## Noncommutative geometry

### ACADEMIC YEAR 2019/2020

### 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.*

### 21 October 2019, 10:30

**Z/2Z-EXTENSIONS OF COMPACT MATRIX QUANTUM GROUPS**

In 2009, Banica and Speicher introduced categories of partitions in order to study a class of compact matrix quantum groups and their representation categories. Later, it turned out that these structures are a valuable tool that can be used to construct new examples of quantum groups and better understand their structure. Categories of partitions, in the sense of the original definition of Banica and Speicher, were already classified. Nevertheless, many different generalizations of this concept were introduced, bringing new inputs and inspiration to the theory of compact matrix quantum groups. In the first part of the talk, we briefly introduce the theory of categories of partitions and describe some generalizations. Then we focus in more detail on a new result regarding construction of Z/2Z-extensions of compact matrix quantum groups, using so-called partitions with extra singletons. We introduce a series of new quantum group products interpolating the tensor product and the free product.

DANIEL GROMADA (Universität des Saarlandes, Saarbrücken)

### 21 October 2019, 14:00

**EQUIVARIANT DIMENSIONS OF GRAPH C*-ALGEBRAS**

We explore the recently introduced local-triviality dimensions by studying gauge actions on graph *C**-algebras, as well as the restrictions of the gauge action to finite cyclic subgroups. For *C**-algebras of finite acyclic graphs and finite cycles, we characterize the finiteness of these dimensions, and we further study the gauge actions on many examples of graph *C**-algebras. These include the Toeplitz algebra, Cuntz algebras, and *q*-deformed spheres. (Based on joint work with A. Chirvasitu and B. Passer.)

MARIUSZ TOBOLSKI (IMPAN)

### 28 October 2019, 10:30

**PODLEŚ SPHERES FOR THE BRAIDED QUANTUM SU(2)**

The construction of "quantum spheres" was first accomplished by Podleś , starting with the quantum group SU*q*(2) for real *q* such that 0<|*q*|<1. Recently, a version of this quantum group was defined by Kasprzak, Meyer, Roy and Woronowicz for complex *q* satisfying 0<|*q*|<1. Here the crucial difference is that, if* q* is not real, the comultiplication takes values in the "braided tensor product" of the C*-algebra C(SU* _{q}*(2)) with itself. I will discuss the result that, despite this difference, the quantum spheres in the braided case (

*q*not real) are exactly the same as those found by Podleś for |

*q*|.

PIOTR M. SOŁTAN (Uniwersytet Warszawski)

### 28 October 2019, 14:00

**TOWARDS THE CLASSIFICATION OF LOCALLY TRIVIAL NONCOMMUTATIVE PRINCIPAL BUNDLES**

A classical result from topology states that if *X* is a numerable principal *G*-bundle, then there exists a map from *X/G* to the classifying space *BG*, and all principal* G*-bundles isomorphic to *X* are classified by the homotopy class of this map. The aim of this talk is to find an analog of this result in noncommutative

topology. First, we introduce the notion of a numerable noncommutative principal bundle in the setting of compact group actions on unital σ-C*-algebras. Then, for a compact group *G*, we define the unital σ-C*-algebra of functions on the noncommutative classifying space, and prove that it classifies all numerable noncommutative principal *G*-bundles. (Based on joint work with Alexandru Chirvasitu)

MARIUSZ TOBOLSKI (IMPAN)

### 4 November 2019, 10:30

**RELATIVE DOUBLE COMMUTANTS IN CORONAS OF SEPARATABLE C*-ALGEBRAS **

Given a subalgebra *A* of a C*-algebraic corona algebra, one can define a relative commutant. In several cases, it is known that the double relative commutant of* A* is *A* itself. We show that this holds true in a class of corona algebras for separable subalgebras *A* that are unital in a suitable sense. The class of corona algebras considered is the class of coronas of stable, separable, simple, nuclear C*-algebras. Separability is an essential condition, as shown by some counterexamples. (Joint work with Martin Mathieu, QUB.)

DAN KUCEROVSKY (University of New Brunswick, Fredericton)

### 4 November 2019, 14:00

**FROM PUSHOUTS OF GRAPHS TO PULLBACKS OF GRAPH ALGEBRAS**

We search for a new concept of graph morphism that would ensure that the assignment of graph algebras to graphs becomes a contravariant functor translating pushouts of graphs into pullbacks of graph algebras. The case of an injective morphism between row-finite graphs is solved by a known concept of admissible subgraph. The non-injective case is motivated by natural and highly non-trivial examples from noncommutative topology (e.g., quantum weighted projective spaces). To accommodate this naturally occurring non-injectivity, we replace the standard idea of mapping vertices to vertices and edges to edges by the more flexible idea of mapping finite paths to finite paths. (Based on joint works with Alexandru Chirvasitu, Sarah Reznikoff and Mariusz Tobolski.)

PIOTR M HAJAC (IMPAN)