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Simple transitive 2-representations of some 2-categories of projective functors
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
2018 (English)In: Beiträge zur Algebra und Geometrie, ISSN 0138-4821, Vol. 59, no 1, p. 41-50Article in journal (Refereed) Published
Abstract [en]

We show that every simple transitive 2-representation 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.

Place, publisher, year, edition, pages
Berlin/Heidelberg: Springer Berlin/Heidelberg, 2018. Vol. 59, no 1, p. 41-50
Keywords [en]
Representation theory, 2-category, Simple transitive 2-representation, Cell 2-representation
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-354581DOI: 10.1007/s13366-017-0348-4ISI: 000431552200004OAI: oai:DiVA.org:uu-354581DiVA, id: diva2:1221839
Available from: 2018-06-20 Created: 2018-06-20 Last updated: 2018-07-04Bibliographically 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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Zimmermann, Jakob

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