Prof. Dr. Moritz Weber




List of publications (see below)

Research interest

Keywords:
  • functional analysis and operator algebras (C*-algebras, von Neumann algebras)
  • quantum symmetries (compact quantum groups, compact matrix quantum groups, Banica-Speicher quantum groups/ easy quantum groups, classification of quantum subgroups, quantum automorphism groups of finite graphs)
  • free probability theory (de Finetti theorems, noncommutative distributions, bi-free probability)
  • combinatorics (set partitions, noncrossing partitions, finite graphs)
My research is in functional analysis, with links to algebra, combinatorics and (free) probability theory. One of the main motivations is to study structures in mathematics when noncommutativity of the multiplication plays a role, for instance when dealing with matrices in linear algebra, operators on a Hilbert space in functional analysis, or with observables in quantum physics.
In such noncommutative theories, symmetries are no longer given only by groups, but by more general objects: by quantum groups. In 1987, Woronowicz gave an analytic definition based on C*-algebras. In the 1990's, Wang introduced free quantum versions of the orthogonal group On, the unitary group Un (both over the complex numbers) and the symmetric group Sn. His work was extended and put into a systematic framework - the one of Banica-Speicher quantum groups (also called easy quantum groups) - by Banica and Speicher in 2009. Their approach is based on the combinatorics of set partitions. These partitions also appear in Voiculescu's free probability theory. There are deep links between Banica-Speicher quantum groups and free probability theory for instance via de Finetti theorems.

Amongst other things, a main focus of my past research has been on the classification of Banica-Speicher quantum groups. This has been completed in the orthogonal case (partially joint work with Raum) and has been begun in the unitary case (joint work with Tarrago). Furthermore, I investigated the combinatorial nature of the representation theory of Banica-Speicher quantum groups and proved a de Finetti theorem for bifreeness (both in joint work with Freslon). New examples of partially quantized quantum groups (similar to the graph product of groups) have been given by Speicher and myself, building on partially quantized real spheres. This can be seen as the quantum isometry groups of some quantum spheres in the sense of Connes's noncommutative geometry. With Gabriel, I investigated fixed point algebras of the actions of Banica-Speicher quantum groups on the Cuntz algebra and we computed their K-groups. With my PhD student Stefan Jung, I am working on quantum spaces fitting with Banica-Speicher quantum groups. With my PhD student Simon Schmidt, I am working on quantum automorphism groups of finite graphs. With my PhD student Daniel Gromada, I am developping computer algebraic methods for the classification of compact matrix quantum groups. With my PhD student Laura Maaßen, I am exploring the links between certain reflection groups and quantum groups.

Here is a little (slightly outdated) introduction into Banica-Speicher/easy quantum groups and their links to free probability:
What actually are Banica-Speicher quantum groups?

Member (PI) of the SFB-TRR 195

Since 1 January 2017, I am one of the PI's of the SFB-TRR 195 Symbolic Tools in Mathematics and their Applications (RWTH Aachen, TU Kaiserslautern, Saarland University, and others), funded by the DFG (German Research Foundation). My project is: I.13 - Computational classification of orthogonal quantum groups.

Publications

Preprints

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  • Weber, M.
    Partition C*-algebras
    arXiv:1710.06199 [math.OA, math.CO, math.QA], 19 pages + 11 pages of appendix and references (2017).

  • Weber, M.
    Partition C*-algebras II - links to compact matrix quantum groups
    arXiv:1710.08662 [math.OA, math.CO, math.QA], 27 pages (2017).

  • Cébron, Guillaume; Weber, M.
    Quantum groups based on spatial partitions
    arXiv:1609.02321 [math.QA, math.OA], 32 pages (2016).

  • Speicher, Roland; Weber, M.
    Quantum groups with partial commutation relations
    arXiv:1603.09192 [math.QA, math.OA], 44 pages (2016).


Monograph

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Voiculescu, Dan-Virgil; Stammeier, Nicolai; Weber, M. (eds)
Free probability and operator algebras
Münster Lecture Notes in Mathematics
European Mathematical Society (EMS)
132 pages
Zürich, 2016

Chapter in monographs

  • Weber, M.
    Basics in free probability, 6 pages
    in Free probability and operator algebras, ed. by Dan-V. Voiculescu, N. Stammeier, M. Weber, EMS, 2016.

  • Weber, M.
    Easy quantum groups, 23 pages
    in Free probability and operator algebras, ed. by Dan-V. Voiculescu, N. Stammeier, M. Weber, EMS, 2016.


in peer reviewed journals

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Google Scholar Account




updated: 30 November 2017   Moritz Weber Impressum