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Simple transitive 2-representations of small quotients of Soergel bimodules
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Inst Super Tecn, Dept Matemat, Ctr Math Anal Geometry & Dynam Syst, P-1049001 Lisbon, Portugal.;Univ Algarve, FCT, Dept Matemat, Campus Gambelas, P-8005139 Faro, Portugal..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.ORCID iD: 0000-0002-4633-6218
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
2019 (English)In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 371, no 8, p. 5551-5590Article in journal (Refereed) Published
Abstract [en]

In all finite Coxeter types but I2(12), I2(18) and I2(30), we classify simple transitive 2-representations for the quotient of the 2-category of Soergel bimodules over the coinvariant algebra which is associated to the two-sided cell that is the closest one to the two-sided cell  containing the identity element. It turns out that, in most of the cases, simple transitive 2-representations are exhausted by cell 2-representations. However, in Coxeter types I2(2k), where k ≥ 3, there exist simple transitive 2-representations which are not equivalent to cell 2-representations.

Place, publisher, year, edition, pages
2019. Vol. 371, no 8, p. 5551-5590
National Category
Algebra and Logic
Identifiers
URN: urn:nbn:se:uu:diva-354594DOI: 10.1090/tran/7456ISI: 000464034200013OAI: oai:DiVA.org:uu-354594DiVA, id: diva2:1221871
Funder
Knut and Alice Wallenberg FoundationSwedish Research CouncilGöran Gustafsson Foundation for Research in Natural Sciences and MedicineAvailable from: 2018-06-20 Created: 2018-06-20 Last updated: 2019-05-10Bibliographically approved
In thesis
1. Classification of simple transitive 2-representations
Open this publication in new window or tab >>Classification of simple transitive 2-representations
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The representation theory of finitary 2-categories is a generalization of the classical representation theory of finite dimensional associative algebras. A key notion in classical representation theory is the notion of simple modules as those are in some sense the building blocks of all modules. A correct analogue of simple modules in the realm of 2-representations is the notion of simple transitive 2-representations since those also turn out to be building blocks of 2-representations.

This thesis is concerned with the classification of simple transitive 2-representations for a number of different interesting 2-categories. In Paper I we study simple transitive 2-representations of Soergel bimodules in Coxeter type I2(4) and show that all simple transitive 2-representations in this case are equivalent to cell 2-representations. In Paper II we classify simple transitive 2-representations for the quotient of the 2-category of Soergel bimodules over the coinvariant algebra which is associated to the two-sided cell that is the closest to the two-sided cell containing the identity element, in all Coxeter types but I2(12), I2(18) and I2(30). It turns out that, in most of the cases, simple transitive 2-representations are exhausted by cell 2-representations. However, in Coxeter types I2(2k), where k ≥ 3, there exist simple transitive 2-representations which are not equivalent to cell 2-representations. In Paper III we show that for any complex polynomial p(X) the set of irreducible, integer matrices which are annihilated by p(X) is finite. Moreover, we study the set of irreducible, integral matrices satisfying X² = nX, for n ≥ 1, and count its elements. In Paper IV we show that every simple transitive 2-representations of the 2-category of projective functors for a certain quotient of the quadratic dual of the preprojective algebra associated with a tree is equivalent to a cell 2-representation. Finally, in Paper V we study simple transitive 2-representations of certain 2-subcategories of the 2-categories of projective functors over star algebras. In the simplest case, which is associated with Dynkin type A2, we show that simple transitive 2-representations are classified by cell 2-representations. However, in the general case we conjecture that there exist many more simple transitive 2-representations.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2018. p. 41
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 108
Keywords
2-representation theory, 2-categories, Soergel bimodules, projective functors, cell 2-representations, simple transitive 2-representations
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-354597 (URN)978-91-506-2710-7 (ISBN)
Public defence
2018-09-07, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2018-08-15 Created: 2018-06-20 Last updated: 2018-08-15Bibliographically approved

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Kildetoft, TobiasMazorchuk, VolodymyrZimmermann, Jakob

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