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Mazorchuk, VolodymyrORCID iD iconorcid.org/0000-0002-4633-6218
Publications (10 of 91) Show all publications
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
Keyword
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. (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.

Keyword
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.

Keyword
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.

Keyword
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
Mazorchuk, V. & Stroppel, C. (2016). G(l, k, d)-modules via groupoids. Journal of Algebraic Combinatorics, 43(1), 11-32
Open this publication in new window or tab >>G(l, k, d)-modules via groupoids
2016 (English)In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 43, no 1, p. 11-32Article in journal (Refereed) Published
Abstract [en]

In this note, we describe a seemingly new approach to the complex representation theory of the wreath product , where G is a finite abelian group. The approach is motivated by an appropriate version of Schur-Weyl duality. We construct a combinatorially defined groupoid in which all endomorphism algebras are direct products of symmetric groups and prove that the groupoid algebra is isomorphic to the group algebra of . This directly implies a classification of simple modules. As an application, we get a Gelfand model for from the classical involutive Gelfand model for the symmetric group. We describe the Schur-Weyl duality which motivates our approach and relate it to various Schur-Weyl dualities in the literature. Finally, we discuss an extension of these methods to all complex reflection groups of type G(l, k, d).

Keyword
Schur-Weyl duality, Wreath product, Simple module, Groupoid
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-275534 (URN)10.1007/s10801-015-0623-0 (DOI)000367611700002 ()
Funder
Swedish Research CouncilKnut and Alice Wallenberg FoundationThe Royal Swedish Academy of Sciences
Available from: 2016-02-04 Created: 2016-02-04 Last updated: 2017-11-30Bibliographically approved
Mazorchuk, V. & Miemietz, V. (2016). Isotypic faithful 2-representations of -simple fiat 2-categories. Mathematische Zeitschrift, 282(1-2), 411-434
Open this publication in new window or tab >>Isotypic faithful 2-representations of -simple fiat 2-categories
2016 (English)In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 282, no 1-2, p. 411-434Article in journal (Refereed) Published
Abstract [en]

We introduce the class of isotypic 2-representations for finitary 2-categories and the notion of inflation of 2-representations. Under some natural assumptions we show that isotypic 2-representations are equivalent to inflations of cell 2-representations.

National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-277996 (URN)10.1007/s00209-015-1546-0 (DOI)000368643400018 ()
Funder
Swedish Research Council
Available from: 2016-02-23 Created: 2016-02-23 Last updated: 2017-11-30Bibliographically approved
Mazorchuk, V. & Miemietz, V. (2016). Morita theory for finitary 2-categories. Quantum Topology, 7(1), 1-28
Open this publication in new window or tab >>Morita theory for finitary 2-categories
2016 (English)In: Quantum Topology, ISSN 1663-487X, E-ISSN 1664-073X, Vol. 7, no 1, p. 1-28Article in journal (Refereed) Published
Abstract [en]

We develop Morita theory for finitary additive 2-representations of finitary 2-categories. As an application we describe Morita equivalence classes for 2-categories of projective functors associated to finite dimensional algebras and for 2-categories of Soergel bimodules.

Keyword
2-representation theory, finitary 2-category, Morita equivalence, projective functor
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-298710 (URN)10.4171/QT/72 (DOI)000376406500001 ()
Available from: 2016-07-06 Created: 2016-07-06 Last updated: 2017-11-28Bibliographically approved
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Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-4633-6218

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