NCG&T Prague

usually takes place each Tuesday at 16:00 Institute of Mathematics CAS, Blue lecture room, Zitna 25, Praha 1
Chair: Tristan Bice, Karen Stung

What is noncommutative geometry and topology? The idea stems from the Gelfand theorem which states that the category of compact Hausdorff spaces and commutative C*-algebras are dual. If we drop the condition of commutativity from our C*-algebras, we arrive at the notion of a noncommutative topological space. This can be carried further into the realm of noncommutative of geometry by equipping *-algebras with geometric structures.
Our research focusses on both quantum algebraic and operator algebraic aspects of noncommutative geometry and topology. This includes research in Hopf algebras, quantum groups, and noncommutative complex geometry, while on the operator algebra side, we study C*-algebras, with particular focus on C*-algebras arising from dynamical constructions such as minimal actions, groupoids, and semigroups.
Partially supported by GAČR project 20-17488Y Applications of C*-algebra classification: dynamics, geometry, and their quantum analogues and PRIMUS grant Spectral Noncommutative Geometry of Quantum Flag Manifolds.

Folding and Nichols algebras over nonabelian groups

Simon Lentner

University of Hamburg

Tuesday, 3. May 2022 - 16:00 to 17:00

This talk will take place on Zoom:
https://cesnet.zoom.us/j/91975183920?pwd=SGdlRUhLS21HUnVFTXBKcE1vYXlrQT09
Meeting ID: 919 7518 3920
Passcode: 102707

I will first give an introduction to Nichols algebras, also in the non-diagonal setting. I will then explain my folding construction, which turns a Nichols algebra with a diagram automorphism into a new Nichols algebra, and by which I was able to construct new families of Nichols algebras over nonabelian groups. I also put this construction into a broader systematic context of category extension, and I explain recent joint work with Angiono and Sanmarco building on work by Heckenberger and Vendramin, in which we show that all Nichols algebras over nonabelian groups with rank larger than 3 arise in this way.

Principal differential calculi over projective bases

Thomas Weber

University of Eastern Piedmont “Amedeo Avogadro”

Friday, 6. May 2022 - 16:00

This talk will take place in the blue seminar room, back building, Žitná 25.

It will also be broadcast on Zoom:
https://cesnet.zoom.us/j/91975183920?pwd=SGdlRUhLS21HUnVFTXBKcE1vYXlrQT09
Meeting ID: 919 7518 3920
Passcode: 102707

In noncommutative geometry Hopf–Galois extensions generalize the concept of principal bundle. Given a covariant first order differential calculus on the total space algebra there are natural notions of base forms and horizontal forms. If the covariant calculus on the structure Hopf algebra is compatible with the former, in the sense that the vertical map constitutes a short exact sequence, we have a principal differential calculus. Then, the faithfully flat Hopf–Galois extension amplifies to a Hopf–Galois extensions of graded objects. In this talk we give a sheaf-theoretic approach to principal differential calculi. This is required in the presence of projective bases, where global sections are trivial and the local information is crucial. The theory is elucidated by the explicit example of q-deformed SL(2,\mathbb{C}) endowed with its covariant calculi.

Ring-theoretic (in)finiteness in ultraproducts of Banach algebras

Bence Horváth

Institute of Mathematics of the Czech Academy of Sciences

Tuesday, 10. May 2022 - 16:00 to 17:00

We say that a unital ring is “Dedekind-finite” (or “directly finite” or “DF”) if every left-invertible element is also right-invertible. In other words, a ring is DF if and only if the only idempotent which is algebraically Murray–von Neumann equivalent to the unit of the ring is the unit itself. A related notion is that of the so-called “proper infiniteness” and “pure infiniteness”. These notions are well-studied in non-commutative ring theory, as well as in the theory of C*-algebras, but not that much in the Banach algebraic setting.

In our talk we outline how these properties are preserved under taking ultraproducts of Banach algebras, and vice versa. As one might expect, in the general Banach algebraic setting the situation differs quite a bit to the C*-algebraic one. Time permitting we say a few words about the related notion of having “stable rank one” and connections to the area of Continuous Model Theory. The talk is based on joint work with Matthew Daws (UCLAN,... more

C*-irreducible regular inclusions

Bartosz Kwaśniewski

Uniwersytet w Białymstok

Friday, 13. May 2022 - 16:00

Recently there has been a lot of research on C*-inclusions with the property that every intermediate C*-algebra is simple. Last year, Mikael Rørdam initiated a systematic study of such inclusions, which he called C*-irreducible. I will use this notion as an excuse to present some of the results of my recent work with Ralph Meyer. I will discuss the relationship between C*-irreducibility, aperiodicity, the almost extension property and noncommutative Cartan inclusions in the sense of Ruy Exel. This leads to a characterisation of regular C*-irreducible inclusions by a long list of equivalent conditions, one being that the inclusion comes from an outer Fell bundle over a discrete group.

Institute of Mathematics

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