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Mazorchuk, VolodymyrORCID iD iconorcid.org/0000-0002-4633-6218
Publications (10 of 126) Show all publications
Ahmed, C., Martin, P. & Mazorchuk, V. (2024). Tonal partition algebras: fundamental and geometrical aspects of representation theory. Communications in Algebra, 52(1), 233-271
Open this publication in new window or tab >>Tonal partition algebras: fundamental and geometrical aspects of representation theory
2024 (English)In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 52, no 1, p. 233-271Article in journal (Refereed) Published
Abstract [en]

For l, n is an element of N we define tonal partition algebra P-l (n) over Z[delta]. We construct modules {Delta mu} mu for P-l (n) over Z[delta], and hence over any integral domain containing Z[delta] (such as C[delta]), that pass to a complete set of irreducible modules over the field of fractions. We show that P-l (n) is semisimple there. That is, we construct for the tonal partition algebras a modular system in the sense of Brauer. Using a "geometrical" index set for the Delta-modules, we give an order with respect to which the decomposition matrix over C (with d. C-x) is upper-unitriangular. We establish several crucial properties of the Delta-modules. These include a tower property, with respect to n, in the sense of Green and Cox-Martin-Parker-Xi; contravariant forms with respect to a natural involutive antiautomorphism; a highest weight category property; and branching rules.

Place, publisher, year, edition, pages
Taylor & Francis, 2024
Keywords
Decomposition matrix, diagram algebras, finite dimensional algebras, highest weight category, partition algebra
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-523225 (URN)10.1080/00927872.2023.2239357 (DOI)001043536800001 ()
Funder
Swedish Research Council
Available from: 2024-02-19 Created: 2024-02-19 Last updated: 2024-02-19Bibliographically approved
Ko, H. & Mazorchuk, V. (2023). 2-representations of small quotients of Soergel bimodules in infinite types. Proceedings of the American Mathematical Society, 151(6), 2277-2290
Open this publication in new window or tab >>2-representations of small quotients of Soergel bimodules in infinite types
2023 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 151, no 6, p. 2277-2290Article in journal (Refereed) Published
Abstract [en]

We determine for which Coxeter types the associated small quo-tient of the 2-category of Soergel bimodules is finitary and, for such a small quotient, classify the simple transitive 2-representations (sometimes under the additional assumption of gradability). We also describe the underlying cat-egories of the simple transitive 2-representations. For the small quotients of general Coxeter types, we give a description for the cell 2-representations.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2023
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-511027 (URN)10.1090/proc/14584 (DOI)000948448700001 ()
Funder
Swedish Research CouncilVergstiftelsen
Available from: 2023-09-13 Created: 2023-09-13 Last updated: 2023-09-13Bibliographically approved
Mackaay, M., Mazorchuk, V., Miemietz, V., Tubbenhauer, D. & Zhang, X. (2023). Simple transitive 2-representations of Soergel bimodules for finite Coxeter types. Proceedings of the London Mathematical Society, 126(5), 1585-1655
Open this publication in new window or tab >>Simple transitive 2-representations of Soergel bimodules for finite Coxeter types
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2023 (English)In: Proceedings of the London Mathematical Society, ISSN 0024-6115, E-ISSN 1460-244X, Vol. 126, no 5, p. 1585-1655Article in journal (Refereed) Published
Abstract [en]

In this paper, we show that Soergel bimodules for finite Coxeter types have only finitely many equivalence classes of simple transitive 2-representations and we complete their classification in all types but H3 and H4.

Place, publisher, year, edition, pages
Wiley-Blackwell, 2023
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-512581 (URN)10.1112/plms.12515 (DOI)000933757000001 ()
Funder
Swedish Research CouncilGöran Gustafsson Foundation for promotion of scientific research at Uppala University and Royal Institute of Technology
Available from: 2023-09-28 Created: 2023-09-28 Last updated: 2023-09-28Bibliographically approved
Chen, C.-W. & Mazorchuk, V. (2023). Some Homological Properties of Category O for Lie Superalgebras. Journal of the Australian Mathematical Society, 114(1), 50-77
Open this publication in new window or tab >>Some Homological Properties of Category O for Lie Superalgebras
2023 (English)In: Journal of the Australian Mathematical Society, ISSN 1446-7887, E-ISSN 1446-8107, Vol. 114, no 1, p. 50-77Article in journal (Refereed) Published
Abstract [en]

For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule Delta(lambda) to be such that every nonzero homomorphism from another Verma supermodule to Delta(lambda) is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras pe(n) and, furthermore, to reduce the problem of description of Ext(O)(1)(L(mu), Delta(lambda)) for pe(n) to the similar problem for the Lie algebra gl(n). Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category O-p for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra pe(n) and the orthosymplectic Lie superalgebra osp(2 vertical bar 2n).

Place, publisher, year, edition, pages
Cambridge University Press, 2023
Keywords
Lie algebra, Lie superalgebra, module, projective dimension, socle
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-496937 (URN)10.1017/S1446788721000239 (DOI)000745504500001 ()
Funder
Swedish Research CouncilGöran Gustafsson Foundation for promotion of scientific research at Uppala University and Royal Institute of Technology
Available from: 2023-02-22 Created: 2023-02-22 Last updated: 2023-02-22Bibliographically approved
Ko, H., Mazorchuk, V. & Mrden, R. (2023). Some Homological Properties of Category O, V. International mathematics research notices, 2023(4), 3329-3373
Open this publication in new window or tab >>Some Homological Properties of Category O, V
2023 (English)In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, Vol. 2023, no 4, p. 3329-3373Article in journal (Refereed) Published
Abstract [en]

We compute projective dimension of translated simple modules in the regular block of the Bernstein–Gelfand–Gelfand category O in terms of Kazhdan–Lusztig combinatorics. This allows us to determine which projectives can appear at the last step of a minimal projective resolution for a translated simple module, confirming a conjecture by Johan Kåhrström. We also derive some inequalities, in terms of Lusztig’s a-function, for possible degrees in which the top (or socle) of a translated simple module can live. Finally, we prove that Kostant’s problem is equivalent to a homological problem of decomposing translated simple modules in O⁠. This gives a conjectural answer to Kostant’s problem in terms of the Kazhdan–Lusztig basis and addresses yet another conjecture by Johan Kåhrström.

Place, publisher, year, edition, pages
Oxford University Press, 2023
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-501599 (URN)10.1093/imrn/rnab330 (DOI)000789469700001 ()
Funder
Swedish Research CouncilVergstiftelsenGöran Gustafsson Foundation for promotion of scientific research at Uppala University and Royal Institute of Technology
Available from: 2023-05-11 Created: 2023-05-11 Last updated: 2023-05-11Bibliographically approved
Chen, C.-W., Cheng, S.-J. & Mazorchuk, V. (2023). Whittaker Categories, Properly Stratified Categories and Fock Space Categorification for Lie Superalgebras. Communications in Mathematical Physics, 401(1), 717-768
Open this publication in new window or tab >>Whittaker Categories, Properly Stratified Categories and Fock Space Categorification for Lie Superalgebras
2023 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 401, no 1, p. 717-768Article in journal (Refereed) Published
Abstract [en]

We study various categories of Whittaker modules over a type I Lie super algebra realized as cokernel categories that fit into the framework of properly stratified categories. These categories are the target of the Backelin functor gamma(zeta) .We show that these categories can be described, up to equivalence, as Serre quotients of the BGG category O and of certain singular categories of Harish-Chandra (g, g(0)& macr;)-bimodules. We also show that gamma(zeta) is a realization of the Serre quotient functor. We further investigate a q-symmetrized Fock space over a quantum group of type A and prove that, for general linear Lie superalgebras our Whittaker categories, the functor gamma(zeta) and various realizations of Serre quotients and Serre quotient functors categorify this q-symmetrized Fock space and its q-symmetrizer. In this picture, the canonical and dual canonical bases in this q-symmetrized Fock space correspond to tilting and simple objects in these Whittaker categories, respectively.

Place, publisher, year, edition, pages
Springer, 2023
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-512254 (URN)10.1007/s00220-023-04652-6 (DOI)000934917400003 ()
Funder
Swedish Research Council
Available from: 2023-09-29 Created: 2023-09-29 Last updated: 2023-09-29Bibliographically approved
Ko, H., Mazorchuk, V. & Zhang, X. (2022). Adjunction in the Absence of Identity. Applied Categorical Structures, 30, 123-172
Open this publication in new window or tab >>Adjunction in the Absence of Identity
2022 (English)In: Applied Categorical Structures, ISSN 0927-2852, E-ISSN 1572-9095, Vol. 30, p. 123-172Article in journal (Refereed) Published
Abstract [en]

We develop a bicategorical setup in which one can speak about adjoint 1-morphisms even in the absence of genuine identity 1-morphisms. We also investigate which part of 2-representation theory of 2-categories extends to this new setup.

Place, publisher, year, edition, pages
Springer NatureSpringer Nature, 2022
Keywords
(Op)lax units, Adjunctions, Fiax categories, Bilax 2-representations
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-469128 (URN)10.1007/s10485-021-09652-y (DOI)000673693500001 ()
Funder
Swedish Research CouncilGöran Gustafsson Foundation for promotion of scientific research at Uppala University and Royal Institute of Technology
Available from: 2022-03-07 Created: 2022-03-07 Last updated: 2024-01-15Bibliographically approved
Mazorchuk, V. & Persson Westin, E. (2022). Essential orders on stratified algebras with duality and S-subcategories in O. Journal of Pure and Applied Algebra, 226(10), Article ID 107083.
Open this publication in new window or tab >>Essential orders on stratified algebras with duality and S-subcategories in O
2022 (English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 226, no 10, article id 107083Article in journal (Refereed) Published
Abstract [en]

We prove uniqueness of the essential order for stratified algebras having simple preserving duality, generalizing a recent result of Coulembier for quasi-hereditary algebras. We apply this to classify, up to equivalence, regular integral blocks of S-subcategories in the BGG category O. We also describe various homological invariants of these blocks.

Place, publisher, year, edition, pages
Elsevier, 2022
Keywords
Stratified algebra, Essential order, Category O, Projective dimension, Finitistic dimension
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-426119 (URN)10.1016/j.jpaa.2022.107083 (DOI)000793066800006 ()
Funder
Swedish Research Council, 2017-03704Göran Gustafsson Foundation for Research in Natural Sciences and Medicine
Available from: 2020-11-24 Created: 2020-11-24 Last updated: 2024-01-15Bibliographically approved
Mazorchuk, v. & Srivastava, S. (2022). Jucys–Murphy elements and Grothendieck groups for generalized rook monoids. Journal of Combinatorial Algebra, 6(1/2), 185-222
Open this publication in new window or tab >>Jucys–Murphy elements and Grothendieck groups for generalized rook monoids
2022 (English)In: Journal of Combinatorial Algebra, ISSN 2415-6302, E-ISSN 2415-6310 , Vol. 6, no 1/2, p. 185-222Article, review/survey (Refereed) Published
Abstract [en]

We consider a tower of generalized rook monoid algebras over the field C of complex numbers and observe that the Bratteli diagram associated to this tower is a simple graph. We construct simple modules and describe Jucys-Murphy elements for generalized rook monoid algebras. Over an algebraically closed field k of positive characteristic p, utilizing Jucys-Murphy elements of rook monoid algebras, for 0 <= i <= p -1 we define the corresponding i-restriction and i-induction functors along with two extra functors. On the direct sum GC of the Grothendieck groups of module categories over rook monoid algebras over k, these functors induce an action of the tensor product of the universal enveloping algebra U(% euro lp(C)) and the monoid algebra C [B] of the bicyclic monoid B. Furthermore, we prove that GC is isomorphic to the tensor product of the basic representation of U(% euro lp(C)) and the unique infinite-dimensional simple module over C[B], and also exhibit that GC is a bialgebra. Under some natural restrictions on the characteristic of k, we outline the corresponding result for generalized rook monoids.

Place, publisher, year, edition, pages
European Mathematical Society Publishing HouseEuropean Mathematical Society - EMS, 2022
Keywords
Generalized rook monoids, Jucys-Murphy elements, Gelfand-Zeitlin basis, bicyclic monoid, Grothendieck group
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-483792 (URN)10.4171/JCA/65 (DOI)000815084000006 ()
Funder
Swedish Research CouncilGöran Gustafsson Foundation for promotion of scientific research at Uppala University and Royal Institute of Technology
Available from: 2022-09-06 Created: 2022-09-06 Last updated: 2024-01-15Bibliographically approved
Mazorchuk, V. & Mrden, R. (2022). LIE ALGEBRA MODULES WHICH ARE LOCALLY FINITE AND WITH FINITE MULTIPLICITIES OVER THE SEMISIMPLE PART. Nagoya mathematical journal, 246, 430-470
Open this publication in new window or tab >>LIE ALGEBRA MODULES WHICH ARE LOCALLY FINITE AND WITH FINITE MULTIPLICITIES OVER THE SEMISIMPLE PART
2022 (English)In: Nagoya mathematical journal, ISSN 0027-7630, E-ISSN 2152-6842, Vol. 246, p. 430-470Article in journal (Refereed) Published
Abstract [en]

For a finite-dimensional Lie algebra L over C with a fixed Levi decomposition L = g proportional to tau , where g is semisimple, we investigate L-modules which decompose, as g-modules, into a direct sum of simple finite-dimensional g-modules with finite multiplicities. We call such modules g-Harish-Chandra modules. We give a complete classification of simple g-Harish-Chandra modules for the Takiff Lie algebra associated to g = sl(2), and for the Schrodinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright's and Arkhipov's completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple g-Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff sl(2) and the Schrodinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple g-Harish-Chandra modules.

Place, publisher, year, edition, pages
Cambridge University PressCambridge University Press (CUP), 2022
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-477818 (URN)10.1017/nmj.2021.8 (DOI)000776795400001 ()
Funder
Swedish Research CouncilVergstiftelsen
Available from: 2022-06-21 Created: 2022-06-21 Last updated: 2024-01-15Bibliographically approved
Projects
Higher representation and invariant theory of associative and Lie (super)algebras [2010-02748_VR]; Uppsala University2-categories, 2-representations and applications [2013-04743_VR]; Uppsala UniversityClassification probblems in higher representation theory [2017-03704_VR]; Uppsala UniversityHigher representation theory: a view towards applications [2021-03731_VR]; Uppsala University
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-4633-6218

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