Hopf algebras representations and tensor categories, PN-III-P4-ID-PCE-2016-0157
Hopf algebras representations and tensor categories, PN-III-P4-ID-PCE-2016-0157
Scopul acestui proiect este de a studia structura si reprezentarile algebrelor Hopf si a categoriilor tensoriale modulare. Se crede ca algebrele Hopf semisimple constituie o generalizare a algebrelor grupale finite semsimple. Din acest punct de vedere, recent, mai multe rezultate interesante în ceea ce privește reprezentările grupurilor finite din literatura de specialitate au fost transferate la algebrele Hopf. Proiectul de fata urmează această direcție principală de cercetare și este o continuare naturala a studiului efectuat de către liderul de proiect în proiectele sale anterioare. Prezenta propunere este formata din trei părți. Prima parte a proiectului se ocupă cu studiul acțiunilor de grupuri finite asupra categoriilor tensoriale. Notiunile de varf si sursa pentru obiectele unei categorii tensoriale equivariantizate vor fi introduse. Cu ajutorul acestor notiuni este de asteptat ca o teorie de reprezentare asemanatoare teoriei reprezentarilor modulare pentru grupurile finite sa fie dezvoltata. Cea de a doua parte a proiectului studiaza centralizatorul Müger pentru categoriile de reprezentari ale unui Drinfeld dublu semisimplu. Tot in acesata parte vom studia proprietatile aritmetice ale categoriilor tensoriale modulare cu scopul de a clasifica astfel de categorii cu un numar mic de obiecte simple. Teoria caracterelor pentru categoriile tensoriale a fost recent introdusa de catre Shimizu. In cea de a treia parte a proiectului vom incerca transferarea anumitor rezultate de teoria caracterelor de la algebrele Hopf semisimple la categoriile de fuziune. Vom incerca de asemenea sa extindem teoria de tip Clifford dezvoltata anterior de catre directorul de proiect de la subalgebre Hopf normale la subalgebre coideale stangi normale. De asemenea vom generaliza notiunea de nucleu al unui obiect simplu de la o categorie de fuziune la o categorie tensoriala arbitrara incercand sa obtinem o versiune mai generala a teoremei lui Brauer in acest context.
The goal of this project is to study the structure and representations Hopf algebra and modular tensor categories. It is commonly believed that semisimple Hopf algebras are a generalization of semsimple finite group algebras. From this point of view, recently, several interesting results from finite group representations were transferred to Hopf algebras. This project follows this main direction of research and it is a natural continuation of the study conducted by the project leader in his previous projects. This proposal consists of three parts. The first part of the project studies the actions of finite groups on tensor categories. Notions of vertices and source for objects of equivariantized tensor categories will be introduced. With these notions it is expected that a similar representation theory to the modular representations theory of finite groups to be developed. The second part of the project studies Müger's centralizer for the categories of representations of a semisimple Drinfeld double. Also in this part we will study the arithmetic properties of modular tensor categories in order to classify such categories with a small number of simple objects. Character theory for tensor categories was recently introduced by Shimizu. In the third part of the project we will try to transfer some results from the character theory of semisimple Hopf algebras to fusion categories. We will also try to extend the Clifford type theory developed earlier by the project leader from normal Hopf subalgebras to normal left coideal subalgebras. We will also generalize the notion of a kernel of a simple object from fusion categories to arbitrary tensor categories. We do this in order to be able to obtain a more general version of Brauer's theorem in this context.
Sebastian Burciu, project leader, IMAR.
Bogdan Ichim, senior researcher, IMAR
Vicentiu Pasol, senior researcher, IMAR
Alexandru Popa, senior researcher, IMAR.
Anca Baltariga, cercetator in formare.
Alexandru Baltariga, cercetator in formare.
Group actions on tensor categories.
Braided tensor categories and modular tensor categories, arithmetic properties.
Representations of Hopf algebras and tensor categories.
Publications:
S. Burciu, Representations and conjugacy classes of semisimple quasitriangular Hopf algebras, SIGMA Symmetry Integrability and Geometry: Methods and Applications 16 (2020), 039, 20 pages..
A. A. Popa and D. Zagier, A simple proof of the Eichler-Selberg trace formula, to appear, Journal für die reine und angewandte Mathematik, (2018), doi:10.1515/crelle-2018-0035, arxiv: 1711.00327.
S. Burciu, On the Grothendieck rings of generalized Drinfeld doubles, Journal of Algebra, Volume 486, 2017, Pages 14-35.
Dissemination:
Sebastian Burciu - "Muger centralizer for representations of factorizable Hopf algebras" at The Brussels Hopf Algebra Workshop 2017, Universite Libre de Bruxelles, Belgium, 29 - 30 August 2017.
Vicentiu Pasol a efectuat o vizita de colaborare la departamentul de Matematica al Radboud Univesity, intalnind matematicieni importanti in tematica proiectului, ca Michael Müger si Wadim Zudilin. Aici a tinut o prezentare coloquium intitulata ``Character sums, multiple Dirichlet series and moduli spaces of curves''
Scientific report 2017:
Foreign visitors:
Glenn Stevens, Boston University, visited IMAR and started new collaborations with V. Pasol.
Publications:
W. Bruns and B. Ichim, Polytope Volume By Descent In The Face Lattice And Applications In Social Choice, submitted, https://arxiv.org/abs/1807.02835.2018.
V. Pasol and W. Zudilin, A study of elliptic gamma function and allies, Research in the Mathematical Sciences, 5, (2018), No. 39
Dissemination:
A fost organizata in cadrul proiectului conferinta "Sixth Bucharest Number Theory Day", July 24, 2018.
Bogdan Ichim va vizita Osnabrück University unde va tine prezentarea intitulata ”Polytope volume by descent in the face lattice and applications in social choice” in decembrie 2018.
De asemenea a fost organizat workshopul "Workshop on L-functions and Galois Representations", June 19, 2018, http://imar.ro/~apopa/workshop2018.html unde au tinut prezentari, Adrian Diaconu, University of Minnesota and IMAR, Adrian Iovița, Concordia University, Montreal si Glenn Stevens, Boston University.
Scientific report 2018:
Foreign visitors:
Ramin-Takloo Bighash, September 16-20 2019 . A tinut conferinta lunara: Applications of Complex Analysis to Counting Arithmetic Objects, September 18 2019.
Publications:
S. Burciu, On monoidal group actions on tensor categories and their Green functors, Monatshefte für Mathematik, March 2019, Volume 188, Issue 3, pp 431–459.
W. Bruns, B. Ichim and C. Söger, Computations of volumes and Ehrhart series in four candidates elections, Annals of Operations Research, September 2019, Volume 280, Issue 1–2, pp 241–265.
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S. Burciu, Structure constants for pre-modular categories, to appear in Bulletin of the London Mathematical Society, arXiv:2002.05483, 2020.
Dissemination:
Alex Popa presented "A combinatorial refinement of the Kronecker-Hurwitz class number relation", at Transient Transcendence in Transylvania, 13-17 May 2019, Brasov
A fost organizata in cadrul proiectului conferinta "Seventh Bucharest Number Theory Day", August 8, 2019", July 24, 2018.
Vicentiu Pasol presented "Structure of finite rings" at The Ninth Congress of Romanian Mathematicians, June 28 - July 3, 2019, Galați.
Vicentiu Pasol and Alex Popa (together with Alexandru Zaharescu) have organised a special session "Number Theory "(Section 1) at The Ninth Congress of Romanian Mathematicians.
Alex Popa presented "Multiple Dirichlet series for affine Weyl groups", at The Ninth Congress of Romanian Mathematicians, June 28 - July 3, 2019, Galați
Sebastian Burciu presented "Conjugacy classes and centralizers for modular tensor categories", at The Ninth Congress of Romanian Mathematicians, June 28 - July 3, 2019, Galați.
Ichim Bogdan presented "An algorithm for volumes of polytopes with applications to social choice", at The Ninth Congress of Romanian Mathematicians, June 28 - July 3, 2019, Galați.
Bogdan Ichim va vizita Osnabrück University unde va tine prezentarea intitulata ”On a class of Gorenstein polytopes” in 28 noiembrie 2019.
Raport stiintific aferent perioadei 2017- septembrie 2019
Final scientific report 2019: