00

**NORM-STABILITY OF DISCRETE GROUPS**

Questions of stability for groups ask whether almost representations of a group have to be close to actual representations. Here, "almost" and "close" are measured by different norms. Such questions are related to approximation problems for groups. While for Hilbert-Schmidt-type norms, a lot of work had been done in recent years, for the operator norm, almost nothing had been known until my recent joint work with Søren Eilers and Adam Sørensen in which we initiated a systematic study of the operator-norm stability. The goal of this talk is to explain the new techniques that we developed, and to show how they allow us to obtain results (in some cases quite surprising) on the stability/non-stability of various classes of discrete groups.

TATIANA SHULMAN (IMPAN)

### 17 December 2018, 10:30-12:00

**COMPACT QUANTUM GROUPS WITH REPRESENTATIONS OF BOUNDED DEGREE**

Compact quantum groups are generalizations of compact Hausdorff groups into the realm of noncommutative geometry. Unlike in the classical group theory, the coinverse (antipode) need not be an involution. If it is, we call such a compact quantum group of Kac type. This involution property has various equivalent formulations in terms of representation theory. In the talk, I will present a theorem, which states that if every irreducible representation of a compact quantum group has dimension less than a fixed constant, then it has to be of Kac type. Our proof is based on a certain equality involving spectral projections.

JACEK KRAJCZOK (IMPAN)

### 17 December 2018, 14:00-15:30

**MODULAR PAIRS IN INVOLUTION FOR MULTIPLIER HOPF ALGEBRAS**

Multiplier Hopf algebras generalize Hopf algebras to the setting of nonunital algebras. In this talk, I shall discuss the definition and the existence of modular pairs in involution for a class of regular multiplier Hopf algebras. Based on joint work with Daniel Wysocki.

ANDRZEJ SITARZ (Uniwersytet Jagielloński / IMPAN)