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Mazorchuk, VolodymyrORCID iD iconorcid.org/0000-0002-4633-6218
Publications (10 of 95) Show all publications
Chan, A. & Mazorchuk, V. (2019). Diagrams and discrete extensions for finitary 2-representations. Mathematical proceedings of the Cambridge Philosophical Society (Print), 166(2), 325-352
Open this publication in new window or tab >>Diagrams and discrete extensions for finitary 2-representations
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]

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.

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 ()
Funder
Swedish Research Council
Available from: 2019-02-11 Created: 2019-02-11 Last updated: 2019-02-11Bibliographically approved
Martin, P. & Mazorchuk, V. (2018). Fiat categorification of the symmetric inverse semigroup ISn and the semigroup Fn*. Semigroup Forum, 96(1), 142-159
Open this publication in new window or tab >>Fiat categorification of the symmetric inverse semigroup ISn and the semigroup Fn*
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]

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.

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 ()
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
Mazorchuk, V. & Zhao, K. (2018). Graded simple Lie algebras and graded simple representations. Manuscripta mathematica, 156(1-2), 215-240
Open this publication in new window or tab >>Graded simple Lie algebras and graded simple representations
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]

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.

National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-351617 (URN)10.1007/s00229-017-0960-5 (DOI)000429343400012 ()
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
Coulembier, K. & Mazorchuk, V. (2018). The G-Centre and Gradable Derived Equivalences. Journal of the Australian Mathematical Society, 105(3), 289-315
Open this publication in new window or tab >>The G-Centre and Gradable Derived Equivalences
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]

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.

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 ()
Funder
Australian Research Council, DP140103239Swedish Research Council
Available from: 2019-01-15 Created: 2019-01-15 Last updated: 2019-01-15Bibliographically approved
Mazorchuk, V. (2017). Classification problems in 2-representation theory. São Paulo Journal of Mathematical Sciences, 11(1), 1-22
Open this publication in new window or tab >>Classification problems in 2-representation theory
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]

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.

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 ()
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
Coulembier, K. & Mazorchuk, V. (2017). Dualities and derived equivalences for category O. Israel Journal of Mathematics, 219(2), 661-706
Open this publication in new window or tab >>Dualities and derived equivalences for category O
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]

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.

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 ()
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
Grensing, A.-L. & Mazorchuk, V. (2017). Finitary 2-categories associated with dual projection functors. Communications in Contemporary Mathematics, 19(3), Article ID 1650016.
Open this publication in new window or tab >>Finitary 2-categories associated with dual projection functors
2017 (English)In: Communications in Contemporary Mathematics, ISSN 0219-1997, Vol. 19, no 3, article id 1650016Article in journal (Refereed) Published
Abstract [en]

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.

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 ()
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
Greenstein, J. & Mazorchuk, V. (2017). Koszul Duality for Semidirect Products and Generalized Takiff Algebras. Algebras and Representation Theory, 20(3), 675-694
Open this publication in new window or tab >>Koszul Duality for Semidirect Products and Generalized Takiff Algebras
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]

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.

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 ()
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
Coulembier, K. & Mazorchuk, V. (2017). Some homological properties of category O. IV. Forum mathematicum, 29(5), 1083-1124
Open this publication in new window or tab >>Some homological properties of category O. IV
2017 (English)In: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337, Vol. 29, no 5, p. 1083-1124Article in journal (Refereed) Published
Abstract [en]

We study projective dimension and graded length of structural modules in parabolic-singular blocks of the BGG category O. Some of these are calculated explicitly, others are expressed in terms of two functions. We also obtain several partial results and estimates for these two functions and relate them to monotonicity properties for quasi-hereditary algebras. The results are then applied to study blocks of O in the context of Guichardet categories, in particular, we show that blocks of O are not always weakly Guichardet.

Keywords
Projective dimension, graded length, quasi-hereditary algebra, parabolic category O
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-334924 (URN)10.1515/forum-2016-0108 (DOI)000408650800005 ()
Funder
Swedish Research Council
Available from: 2017-11-29 Created: 2017-11-29 Last updated: 2017-11-29Bibliographically approved
Mazorchuk, V. & Miemietz, V. (2016). Endomorphisms of Cell 2-Representations. International mathematics research notices (24), 7471-7498
Open this publication in new window or tab >>Endomorphisms of Cell 2-Representations
2016 (English)In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, no 24, p. 7471-7498Article in journal (Refereed) Published
Abstract [en]

We determine the endomorphism categories of cell 2-representations of fiat 2-categories associated with strongly regular two-sided cells and classify, up to biequivalence, J-simple fiat 2-categories which have only one two-sided cell J apart from the identities.

National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-315940 (URN)10.1093/imrn/rnw025 (DOI)000392188600004 ()
Funder
Swedish Research CouncilThe Royal Swedish Academy of SciencesEU, European Research Council, PERG07-GA-2010-268109
Available from: 2017-02-22 Created: 2017-02-22 Last updated: 2017-11-29Bibliographically approved
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ORCID iD: ORCID iD iconorcid.org/0000-0002-4633-6218

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