Cohomology of quantum groups and quantum automorphism groups of finite graphs
Workshop in Saarbrücken, 8 - 12 October, 2018
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Funded by the SFB TRR 195 Symbolic Tools in Mathematics and their Applications and the German Research Foundation. |
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Organized by Simon Schmidt and Moritz Weber.
Aim of the workshop
Participants
Schedule
Preprogram
Accommodation
Abstracts of the talks
Aim of the workshop
The idea of the workshop is to bring together people who are interested in- Cohomology of quantum groups
- Quantum automorphism groups of finite graphs.
related to those topics. We aim for about 12 talks and enough time for private discussions. Note
that the workshop features a lecture series on cohomology of quantum groups by Julien Bichon,
see also here . On Tuesday, October 2 in the afternoon, there will be a preprogram for local students.
Participants
- Isabelle Baraquin
- Julien Bichon
- Arthur Chassaniol
- Kari Eifler
- Uwe Franz
- Amaury Freslon
- Daniel Gromada
- Stefan Jung
- Luca Junk
- Laura Maaßen
- Myriam Mahaman
- Alexander Mang
- Manuel Martins
- Vincent Preiß
- David Reutter
- David Roberson
- Julien Schanz
- Simon Schmidt
- Adam Skalski
- Roland Speicher
- Mirko Stappert
- Monika Varšo
- Dominic Verdon
- Xumin Wang
- Mateusz Wasilewski
- Moritz Weber
- Alexander Wendel
- Andreas Widenka
- Anna Wysoczańska-Kula
Schedule
The workshop is anticipated to start on monday morning and close on friday around lunchtime.Each talk consists of 50 minutes plus 10 minutes discussion. All lectures take place in lecture hall IV,
building E2 4. The seminar rooms SR7(203), SR6(217), SR10(316) and lecture hall IV can be used
for private discussions. You can find the schedule below.
Monday, 8 October
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9:30 am 10:30 am 11:00 am 12:00 am 1:00 pm 2:00 pm 3:00 pm 3:30 pm |
Julien Bichon Coffee Anna Kula Daniel Gromada Lunch Manuel Martins Coffee Private Discussions |
Tuesday, 9 October
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9:30 am 10:30 am 11:00 am 12:00 am 1:00 pm 2:00 pm 3:00 pm 3:30 pm |
Julien Bichon Coffee Simon Schmidt Arthur Chassaniol Lunch Snapshot session (Isabelle Baraquin, Kari Eifler, Myriam Mahaman, Adam Skalski, Monika Varšo, Mateusz Wasilewski) Coffee Private Discussions |
Wednesday, 10 October
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9:30 am 10:30 am 11:00 am 12:00 am 1:00 pm 2:00 pm 7:00 pm |
Julien Bichon Coffee Laura Maaßen Alexander Mang Lunch Free Afternoon/Excursion (to Völklinger Hütte) Dinner at Stiefel Bräu . |
Thursday, 11 October
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9:30 am 10:30 am 11:00 am 12:00 am 1:00 pm 2:00 pm 3:00 pm 3:30 pm |
David Roberson Coffee David Reutter Dominic Verdon Lunch Amaury Freslon Coffee Private Discussions |
Friday, 12 October
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9:30 am 10:30 am 11:00 am 12:00 am 1:00 pm |
Uwe Franz Coffee Private discussions Lunch End of the Workshop |
Preprogram
On Tuesday, 2 October, there will be two preparatory lectures for local students.The lectures take place in seminar room 6, building E2 4.
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2:00 pm 3:00 pm 3:30 pm |
Quantum groups and quantum automorphism groups of finite graphs Coffee Cohomology (notes) |
Accommodation
Accommodation will be provided. For more information on how to get to Saarbrücken andhow to get to the math department, see our webpage with informations for visitors.
Abstracts of the talks
- Arthur Chassaniol
Quantum symmetries of vertex-transitive graphs using Tannaka-Krein dualityAs T. Banica and R. Speicher did for the liberation of othogonal Lie groups we use Tannaka-Krein duality to establish a correspondence between quantum automorphism groups of graphs and symmetric tensor categories made of intertwinners spaces. Then we study these spaces for vertex-transitive graphs to characterize graphs having quantum symmetries. We finally give some applications for circulant graphs. - Uwe Franz
Hochschild cohomology of universal quantum groups and related topicsWe compute the residually finite dimensional quotient of the universal unitary quantum groups UQ+ of Wang and Van Daele and present partial results about Hochschild cohomology groups of this quantum group. The talk is based on joint work with Biswarup Das, Anna Kula, and Adam Skalski. - Daniel Gromada
Intertwiner spaces of quantum group subrepresentations
Easy quantum groups are quantum groups lying between SN and ON+, whose intertwiner spaces can be described using partitions. If we want to describe all quantum groups between SN and ON+, we have to use linear combinations of partitions. One of the goals of our research is to find examples of non-easy quantum groups between SN and ON+ and to describe their intertwiner spaces.
In the talk we present a way how to obtain certain examples. Every quantum group G such that SN ⊂ G ⊂ BN#+ has an (N − 1)-dimensional subrepresentation. It turns out that this subrepresentation can be put in such a form that it generates a quantum group H such that SN-1 ⊂ H ⊂ ON-1+ ,so it can be described by linear combinations of partitions and we are able to derive the corresponding category explicitly. Additional examples can be constructed by going back and extending this subrepresentation by some one-dimensional representation.
In the end of the talk, we indicate the direction of our further research, which is to attack the last question of extending a given quantum group H by Z2 using the A. Freslon’s colored partitions. - Laura Maaßen
The intertwiner spaces of non-easy group-theoretical quantum groupsIn 2015, Raum and Weber definied group-theoretical quantum groups as compact matrix quantum groups whose squared generators are central projections. They showed that group-theoretical quantum groups have a presentation as semi-direct product quantum groups and studied the case of easy group-theoretical quantum groups. In this talk we determine the intertwiner spaces of non-easy group-theoretical quantum groups. We generalise group-theoretical categories of partitions and use a fiber functor to map partitions to linear maps which is slightly different from the one for easy quantum groups. We show that this construction provides the intertwiner spaces of group-theoretical quantum groups in general. - Alexander Mang
Half-Liberations of the Unitary GroupCategories of colored pair partitions correspond to easy quantum groups interpolating either the orthogonal or the unitary group and their respective free quantum versions, depending on how many colors are used. Only one non-trivial category of one-colored pair partitions exists, giving rise to the half-liberated orthogonal quantum group. However, multiple unitary half-liberations emerge when applying two colors instead of one. By choosing a certain class of projective universal generators as a starting point, all such categories can be determined, both in terms of their abstract descriptions and their generating partitions. Doing so enlightens features of the liberation process invisible in the orthogonal case. In particular, two distinct stages become apparent, including a unique category at the transition mark. This is joint work with Moritz Weber. - Manuel Martins
Quantum groups acting on the nodal cubicSome singular curves in the plane have recently been shown to admit a quantum group action, which turns them into quantum homogeneous spaces. The principal bundle over the curve correspond to a coalgebra Galois extension arising in this setting and the associated vector bundles become candidates to generate the K_0 group of the curve, if the extension is not cleft. One concrete example of such singular curves is the nodal cubic. In this talk we understand the structure of the corresponding Hopf algebra in terms of small quantum groups at roots of 1 and use this to show that the resulting coalgebra Galois extension is minimal but also cleft. Masuoka's study of Hopf algebras with cocommutative coradical offers an alternative proof of the cleftness. We also discuss some ring-theoretic properties of the Hopf algebra and describe its group-likes and twisted primitives. - David Reutter
From quantum automorphisms to pseudo-telepathyQuantum pseudo-telepathy is a phenomenon in quantum information theory, where non-communicating parties can use pre-shared entanglement to perform a task classically impossible without communication. Instances of quantum pseudo-telepathy in the graph isomorphism game (described in the previous talk) correspond to quantum isomorphisms between non-isomorphic graphs. Superficially, this seems to be at odds with the study of quantum automorphism groups of graphs which, after all, are about quantum isomorphisms from a graph to itself.
However, in this talk, I will explain how the quantum automorphism group of a graph completely determines the quantum isomorphisms out of it. Explicitly, graphs quantum isomorphic to a given graph G can be classified in terms of certain algebraic structures in the category of representations of the algebra dual to the quantum automorphism group of G. In the case that G has no quantum symmetries, this classification can be expressed purely in terms of classical group theory and symplectic linear algebra.
This work is based on a notion of `quantum function' and a 2-categorical framework for finite quantum set and quantum graph theory, which `quantizes' the usual categories of sets and graphs and which combines quantum isomorphisms and quantum automorphisms in a single algebraic framework. - David Roberson
Quantum Isomorphisms and AutomorphismsWe introduce the notion of quantum isomorphism of two graphs. The definition is based on a game in which two parties cooperate in an attempt to convince a referee that the two graphs are isomorphic. Perfect classical strategies correspond to isomorphisms of the graphs, and thus quantum isomorphisms are defined in terms of perfect "quantum" strategies for this game, i.e., strategies in which the parties are allowed to perform local quantum mechanical measurements on a shared quantum state. Though the definition seems far removed from the C*-algebraic notion of quantum automorphism groups of graphs, we will see that they are related in the same way as classical isomorphisms and automorphism groups of graphs. This provides a link between quantum permutation groups and physical behaviors that are observable in the real world.
We also define orbits and orbitals (orbits on ordered pairs) of quantum permutation groups. We show that, as in the classical case, the orbitals form a coherent configuration which provides information about the quantum permutation group. This also allows us to rephrase the existence of a quantum isomorphism between two graphs in terms of the quantum automorphism group of their disjoint union.
Finally, we introduce a construction of non-isomorphic yet quantum isomorphic graphs, the existence of which is nontrivial. - Simon Schmidt
On the quantum symmetry of graphsTo 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 is now to see if a graph has quantum symmetry or not, i.e. if its automorphism group and its quantum automorphism group coincide.
In this talk, we show that the Petersen graph has no quantum symmetry by using the fact that the Petersen graph is a strongly regular graph. Furthermore, we show that if the automorphism group of a graph contains a certain pair of automorphisms, this graph has quantum symmetry. - Dominic Verdon
Quantum symmetries from classical symmetriesUsing the classification from the previous talk, we exhibit a construction of quantum graph isomorphisms into a given graph from certain subgroups of its classical automorphism group. If a graph has no quantum symmetries, this construction gives rise to all quantum isomorphisms into it. We can therefore apply recent results about graphs with no quantum symmetries to quantum information theory; we show that no vertex transitive graph of order less than or equal to 11 can exhibit pseudo-telepathy in the graph isomorphism game. (Interestingly, some of these graphs do admit quantum isomorphisms to certain noncommutative graphs, of possible interest in zero-error communication.) We also show how our construction may also be used to obtain quantum symmetries of a graph (i.e. irreducible representations of the quantum automorphism group algebra) solely from its classical automorphism group. - Anna Wysoczańska-Kula
The second cohomology group with trivial coefficeints of the free unitary quantum groupI will present the main ideas of the method that served to compute H^2(U_n^+, {_\varepsilon}\mathbb{C}_\varepsilon), second cohomology group with trivial coefficeints of the free unitary quantum group. If time permits, I will present some proofs and make some comments on H^2 for the freely modified bistochastic. The talk is based on joint work with Biswarup Das, Uwe Franz, and Adam Skalski.
| Updated: 15 October 2018 Simon Schmidt | Impressum |
