Pivotal fusion categories: character theory and Galois symmetries

Abstract (Romanian)

Scopul acestui proiect este studiul categoriilor de fuziune si al algebrelor Hopf semisimple punand accent pe studiul categoriilor de fuziune braided.

In prima parte a proiectului vom incerca sa transferam cateva rezultate din teoria caracterelor algebrelor Hopf semisimple la categoriile de fuziune pivotale. Pentru aceasta, vom folosi teoria caracterelor pentru categoriile de fuziune pivotale recent dezvoltata de catre Shimizu. De asemenea, vom generaliza notiunea de nucleu a unui obiect simplu si vom explora conexiunile acesteia cu subalgebrele etale ale algebrei adjuncte. In raport cu o subcategorie de fuziune vom introduce notiunea de clasa de caractere similara celei folosite recent de catre Bantay pentru categoriile tensoriale modulare.

A doua parte a proiectului studiaza grupul Galois determinat de tabla de caractere a unei categorii de fuziune impreună cu actiunea sa asupra obiectelor simple. Clasele de conjugare pentru o categorie de fuziune au fost de asemenea introduse recent de catre Shimizu. Multiplicarea a doua sume de clase de conjugare determina anumite constante de structura care dau informatii noi despre structura categoriilor de fuziune.

A treia parte studiaza centralizatorul Müger pentru categoriile de fuziune braided. Vor fi explorate noi conexiuni intre clasele de conjugare, centralizatorul Müger si transformata Fourier a unei categorii de fuziune braided, subiect dezvoltat anterior de directorul de proiect pentru categoriile tensoriale modulare.

Abstract (English)

The goal of this project is to study the structure of fusion categories and semisimple Hopf algebras with a special attention to braided fusion categories.

In the first part of the project we will try to transfer some results from the character theory of semisimple Hopf algebras to pivotal fusion categories. In order to do this we will use the character theory for pivotal fusion categories recently developed by Shimizu. We will also generalize the notion of a kernel of a simple object and explore its connections with etale subalgebras of the adjoint subalgebra. The notion of classes of characters will be introduced relative to a fusion subcategory of a pivotal fusion category, similarly to the classes given recently by Bantay for modular tensor categories.

The second part of the project studies the Galois group determined by the character table of a fusion category together with its action on simple objects. Conjugacy classes for fusion categories were also recently introduced by Shimizu. The multiplication of two conjugacy class sums determines some constants that give new information on the structure of the fusion category.

The third part of the project studies Müger's centralizer for braided fusion categories. Connections between the Fourier transform, conjugacy classes and the Müger centralizer of a braided fusion category, as previously developed by the project leader in the settings of modular categories will also be explored.

Team members:

Experienced researchers:

• Sebastian Burciu,  project leader, IMAR.

• Radu Popescu,  researcher,  IMAR.

• Daniel Beltita, researcher, IMAR.

• Bogdan Ichim, researcher, IMAR.

• Vlad Matei, postdoctoral researcher, IMAR.

Young  researchers:

• Alexandra Andriciuc, Ph. D. student, IMAR.

• Ionut Simon, Ph. D. student,  IMAR (former member).

Research objectives:

1.  Character theory for fusion categories.  

2.  Galois symmetries and arithmetical properties of pivotal fusion categories.

3.  Müger’s centralizer in braided fusion categories.

2021

  Publications

1. S. Burciu, "On the Galois symmetries for the character table of an integral fusion category", Journal of Algebra and Its Applications, 22, (2023) no. 1, 2350026,  https://doi.org/10.1142/S0219498823500263

2.  I. Beltita, D. Beltita, "On square-integrable representations and the coadjoint action of solvable Lie groups", Preprint arXiv: 2111.01034, [math.RT].

3. W. Bruns and  B. Ichim, "Computations of volumes in five candidates elections",  Scientific Reports 13 (2023), 13266, 

https://www.nature.com/articles/s41598-023-39656-8. arxiv: 2109.00473, [math CO], [math AC].

4. D. Beltita, G. Larotonda, "Unitary group orbits versus groupoid orbits of normal operators", Journal of Geometric Analysis 33 (2023), no. 3, Paper No. 95, 44 pp, https://link.springer.com/article/10.1007/s12220-022-01187-5.

5. S. Burciu, "Subalgebras of etale algebras and fusion subcategories", Preprint, arxiv: 2105.11202, [math.QA]. 

Dissemination 

1. Sebastian Burciu, "Subalgebras of etale algebras and fusion subcategories", AMS Fall Western Sectional Meeting, Special Session on Tensor Categories and Applications, October 23-24, 2021 

www.ams.org/meetings/sectional/2283_program_ss9.html#title 

2.  Daniel Beltita, "C*-algebras of amenable non-discrete groups and their AF- embeddability, or not", Great Planes Operator Theory Symposium (GPOTS) (Washington University in St. Louis, USA), 10-14 May 2021 

www.math.wustl.edu/$%5Csim$ylsong/GPOTS.php   

3. Daniel Beltita, "Finite approximation properties of group C*-algebras", 

Seminario ``Mischa Cotlar'' (Instituto Argentino de Matematica ``Alberto P. Calderón'' in Buenos Aires, Argentina), June 18, 2021, 

seminariomcotlar.wordpress.com/2021/05/30/daniel-beltita-1300-bs-as/ 

4. Daniel Beltita, "Poisson geometrical aspects of von Neumann algebras", (mini-course consisting of 3 lectures),  A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability (University of Vienna, Austria), August 16-20, 2021, 

sites.google.com/view/finite-infinite-workshop 

5. Daniel Beltita, "Dynamical systems of nilpotent Lie groups and their corresponding C*-algebras", VII School and Workshop on Lie Theory (Universidade Federal de Juiz de Fora in Brazil), September 8-15, 2021, 

sites.google.com/view/liecongress7 

6. Daniel Beltita, "From closures of unitary group actions to groupoid actions",  VI Congreso Latinoamericano de Matematicos (CLAM 2021)/sesiunea 2. Analysis Funcional y Geometria (Centro de Matematica in Montevideo, Uruguay) September 13-17, 2021, 

https://clam2021.cmat.edu.uy/. 

7. Radu Popescu, Grafical calculus for morphisms in ribbon categories, Topology Seminar IMAR, November 26, 2021

Scientific report 2021

2022

  Publications

6. S. Burciu, S. Palcoux "Structure constants, Isaacs property and Extended-Haagerup fusion categories", Preprint, arxiv: 2210.13936, [math.QA].

7. S. Burciu, "On some Frobenius type divisibility results in a premodular category", Preprint, arxiv: 2209.12530, [math.QA].

8.  B. Ichim, J. J. Moyano-Fernandez, "On the consistency of the score sheets of a round-robin football tournament", Preprint, arxiv: 2208.12372, [math CO], [math AC].

9. I. Beltita, D. Beltita, "Continuous selection of Lagrangian subspaces, arXiv: 2211.02433, [math.RT], to appear Linear and Multilinear Algebra (DOI: 10.1080/03081087.2023.2263626).

10. I. Beltita, D. Beltita, "A note on ideal spaces of Mautner groups", Preprint, arXiv: 2211.04424, [math.OA].

11. I. Beltita, D.Beltita, "Solvable Lie groups whose regular representation is a factor representation",  In: P. Kielanowski, A. Dobrogowska, G.A. Goldin, T. Golinski (eds.), Geometric Methods in Physics XXXIX, XXXIX Workshop, Bialystok, Poland, June 19 - 25, 2022. Birkhauser, Basel, 2023, pp. 137-145. 

  Dissemination 

Online communications: 

8. Sebastian Burciu, "On conjugacy classes and Grothendieck rings of premodular categories", Tsinghua Seminar, BIMSA-Tsinghua Quantum Symmetry Seminar, September 14, 2022.

https://www.bimsa.cn/newsinfo/752541.html 

9. Sebastian Burciu, "Subalgebras of etale algebras and fusion subcategories", Hopf in Turin - 2022, Turin, Italy, September 6-9, 2022 

https://www.hopf-turin-22.it/home.

10. Daniel Beltita, "Transformation groupoids in W*-algebras",  42 International Conference on Quantum Probability and Infinite Dimensional Analysis (QP-42), Indian Statistical Institute, Statistics and Mathematics Unit, Bangalore, India, January 17-20, 2022 https://www.isibang.ac.in/$\sim$statmath/conferences/QPIDA-2022.html

11. Daniel Beltita, "Nilpotent Lie groups and their corresponding C*-algebras", Seminario ``Mischa Cotlar'', Instituto Argentino de Matematica ``Alberto P. Calderón'' in Buenos Aires, September 30, 2022 

    https://seminariomcotlar.wordpress.com/2022/09/27/daniel-beltita/  

Communications with physical presence: 

12. Sebastian Burciu, "On conjugacy classes for premodular categories", Hopf algebras, monoidal categories and related topics, Bucharest, July 27-29, 2022 https://hopfconferencebuch.wixsite.com/website.

13. Daniel Beltita, "On the solvable Lie groups whose regular representation is a factor representation", XXXIX Workshop on Geometric Methods in Physics, Faculty of Mathematics, University in Białystok, Poland, June 19-25, 2022 

https://wgmp.uwb.edu.pl/wgmp39/

14. Daniel Beltita, "Quantization and enveloping algebras of Lie groups," (mini-course consisting of 3 lectures), XI School on Geometry and Physics, Faculty of Mathematics, University in Białystok, Poland, June 27-July 1, 2022, https://wgmp.uwb.edu.pl/wgmp39/

15. Bogdan Ichim, ”Computations of probabilities in five candidates elections,” 23-rd Conference of the Romanian Society of Probability and Statistics (SPSR 2022), November 18-19, 20022, Bucharest, Romania

http://imar.ro/~imar/2022/SPSR/SPSR-2022-Bucharest-Program.pdf

Scientific report 2022

2023

  Publications

12. S. Burciu  and S. Palcoux, "Burnside type results for fusion rings", Preprint https://arxiv.org/abs/2302.07604, [math.QA], [math.RT].

13. S. Burciu, "On Harada's identity and some other consequences of Burnside's vanishing property",Preprint thttps://arxiv.org/abs/2306.11721, [math.QA], [math.RT].

14. I. Beltita, D. Beltita, On stably finiteness for C*-algebras of exponential solvable Lie groups. Mathematische Zeitschrift 304 (2023), no. 1, Paper No. 2, 36 pp.

15. D. Beltita, K.-H. Neeb, Holomorphic extension of one-parameter operator groups, to appear in Pure and Applied Functional Analysis,  arXiv:2304.09597v2 [math.RT].

16. D. Beltita, G. Larotonda, Groupoid techniques in operator theory, to appear in Geometric Methods in Physics XL, Birkhauser, Basel.

17. D. Beltita, Lecture notes on quantization and group C*-algebras. In: P. Kielanowski, A. Dobrogowska, G.A. Goldin, T. Golinski (eds.), Geometric Methods in Physics XXXIX, XXXIX Workshop, Bialystok, Poland, June 19 - 25, 2022. Birkhauser, Basel, 2023, pp. 319--327.

18. I. Beltita, D. Beltita, On the regular representation of solvable Lie groups with open coadjoint quasi-orbits, Preprint arXiv: 2305.04452 [math.RT]. 

Dissemination 

Online communications: 

16. Daniel Beltita, "Lie group representations and holomorphic extension of one-parameter operator groups", 50 Anniversario del IAM, Instituto Argentino de Matematica ”Alberto P. Calderón” (IAM-CONICET), Buenos Aires, Argentina, 23–27 October 2023 (online: https://iam50aniversario.wixsite.com/iam50aniversario)

17. Daniel Beltita, "Representation theory of solvable Lie groups and finite approximations of C^∗-algebras", Representation Theory & Noncommutative Geometry, An AIM Research Community, 13 December 2023 (online: https://sites.google.com/view/rtncg/what-we-do).

Communications with physical presence: 

18. Radu Popescu, "On a theorem of Ph. Hall ", Topology seminar IMAR, Topology Seminar, IMAR, March 3, 2023

19. Daniel Beltita, "On the primitive ideal spectrum of solvable Lie groups", 58th Seminar Sophus Lie, Friedrich-Alexander-Unversitat Erlangen-Nurnberg, Department Mathematik, Erlangen, Germany, 9 – 10 March 2023  (https://sites.google.com/view/ssl58)

20. Sebastian Burciu, "Burnside vanishing type results for fusion categories", The Tenth Congress of the Romanian Mathematicians (June 30 – July 5, 2023), University of Pitești, Pitești, Romania (http://www.imar.ro/~congmatro10/sessions.html) 

21. Vlad Matei, Quantitative Hilbert irreducibility for  tori, The Tenth Congress of the Romanian Mathematicians (June 30 – July 5, 2023), University of Pitești, Pitești, Romania (http://www.imar.ro/~congmatro10/sessions.html) 

22. Daniel Beltita, "Groupoid techniques in Hilbert space operator theory", XL Workshop on Geometric Methods in Physics, Bia lowie˙za, Poland, 2 – 8 July 2023 (https://wgmp.uwb.edu.pl/wgmp40/)

23. Sebastian Burciu, "On the zero entries in the character table of a fusion category", Groups, Rings, Lie and Hopf Algebras. V, Aug 21 - Aug 25, 2023, Memorial University of Newfoundland Harlow, UK  (https://www.mun.ca/aac/workshops/future-and-recent-workshops/)

24. Daniel Beltita, "Holomorphic extension of one-parameter operator groups and standard subspaces", Mini-Workshop: Standard Subspaces in Quantum Field Theory and Representation Theory, Mathematisches Forschungsinstitut Oberwolfach, Germany,  29 October – 3 November 2023 (https://www.mfo.de/occasion/2344b)

25. Daniel Beltita, "Holomorphic extension of one-parameter operator groups with applications to Lie group representations", International Conference on Applied and Pure Mathematics (ICAPM 2023), Technical University “Gheorghe Asachi”, Iasi, Romania, 9 – 12 November 2023 (http://math.etti.tuiasi.ro/apm2023/)

26. Radu Popescu, "Pasch geometries and hypergroups", Topology Seminar, IMAR, November 3, 2023

27. Bogdan Ichim, "On Applications of Combinatorial Learning”, Universität Osnabrück, Germany, November 30, 2023

28. Vlad Matei, Around Hilbert Irreducibility, Nicolae Popescu Number Theory Seminar, IMAR, December 6, 2023. 

Events (co-)organized:

1.  Geometry and Topology of Algebraic Varieties, Special session at The Tenth Congress of the Romanian Mathematicians (June 30 – July 5, 2023), University of Pitești, Pitești, Romania, Organizers: Laurențiu Maxim, Mihai Tibăr, Radu Popescu. 

2. Geometry, Topology and Algebra of Singular Spaces, celebrating the 70th anniversary of Prof. Alex Suciu and Prof. Laurențiu Păunescu, (July 7 – 11, 2023), IMAR, Bucharest, Romania, Organizers: Laurențiu Maxim, Mihai Tibăr, Radu Popescu.  

3.  Special session on Hopf algebras, tensor categories and related topics, Special session at The Tenth Congress of the Romanian Mathematicians (June 30 – July 5, 2023), University of Pitești, Pitești, Romania, Organizers: A. Agore, S. Burciu, S. Caenepeel.

4. XL Workshop on Geometric Methods in Physics, (July 2-8, 2023), Białowieża and XII School on Geometry and Physics, (June 26-30, 2023) Białystok, Organizers: A. Dobrogowska, T. Goliński, D. Beltiţă, T. Czyżycki, G. Goldin, P. Kielanowski, T. Ratiu, A. Sliżewska.

Scientific report 2023