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Mazorchuk, Volodymyrorcid.org/0000-0002-4633-6218

Open this publication in new window or tab >>Diagrams and discrete extensions for finitary 2-representations### Chan, Aaron

### Mazorchuk, Volodymyr

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_some",{id:"formSmash:j_idt184:0:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_otherAuthors",{id:"formSmash:j_idt184:0:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Mathematical proceedings of the Cambridge Philosophical Society (Print), ISSN 0305-0041, E-ISSN 1469-8064, Vol. 166, no 2, p. 325-352Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Cambridge University Press, 2019
##### National Category

Computer Systems
##### Identifiers

urn:nbn:se:uu:diva-376726 (URN)10.1017/S0305004117000858 (DOI)000456596000007 ()
#####

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##### Funder

Swedish Research Council
Available from: 2019-02-11 Created: 2019-02-11 Last updated: 2019-02-11Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

In this paper we introduce and investigate the notions of diagrams and discrete extensions in the study of finitary 2-representations of finitary 2-categories.

Open this publication in new window or tab >>Simple transitive 2-representations of small quotients of Soergel bimodules### Kildetoft, Tobias

### Mackaay, Marco

### Mazorchuk, Volodymyr

### Zimmermann, Jakob

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 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]

##### National Category

Algebra and Logic
##### Identifiers

urn:nbn:se:uu:diva-354594 (URN)10.1090/tran/7456 (DOI)000464034200013 ()
#####

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##### Funder

Knut and Alice Wallenberg FoundationSwedish Research CouncilGöran Gustafsson Foundation for Research in Natural Sciences and Medicine
Available from: 2018-06-20 Created: 2018-06-20 Last updated: 2019-05-10Bibliographically approved

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.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.

In all finite Coxeter types but I_{2}(12), I_{2}(18) and I_{2}(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 I_{2}(2k), where k ≥ 3, there exist simple transitive 2-representations which are not equivalent to cell 2-representations.

Open this publication in new window or tab >>Simple Transitive 2-Representations via (Co-)Algebra 1-Morphisms### Mackaay, Marco

### Mazorchuk, Volodymyr

### Miemietz, Vanessa

### Tubbenhauer, Daniel

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 68, no 1, p. 1-33Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

INDIANA UNIV MATH JOURNAL, 2019
##### National Category

Algebra and Logic
##### Identifiers

urn:nbn:se:uu:diva-379778 (URN)10.1512/iumj.2019.68.7554 (DOI)000460124100001 ()
#####

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##### Funder

Swedish Research CouncilKnut and Alice Wallenberg FoundationGöran Gustafsson Foundation for promotion of scientific research at Uppala University and Royal Institute of Technology
Available from: 2019-03-21 Created: 2019-03-21 Last updated: 2019-03-21Bibliographically approved

Inst Super Tecn, Ctr Math Anal Geometry & Dynam Syst, Dept Matemat, P-1049001 Lisbon, Portugal;Univ Algarve, Dept Matemat, FCT, Campus Gambelas, P-8005139 Faro, Portugal.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

Univ East Anglia, Sch Math, Norwich NR4 7TJ, Norfolk, England.

Univ Bonn, Math Inst, Endenicher Allee 60,Room 1-003, D-53115 Bonn, Germany;Univ Zurich, Inst Math, Winterthurerstr 190,Campus Irchel,Off Y27J32, CH-8057 Zurich, Switzerland.

For any fiat 2-category C, we show how its simple transitive 2-representations can be constructed using co-algebra 1-morphisms in the injective abelianization of C. Dually, we show that these can also be constructed using algebra 1-morphisms in the projective abelianization of C. We also extend Morita-Takeuchi theory to our setup and work out several examples, including that of Soergel bimodules for dihedral groups, explicitly.

Open this publication in new window or tab >>Fiat categorification of the symmetric inverse semigroup IS_{n} and the semigroup F_{n*}### Martin, Paul

### Mazorchuk, Volodymyr

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 2018 (English)In: Semigroup Forum, ISSN 0037-1912, E-ISSN 1432-2137, Vol. 96, no 1, p. 142-159Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

SPRINGER, 2018
##### Keywords

Categorification, 2-category, Symmetric inverse semigroup, Dual symmetric inverse semigroup
##### National Category

Algebra and Logic
##### Identifiers

urn:nbn:se:uu:diva-343777 (URN)10.1007/s00233-017-9866-5 (DOI)000423372400009 ()
#####

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##### Funder

Swedish Research CouncilGöran Gustafsson Foundation for Research in Natural Sciences and Medicine
Available from: 2018-03-07 Created: 2018-03-07 Last updated: 2018-03-07Bibliographically approved

Univ Leeds, Dept Pure Math, Leeds LS2 9JT, W Yorkshire, England..

Starting from the symmetric group , we construct two fiat 2-categories. One of them can be viewed as the fiat "extension" of the natural 2-category associated with the symmetric inverse semigroup (considered as an ordered semigroup with respect to the natural order). This 2-category provides a fiat categorification for the integral semigroup algebra of the symmetric inverse semigroup. The other 2-category can be viewed as the fiat "extension" of the 2-category associated with the maximal factorizable subsemigroup of the dual symmetric inverse semigroup (again, considered as an ordered semigroup with respect to the natural order). This 2-category provides a fiat categorification for the integral semigroup algebra of the maximal factorizable subsemigroup of the dual symmetric inverse semigroup.

Open this publication in new window or tab >>Graded simple Lie algebras and graded simple representations### Mazorchuk, Volodymyr

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.### Zhao, Kaiming

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); 2018 (English)In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 156, no 1-2, p. 215-240Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Algebra and Logic
##### Identifiers

urn:nbn:se:uu:diva-351617 (URN)10.1007/s00229-017-0960-5 (DOI)000429343400012 ()
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##### Funder

Swedish Research CouncilThe Royal Swedish Academy of SciencesKnut and Alice Wallenberg Foundation
Available from: 2018-06-13 Created: 2018-06-13 Last updated: 2018-06-13Bibliographically approved

Wilfrid Laurier Univ, Dept Math, Waterloo, ON N2L 3C5, Canada.;Hebei Normal Teachers Univ, Coll Math & Informat Sci, Shijiazhuang 050016, Hebei, Peoples R China..

Let Q be an abelian group and a field. We prove that any Q-graded simple Lie algebra over is isomorphic to a loop algebra in case has a primitive root of unity of order |Q|, if Q is finite, or is algebraically closed and (as cardinals). For Q-graded simple modules over any Q-graded Lie algebra , we propose a similar construction of all Q-graded simple modules over any Q-graded Lie algebra over starting from nonextendable gradings of simple -modules. We prove that any Q-graded simple module over is isomorphic to a loop module in case has a primitive root of unity of order |Q| if Q is finite, or is algebraically closed and as above. The isomorphism problem for simple graded modules constructed in this way remains open. For finite-dimensional Q-graded semisimple algebras we obtain a graded analogue of the Weyl Theorem.

Open this publication in new window or tab >>The G-Centre and Gradable Derived Equivalences### Coulembier, Kevin

### Mazorchuk, Volodymyr

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_some",{id:"formSmash:j_idt184:5:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_otherAuthors",{id:"formSmash:j_idt184:5:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_otherAuthors",multiple:true}); 2018 (English)In: Journal of the Australian Mathematical Society, ISSN 1446-7887, E-ISSN 1446-8107, Vol. 105, no 3, p. 289-315Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

CAMBRIDGE UNIV PRESS, 2018
##### Keywords

group actions, gradings, derived equivalences, generalisations of centres, superalgebras
##### National Category

Algebra and Logic
##### Identifiers

urn:nbn:se:uu:diva-369380 (URN)10.1017/S1446788717000404 (DOI)000449418200001 ()
#####

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##### Funder

Australian Research Council, DP140103239Swedish Research Council
Available from: 2019-01-15 Created: 2019-01-15 Last updated: 2019-01-15Bibliographically approved

Univ Sydney, Sch Math & Stat, Sydney, NSW 2006, Australia.

We propose a generalisation for the notion of the centre of an algebra in the setup of algebras graded by an arbitrary abelian group G. Our generalisation, which we call the G-centre, is designed to control the endomorphism category of the grading shift functors. We show that the G-centre is preserved by gradable derived equivalences given by tilting modules. We also discuss links with existing notions in superalgebra theory.

Open this publication in new window or tab >>Classification problems in 2-representation theory### Mazorchuk, Volodymyr

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: São Paulo Journal of Mathematical Sciences, ISSN 1982-6907, E-ISSN 2316-9028, Vol. 11, no 1, p. 1-22Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Categorification, 2-Category, 2-Representation, Simple transitive 2-representation, Classification
##### National Category

Algebra and Logic
##### Identifiers

urn:nbn:se:uu:diva-376850 (URN)10.1007/s40863-017-0059-7 (DOI)000449599300001 ()
#####

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##### Funder

Swedish Research CouncilGöran Gustafsson Foundation for promotion of scientific research at Uppala University and Royal Institute of Technology
Available from: 2019-02-12 Created: 2019-02-12 Last updated: 2019-02-12Bibliographically approved

This article surveys recent advances and future challenges in the 2-representation theory of finitary 2-categories with a particular emphasis on problems related to classification of various classes of 2-representations.

Open this publication in new window or tab >>Dualities and derived equivalences for category O### Coulembier, Kevin

### Mazorchuk, Volodymyr

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_some",{id:"formSmash:j_idt184:7:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_otherAuthors",{id:"formSmash:j_idt184:7:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 219, no 2, p. 661-706Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

HEBREW UNIV MAGNES PRESS, 2017
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-324334 (URN)10.1007/s11856-017-1494-y (DOI)000401257400007 ()
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##### Funder

Swedish Research CouncilKnut and Alice Wallenberg FoundationThe Royal Swedish Academy of Sciences
Available from: 2017-06-16 Created: 2017-06-16 Last updated: 2017-06-16Bibliographically approved

Univ Ghent, Dept Math Anal, Krijgslaan 281, B-9000 Ghent, Belgium..

We determine the Ringel duals for all blocks in the parabolic versions of the BGG category associated to a reductive finite-dimensional Lie algebra. In particular, we find that, contrary to the original category and the specific previously known cases in the parabolic setting, the blocks are not necessarily Ringel self-dual. However, the parabolic category as a whole is still Ringel self-dual. Furthermore, we use generalisations of the Ringel duality functor to obtain large classes of derived equivalences between blocks in parabolic and original category . We subsequently classify all derived equivalence classes of blocks of category in type A which preserve the Koszul grading.

Open this publication in new window or tab >>Finitary 2-categories associated with dual projection functors### Grensing, Anna-Louise

### Mazorchuk, Volodymyr

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_some",{id:"formSmash:j_idt184:8:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_otherAuthors",{id:"formSmash:j_idt184:8:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: Communications in Contemporary Mathematics, ISSN 0219-1997, Vol. 19, no 3, article id 1650016Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Finitary 2-category, projection functor, finite dimensional algebra, bimodule, monoid, generators and relations
##### National Category

Algebra and Logic
##### Identifiers

urn:nbn:se:uu:diva-322078 (URN)10.1142/S0219199716500164 (DOI)000399087200002 ()
#####

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##### Funder

German Research Foundation (DFG), SPP 1388Swedish Research CouncilKnut and Alice Wallenberg FoundationThe Royal Swedish Academy of Sciences
Available from: 2017-05-19 Created: 2017-05-19 Last updated: 2017-05-19Bibliographically approved

Univ Bielefeld, Fac Math, POB 100 131, D-33501 Bielefeld, Germany..

We study finitary 2-categories associated to dual projection functors for finite-dimensional associative algebras. In the case of path algebras of admissible tree quivers (which includes all Dynkin quivers of type A), we show that the monoid generated by dual projection functors is the Hecke-Kiselman monoid of the underlying quiver and also obtain a presentation for the monoid of indecomposable subbimodules of the identity bimodule.

Open this publication in new window or tab >>Koszul Duality for Semidirect Products and Generalized Takiff Algebras### Greenstein, Jacob

### Mazorchuk, Volodymyr

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_some",{id:"formSmash:j_idt184:9:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_otherAuthors",{id:"formSmash:j_idt184:9:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 20, no 3, p. 675-694Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Module algebras, Semi-direct products, Graded algebras, Koszul duality
##### National Category

Algebra and Logic
##### Identifiers

urn:nbn:se:uu:diva-323763 (URN)10.1007/s10468-016-9660-1 (DOI)000401540300008 ()
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##### Funder

Swedish Research CouncilKnut and Alice Wallenberg FoundationThe Royal Swedish Academy of Sciences
Available from: 2017-06-12 Created: 2017-06-12 Last updated: 2017-06-12Bibliographically approved

Univ Calif Riverside, Dept Math, Riverside, CA 92521 USA..

We obtain Koszul-type dualities for categories of graded modules over a graded associative algebra which can be realized as the semidirect product of a bialgebra coinciding with its degree zero part and a graded module algebra for the latter. In particular, this applies to graded representations of the universal enveloping algebra of the Takiff Lie algebra (or the truncated current algebra) and its (super)analogues, and also to semidirect products of quantum groups with braided symmetric and exterior module algebras in case the latter are flat deformations of classical ones.