--- _id: '66293' abstract: - lang: eng text: In 1970, Gelfand posed the problem of classifying the indecomposable objects in a representation category equivalent to the principal block of Harish-Chandra modules for $\mathsf{SL}_2(\mathbb{R})$; explicit solutions were obtained by Bondarenko, and, independently, Crawley-Boevey. In this article, we give a complete answer to Gelfand's problem from a derived category perspective. We classify indecomposable objects in the bounded derived category of nilpotent representations of the Gelfand quiver in terms of band and string complexes, and determine their images under the derived Auslander-Reiten translation, the sign involution, and the contragredient duality. The four main combinatorial classes are characterized in Lie-theoretic as well as homological terms. For the abelian category of nilpotent representations, we provide projective resolutions, standard homological invariants and explicit representation matrices of all indecomposables. Our approach can be extended to arrow ideal completions of path algebras of skew-gentle quivers. author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: Wassilij full_name: Gnedin, Wassilij last_name: Gnedin citation: ama: Burban I, Gnedin W. Representation theory of the Gelfand quiver and Harish-Chandra modules for SL_2(R). arXiv:260400274. Published online 2026. apa: Burban, I., & Gnedin, W. (2026). Representation theory of the Gelfand quiver and Harish-Chandra modules for SL_2(R). In arXiv:2604.00274. bibtex: '@article{Burban_Gnedin_2026, title={Representation theory of the Gelfand quiver and Harish-Chandra modules for SL_2(R)}, journal={arXiv:2604.00274}, author={Burban, Igor and Gnedin, Wassilij}, year={2026} }' chicago: Burban, Igor, and Wassilij Gnedin. “Representation Theory of the Gelfand Quiver and Harish-Chandra Modules for SL_2(R).” ArXiv:2604.00274, 2026. ieee: I. Burban and W. Gnedin, “Representation theory of the Gelfand quiver and Harish-Chandra modules for SL_2(R),” arXiv:2604.00274. 2026. mla: Burban, Igor, and Wassilij Gnedin. “Representation Theory of the Gelfand Quiver and Harish-Chandra Modules for SL_2(R).” ArXiv:2604.00274, 2026. short: I. Burban, W. Gnedin, ArXiv:2604.00274 (2026). date_created: 2026-07-07T06:28:32Z date_updated: 2026-07-07T06:29:40Z external_id: arxiv: - '2604.00274' language: - iso: eng publication: arXiv:2604.00274 status: public title: Representation theory of the Gelfand quiver and Harish-Chandra modules for SL_2(R) type: preprint user_id: '72064' year: '2026' ... --- _id: '66292' abstract: - lang: eng text: In this article we study the principal block of the category of real Harish-Chandra modules for the group $\mathsf{SL}_2(\RR)$ and relate it to the category of finite dimensional modules over the so-called real Gelfand order. We describe several distinguished classes of the corresponding indecomposable representations. author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: Yuriy full_name: Drozd, Yuriy last_name: Drozd citation: ama: Burban I, Drozd Y. Representation theory of the real Gelfand order and real Harish-Chandra modules for SL_2(R). arXiv:260518000. Published online 2026. apa: Burban, I., & Drozd, Y. (2026). Representation theory of the real Gelfand order and real Harish-Chandra modules for SL_2(R). In arXiv:2605.18000. bibtex: '@article{Burban_Drozd_2026, title={Representation theory of the real Gelfand order and real Harish-Chandra modules for SL_2(R)}, journal={arXiv:2605.18000}, author={Burban, Igor and Drozd, Yuriy}, year={2026} }' chicago: Burban, Igor, and Yuriy Drozd. “Representation Theory of the Real Gelfand Order and Real Harish-Chandra Modules for SL_2(R).” ArXiv:2605.18000, 2026. ieee: I. Burban and Y. Drozd, “Representation theory of the real Gelfand order and real Harish-Chandra modules for SL_2(R),” arXiv:2605.18000. 2026. mla: Burban, Igor, and Yuriy Drozd. “Representation Theory of the Real Gelfand Order and Real Harish-Chandra Modules for SL_2(R).” ArXiv:2605.18000, 2026. short: I. Burban, Y. Drozd, ArXiv:2605.18000 (2026). date_created: 2026-07-07T06:22:32Z date_updated: 2026-07-07T06:23:43Z external_id: arxiv: - '2605.18000' language: - iso: eng publication: arXiv:2605.18000 status: public title: Representation theory of the real Gelfand order and real Harish-Chandra modules for SL_2(R) type: preprint user_id: '72064' year: '2026' ... --- _id: '63620' abstract: - lang: eng text: We introduce a new class of reflection groups associated with the canonical bilinear lattices of Lenzing, which we call reflection groups of canonical type. The main result of this work is a categorification of the corresponding poset of non-crossing partitions for any such group, realized via the poset of thick subcategories of the category of coherent sheaves on an exceptional hereditary curve generated by an exceptional sequence. A second principal result, essential for the categorification, is a proof of the transitivity of the Hurwitz action in these reflection groups. author: - first_name: Barbara full_name: Baumeister, Barbara last_name: Baumeister - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: Georges full_name: Neaime, Georges last_name: Neaime - first_name: Charly Merlin full_name: Schwabe, Charly Merlin id: '103440' last_name: Schwabe citation: ama: Baumeister B, Burban I, Neaime G, Schwabe CM. Non-crossing partitions for exceptional hereditary curves. arXiv:251201729. Published online 2025. apa: Baumeister, B., Burban, I., Neaime, G., & Schwabe, C. M. (2025). Non-crossing partitions for exceptional hereditary curves. In arXiv:2512.01729. bibtex: '@article{Baumeister_Burban_Neaime_Schwabe_2025, title={Non-crossing partitions for exceptional hereditary curves}, journal={arXiv:2512.01729}, author={Baumeister, Barbara and Burban, Igor and Neaime, Georges and Schwabe, Charly Merlin}, year={2025} }' chicago: Baumeister, Barbara, Igor Burban, Georges Neaime, and Charly Merlin Schwabe. “Non-Crossing Partitions for Exceptional Hereditary Curves.” ArXiv:2512.01729, 2025. ieee: B. Baumeister, I. Burban, G. Neaime, and C. M. Schwabe, “Non-crossing partitions for exceptional hereditary curves,” arXiv:2512.01729. 2025. mla: Baumeister, Barbara, et al. “Non-Crossing Partitions for Exceptional Hereditary Curves.” ArXiv:2512.01729, 2025. short: B. Baumeister, I. Burban, G. Neaime, C.M. Schwabe, ArXiv:2512.01729 (2025). date_created: 2026-01-15T09:37:34Z date_updated: 2026-07-06T07:56:11Z external_id: arxiv: - '2512.01729' language: - iso: eng publication: arXiv:2512.01729 status: public title: Non-crossing partitions for exceptional hereditary curves type: preprint user_id: '103440' year: '2025' ... --- _id: '66291' abstract: - lang: eng text: "Abstract\r\n In 1993 Keski-Vakkuri and Wen introduced a model for the fractional quantum Hall effect based on multilayer two-dimensional electron systems satisfying quasi-periodic boundary conditions. Such a model is essentially specified by a choice of a complex torus E and a symmetric positively definite matrix K of size g with non-negative integral coefficients, satisfying some further constraints. The space of the corresponding wave functions turns out to be \r\n \r\n $$\\delta $$\r\n \r\n \ δ\r\n \r\n \r\n \ -dimensional, where \r\n \ \r\n $$\\delta $$\r\n \r\n \ δ\r\n \r\n \r\n \ is the determinant of K. We construct a hermitian holomorphic bundle of rank \r\n \ \r\n $$\\delta $$\r\n \r\n \ δ\r\n \r\n \r\n \ on the abelian variety A (which is the g-fold product of the torus E with itself), whose fibres can be identified with the space of wave function of Keski-Vakkuri and Wen. A rigorous construction of this “magnetic bundle” involves the technique of Fourier–Mukai transforms on abelian varieties. The constructed bundle turns out to be simple and semi-homogeneous and it can be equipped with two different (and natural) hermitian metrics: the one coming from the center-of-mass dynamics and the one coming from the Hilbert space of the underlying many-body system. We prove that the canonical Bott–Chern connection of the first hermitian metric is always projectively flat and give sufficient conditions for this property for the second hermitian metric." article_number: '97' author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: Semyon full_name: Klevtsov, Semyon last_name: Klevtsov citation: ama: Burban I, Klevtsov S. Algebraic Geometry of the Multilayer Model of the Fractional Quantum Hall Effect on a Torus. Communications in Mathematical Physics. 2025;406(5). doi:10.1007/s00220-025-05267-9 apa: Burban, I., & Klevtsov, S. (2025). Algebraic Geometry of the Multilayer Model of the Fractional Quantum Hall Effect on a Torus. Communications in Mathematical Physics, 406(5), Article 97. https://doi.org/10.1007/s00220-025-05267-9 bibtex: '@article{Burban_Klevtsov_2025, title={Algebraic Geometry of the Multilayer Model of the Fractional Quantum Hall Effect on a Torus}, volume={406}, DOI={10.1007/s00220-025-05267-9}, number={597}, journal={Communications in Mathematical Physics}, publisher={Springer Science and Business Media LLC}, author={Burban, Igor and Klevtsov, Semyon}, year={2025} }' chicago: Burban, Igor, and Semyon Klevtsov. “Algebraic Geometry of the Multilayer Model of the Fractional Quantum Hall Effect on a Torus.” Communications in Mathematical Physics 406, no. 5 (2025). https://doi.org/10.1007/s00220-025-05267-9. ieee: 'I. Burban and S. Klevtsov, “Algebraic Geometry of the Multilayer Model of the Fractional Quantum Hall Effect on a Torus,” Communications in Mathematical Physics, vol. 406, no. 5, Art. no. 97, 2025, doi: 10.1007/s00220-025-05267-9.' mla: Burban, Igor, and Semyon Klevtsov. “Algebraic Geometry of the Multilayer Model of the Fractional Quantum Hall Effect on a Torus.” Communications in Mathematical Physics, vol. 406, no. 5, 97, Springer Science and Business Media LLC, 2025, doi:10.1007/s00220-025-05267-9. short: I. Burban, S. Klevtsov, Communications in Mathematical Physics 406 (2025). date_created: 2026-07-07T06:18:00Z date_updated: 2026-07-07T06:18:58Z doi: 10.1007/s00220-025-05267-9 intvolume: ' 406' issue: '5' language: - iso: eng publication: Communications in Mathematical Physics publication_identifier: issn: - 0010-3616 - 1432-0916 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Algebraic Geometry of the Multilayer Model of the Fractional Quantum Hall Effect on a Torus type: journal_article user_id: '72064' volume: 406 year: '2025' ... --- _id: '66296' abstract: - lang: eng text:

In this paper, we elaborate ring theoretic properties of nodal orders. In particular, we prove that they are closed under taking crossed products with finite groups.

author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: Yuriy full_name: Drozd, Yuriy last_name: Drozd citation: ama: 'Burban I, Drozd Y. Some aspects of the theory of nodal orders. In: Contemporary Mathematics. American Mathematical Society; 2025. doi:10.1090/conm/829/16543' apa: Burban, I., & Drozd, Y. (2025). Some aspects of the theory of nodal orders. In Contemporary Mathematics. American Mathematical Society. https://doi.org/10.1090/conm/829/16543 bibtex: '@inbook{Burban_Drozd_2025, place={Providence, Rhode Island}, title={Some aspects of the theory of nodal orders}, DOI={10.1090/conm/829/16543}, booktitle={Contemporary Mathematics}, publisher={American Mathematical Society}, author={Burban, Igor and Drozd, Yuriy}, year={2025} }' chicago: 'Burban, Igor, and Yuriy Drozd. “Some Aspects of the Theory of Nodal Orders.” In Contemporary Mathematics. Providence, Rhode Island: American Mathematical Society, 2025. https://doi.org/10.1090/conm/829/16543.' ieee: 'I. Burban and Y. Drozd, “Some aspects of the theory of nodal orders,” in Contemporary Mathematics, Providence, Rhode Island: American Mathematical Society, 2025.' mla: Burban, Igor, and Yuriy Drozd. “Some Aspects of the Theory of Nodal Orders.” Contemporary Mathematics, American Mathematical Society, 2025, doi:10.1090/conm/829/16543. short: 'I. Burban, Y. Drozd, in: Contemporary Mathematics, American Mathematical Society, Providence, Rhode Island, 2025.' date_created: 2026-07-07T06:35:21Z date_updated: 2026-07-07T06:35:36Z doi: 10.1090/conm/829/16543 language: - iso: eng place: Providence, Rhode Island publication: Contemporary Mathematics publication_identifier: isbn: - '9781470481049' - '9781470476199' issn: - 0271-4132 - 1098-3627 publication_status: published publisher: American Mathematical Society status: public title: Some aspects of the theory of nodal orders type: book_chapter user_id: '72064' year: '2025' ... --- _id: '66294' author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban citation: ama: Burban I. Exceptional hereditary curves and real curve orbifolds. Algebra and Discrete Mathematics. 2025;38(2):166-203. doi:10.12958/adm2365 apa: Burban, I. (2025). Exceptional hereditary curves and real curve orbifolds. Algebra and Discrete Mathematics, 38(2), 166–203. https://doi.org/10.12958/adm2365 bibtex: '@article{Burban_2025, title={Exceptional hereditary curves and real curve orbifolds}, volume={38}, DOI={10.12958/adm2365}, number={2}, journal={Algebra and Discrete Mathematics}, publisher={Luhansk Taras Shevchenko National University}, author={Burban, Igor}, year={2025}, pages={166–203} }' chicago: 'Burban, Igor. “Exceptional Hereditary Curves and Real Curve Orbifolds.” Algebra and Discrete Mathematics 38, no. 2 (2025): 166–203. https://doi.org/10.12958/adm2365.' ieee: 'I. Burban, “Exceptional hereditary curves and real curve orbifolds,” Algebra and Discrete Mathematics, vol. 38, no. 2, pp. 166–203, 2025, doi: 10.12958/adm2365.' mla: Burban, Igor. “Exceptional Hereditary Curves and Real Curve Orbifolds.” Algebra and Discrete Mathematics, vol. 38, no. 2, Luhansk Taras Shevchenko National University, 2025, pp. 166–203, doi:10.12958/adm2365. short: I. Burban, Algebra and Discrete Mathematics 38 (2025) 166–203. date_created: 2026-07-07T06:30:50Z date_updated: 2026-07-07T06:31:07Z doi: 10.12958/adm2365 intvolume: ' 38' issue: '2' language: - iso: eng page: 166-203 publication: Algebra and Discrete Mathematics publication_identifier: issn: - 1726-3255 - 2415-721X publication_status: published publisher: Luhansk Taras Shevchenko National University status: public title: Exceptional hereditary curves and real curve orbifolds type: journal_article user_id: '72064' volume: 38 year: '2025' ... --- _id: '66297' abstract: - lang: eng text:

The goal of this paper is to give an explicit computation of the curvature of the magnetic vector bundle of the multi-layer model of the fractional quantum Hall effect on a torus. We also obtain concrete formulae for the norms of the corresponding wave functions arising in such models.

author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: Semyon full_name: Klevtsov, Semyon last_name: Klevtsov citation: ama: 'Burban I, Klevtsov S. Norms of wave functions for FQHE models on a torus. In: Contemporary Mathematics. American Mathematical Society; 2025. doi:10.1090/conm/829/16544' apa: Burban, I., & Klevtsov, S. (2025). Norms of wave functions for FQHE models on a torus. In Contemporary Mathematics. American Mathematical Society. https://doi.org/10.1090/conm/829/16544 bibtex: '@inbook{Burban_Klevtsov_2025, place={Providence, Rhode Island}, title={Norms of wave functions for FQHE models on a torus}, DOI={10.1090/conm/829/16544}, booktitle={Contemporary Mathematics}, publisher={American Mathematical Society}, author={Burban, Igor and Klevtsov, Semyon}, year={2025} }' chicago: 'Burban, Igor, and Semyon Klevtsov. “Norms of Wave Functions for FQHE Models on a Torus.” In Contemporary Mathematics. Providence, Rhode Island: American Mathematical Society, 2025. https://doi.org/10.1090/conm/829/16544.' ieee: 'I. Burban and S. Klevtsov, “Norms of wave functions for FQHE models on a torus,” in Contemporary Mathematics, Providence, Rhode Island: American Mathematical Society, 2025.' mla: Burban, Igor, and Semyon Klevtsov. “Norms of Wave Functions for FQHE Models on a Torus.” Contemporary Mathematics, American Mathematical Society, 2025, doi:10.1090/conm/829/16544. short: 'I. Burban, S. Klevtsov, in: Contemporary Mathematics, American Mathematical Society, Providence, Rhode Island, 2025.' date_created: 2026-07-07T06:37:04Z date_updated: 2026-07-07T06:37:18Z doi: 10.1090/conm/829/16544 language: - iso: eng place: Providence, Rhode Island publication: Contemporary Mathematics publication_identifier: isbn: - '9781470481049' - '9781470476199' issn: - 0271-4132 - 1098-3627 publication_status: published publisher: American Mathematical Society status: public title: Norms of wave functions for FQHE models on a torus type: book_chapter user_id: '72064' year: '2025' ... --- _id: '66295' abstract: - lang: eng text: In this paper, we study properties of nodal orders defined over arbitrary base fields. In particular we give a classification of complete real nodal orders. author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: Yuriy full_name: Drozd, Yuriy last_name: Drozd citation: ama: Burban I, Drozd Y. Classification of real nodal orders. arXiv:241005792. Published online 2024. apa: Burban, I., & Drozd, Y. (2024). Classification of real nodal orders. In arXiv:2410.05792. bibtex: '@article{Burban_Drozd_2024, title={Classification of real nodal orders}, journal={arXiv:2410.05792}, author={Burban, Igor and Drozd, Yuriy}, year={2024} }' chicago: Burban, Igor, and Yuriy Drozd. “Classification of Real Nodal Orders.” ArXiv:2410.05792, 2024. ieee: I. Burban and Y. Drozd, “Classification of real nodal orders,” arXiv:2410.05792. 2024. mla: Burban, Igor, and Yuriy Drozd. “Classification of Real Nodal Orders.” ArXiv:2410.05792, 2024. short: I. Burban, Y. Drozd, ArXiv:2410.05792 (2024). date_created: 2026-07-07T06:32:11Z date_updated: 2026-07-07T06:32:30Z external_id: arxiv: - '2410.05792' language: - iso: eng publication: arXiv:2410.05792 status: public title: Classification of real nodal orders type: preprint user_id: '72064' year: '2024' ... --- _id: '44328' abstract: - lang: eng text: In this paper, we study equivalences between the categories of quasi–coherent sheaves on non–commutative noetherian schemes. In particular, we give a new proof of Căldăraru's conjecture about Morita equivalences of Azumaya algebras on noetherian schemes. Moreover, we derive necessary and sufficient condition for two reduced non–commutative curves to be Morita equivalent. article_number: '108273' author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: Yu. full_name: Drozd, Yu. last_name: Drozd citation: ama: Burban I, Drozd Yu. Morita theory for non-commutative noetherian schemes. Advances in Mathematics. 2022;399. doi:10.1016/j.aim.2022.108273 apa: Burban, I., & Drozd, Yu. (2022). Morita theory for non-commutative noetherian schemes. Advances in Mathematics, 399, Article 108273. https://doi.org/10.1016/j.aim.2022.108273 bibtex: '@article{Burban_Drozd_2022, title={Morita theory for non-commutative noetherian schemes}, volume={399}, DOI={10.1016/j.aim.2022.108273}, number={108273}, journal={Advances in Mathematics}, author={Burban, Igor and Drozd, Yu.}, year={2022} }' chicago: Burban, Igor, and Yu. Drozd. “Morita Theory for Non-Commutative Noetherian Schemes.” Advances in Mathematics 399 (2022). https://doi.org/10.1016/j.aim.2022.108273. ieee: 'I. Burban and Yu. Drozd, “Morita theory for non-commutative noetherian schemes,” Advances in Mathematics, vol. 399, Art. no. 108273, 2022, doi: 10.1016/j.aim.2022.108273.' mla: Burban, Igor, and Yu. Drozd. “Morita Theory for Non-Commutative Noetherian Schemes.” Advances in Mathematics, vol. 399, 108273, 2022, doi:10.1016/j.aim.2022.108273. short: I. Burban, Yu. Drozd, Advances in Mathematics 399 (2022). date_created: 2023-05-02T18:34:25Z date_updated: 2023-05-07T01:35:27Z department: - _id: '602' doi: 10.1016/j.aim.2022.108273 intvolume: ' 399' language: - iso: eng publication: Advances in Mathematics publication_status: published status: public title: Morita theory for non-commutative noetherian schemes type: journal_article user_id: '49063' volume: 399 year: '2022' ... --- _id: '44327' article_number: '104499' author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: A. full_name: Peruzzi, A. last_name: Peruzzi citation: ama: Burban I, Peruzzi A. On elliptic solutions of the associative Yang-Baxter equation. Journal of Geometry and Physics. 2022;176. apa: Burban, I., & Peruzzi, A. (2022). On elliptic solutions of the associative Yang-Baxter equation. Journal of Geometry and Physics, 176, Article 104499. bibtex: '@article{Burban_Peruzzi_2022, title={On elliptic solutions of the associative Yang-Baxter equation}, volume={176}, number={104499}, journal={Journal of Geometry and Physics}, author={Burban, Igor and Peruzzi, A.}, year={2022} }' chicago: Burban, Igor, and A. Peruzzi. “On Elliptic Solutions of the Associative Yang-Baxter Equation.” Journal of Geometry and Physics 176 (2022). ieee: I. Burban and A. Peruzzi, “On elliptic solutions of the associative Yang-Baxter equation,” Journal of Geometry and Physics, vol. 176, Art. no. 104499, 2022. mla: Burban, Igor, and A. Peruzzi. “On Elliptic Solutions of the Associative Yang-Baxter Equation.” Journal of Geometry and Physics, vol. 176, 104499, 2022. short: I. Burban, A. Peruzzi, Journal of Geometry and Physics 176 (2022). date_created: 2023-05-02T18:32:49Z date_updated: 2023-05-07T01:41:35Z department: - _id: '602' intvolume: ' 176' language: - iso: eng publication: Journal of Geometry and Physics publication_status: published status: public title: On elliptic solutions of the associative Yang-Baxter equation type: journal_article user_id: '49063' volume: 176 year: '2022' ... --- _id: '44537' author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: C. full_name: Alfes-Neumann, C. last_name: Alfes-Neumann - first_name: M. full_name: Raum, M. last_name: Raum citation: ama: Burban I, Alfes-Neumann C, Raum M. A classification of polyharmonic Maaß forms via quiver representations. Published online 2022. apa: Burban, I., Alfes-Neumann, C., & Raum, M. (2022). A classification of polyharmonic Maaß forms via quiver representations. bibtex: '@article{Burban_Alfes-Neumann_Raum_2022, title={A classification of polyharmonic Maaß forms via quiver representations}, author={Burban, Igor and Alfes-Neumann, C. and Raum, M.}, year={2022} }' chicago: Burban, Igor, C. Alfes-Neumann, and M. Raum. “A Classification of Polyharmonic Maaß Forms via Quiver Representations,” 2022. ieee: I. Burban, C. Alfes-Neumann, and M. Raum, “A classification of polyharmonic Maaß forms via quiver representations.” 2022. mla: Burban, Igor, et al. A Classification of Polyharmonic Maaß Forms via Quiver Representations. 2022. short: I. Burban, C. Alfes-Neumann, M. Raum, (2022). date_created: 2023-05-07T00:54:50Z date_updated: 2026-07-07T06:20:58Z department: - _id: '602' language: - iso: eng main_file_link: - url: https://doi.org/10.48550/arXiv.2207.02278 publication_status: published status: public title: A classification of polyharmonic Maaß forms via quiver representations type: preprint user_id: '72064' year: '2022' ... --- _id: '44329' abstract: - lang: eng text: This paper is devoted to algebro-geometric study of infinite dimensional Lie bialgebras, which arise from solutions of the classical Yang–Baxter equation. We regard trigonometric solutions of this equation as twists of the standard Lie bialgebra cobracket on an appropriate affine Lie algebra and work out the corresponding theory of Manin triples, putting it into an algebro-geometric context. As a consequence of this approach, we prove that any trigonometric solution of the classical Yang–Baxter equation arises from an appropriate algebro-geometric datum. The developed theory is illustrated by some concrete examples. author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: R. full_name: Abedin, R. last_name: Abedin citation: ama: Burban I, Abedin R. Algebraic geometry of Lie bialgebras defined by solutions of the classical Yang-Baxter equation. Communications in Mathematical Physics. 2021;387(2):1051–1109. doi:10.1007/s00220-021-04188-7 apa: Burban, I., & Abedin, R. (2021). Algebraic geometry of Lie bialgebras defined by solutions of the classical Yang-Baxter equation. Communications in Mathematical Physics, 387(2), 1051–1109. https://doi.org/10.1007/s00220-021-04188-7 bibtex: '@article{Burban_Abedin_2021, title={Algebraic geometry of Lie bialgebras defined by solutions of the classical Yang-Baxter equation}, volume={387}, DOI={10.1007/s00220-021-04188-7}, number={2}, journal={Communications in Mathematical Physics}, author={Burban, Igor and Abedin, R.}, year={2021}, pages={1051–1109} }' chicago: 'Burban, Igor, and R. Abedin. “Algebraic Geometry of Lie Bialgebras Defined by Solutions of the Classical Yang-Baxter Equation.” Communications in Mathematical Physics 387, no. 2 (2021): 1051–1109. https://doi.org/10.1007/s00220-021-04188-7.' ieee: 'I. Burban and R. Abedin, “Algebraic geometry of Lie bialgebras defined by solutions of the classical Yang-Baxter equation,” Communications in Mathematical Physics, vol. 387, no. 2, pp. 1051–1109, 2021, doi: 10.1007/s00220-021-04188-7.' mla: Burban, Igor, and R. Abedin. “Algebraic Geometry of Lie Bialgebras Defined by Solutions of the Classical Yang-Baxter Equation.” Communications in Mathematical Physics, vol. 387, no. 2, 2021, pp. 1051–1109, doi:10.1007/s00220-021-04188-7. short: I. Burban, R. Abedin, Communications in Mathematical Physics 387 (2021) 1051–1109. date_created: 2023-05-02T18:36:54Z date_updated: 2023-05-07T01:35:11Z department: - _id: '602' doi: 10.1007/s00220-021-04188-7 intvolume: ' 387' issue: '2' language: - iso: eng page: 1051–1109 publication: Communications in Mathematical Physics publication_status: published status: public title: Algebraic geometry of Lie bialgebras defined by solutions of the classical Yang-Baxter equation type: journal_article user_id: '49063' volume: 387 year: '2021' ... --- _id: '44331' abstract: - lang: eng text: In this paper, we study properties of the algebras of planar quasi-invariants. These algebras are Cohen–Macaulay and Gorenstein in codimension one. Using the technique of matrix problems, we classify all Cohen–Macaulay modules of rank one over them and determine their Picard groups. In terms of this classification, we describe the spectral modules of the planar rational Calogero–Moser systems. Finally, we elaborate the theory of the algebraic inverse scattering method, providing explicit computations of some ‘isospectral deformations’ of the planar rational Calogero–Moser system in the case of the split rational potential. author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: A. full_name: Zheglov, A. last_name: Zheglov citation: ama: Burban I, Zheglov A. Cohen-Macaulay modules over the algebra of planar quasi-invariants and Calogero-Moser systems. Proceedings of the London Mathematical Society. 2020;121(4):1033–1082. doi:10.1112/plms.12341 apa: Burban, I., & Zheglov, A. (2020). Cohen-Macaulay modules over the algebra of planar quasi-invariants and Calogero-Moser systems. Proceedings of the London Mathematical Society, 121(4), 1033–1082. https://doi.org/10.1112/plms.12341 bibtex: '@article{Burban_Zheglov_2020, title={Cohen-Macaulay modules over the algebra of planar quasi-invariants and Calogero-Moser systems}, volume={121}, DOI={10.1112/plms.12341}, number={4}, journal={Proceedings of the London Mathematical Society}, author={Burban, Igor and Zheglov, A.}, year={2020}, pages={1033–1082} }' chicago: 'Burban, Igor, and A. Zheglov. “Cohen-Macaulay Modules over the Algebra of Planar Quasi-Invariants and Calogero-Moser Systems.” Proceedings of the London Mathematical Society 121, no. 4 (2020): 1033–1082. https://doi.org/10.1112/plms.12341.' ieee: 'I. Burban and A. Zheglov, “Cohen-Macaulay modules over the algebra of planar quasi-invariants and Calogero-Moser systems,” Proceedings of the London Mathematical Society, vol. 121, no. 4, pp. 1033–1082, 2020, doi: 10.1112/plms.12341.' mla: Burban, Igor, and A. Zheglov. “Cohen-Macaulay Modules over the Algebra of Planar Quasi-Invariants and Calogero-Moser Systems.” Proceedings of the London Mathematical Society, vol. 121, no. 4, 2020, pp. 1033–1082, doi:10.1112/plms.12341. short: I. Burban, A. Zheglov, Proceedings of the London Mathematical Society 121 (2020) 1033–1082. date_created: 2023-05-02T18:47:19Z date_updated: 2023-05-07T01:30:54Z department: - _id: '602' doi: 10.1112/plms.12341 intvolume: ' 121' issue: '4' language: - iso: eng page: 1033–1082 publication: Proceedings of the London Mathematical Society publication_status: published status: public title: Cohen-Macaulay modules over the algebra of planar quasi-invariants and Calogero-Moser systems type: journal_article user_id: '49063' volume: 121 year: '2020' ... --- _id: '44333' abstract: - lang: eng text: This work deals with an algebro–geometric theory of solutions of the classical Yang–Baxter equation based on torsion free coherent sheaves of Lie algebras on Weierstraß cubic curves. author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: L. full_name: Galinat, L. last_name: Galinat citation: ama: Burban I, Galinat L. Torsion free sheaves on Weierstraß cubic curves and the classical Yang-Baxter equation. Communications in Mathematical Physics. 2018;364(1):123–169. doi:10.1007/s00220-018-3172-2 apa: Burban, I., & Galinat, L. (2018). Torsion free sheaves on Weierstraß cubic curves and the classical Yang-Baxter equation. Communications in Mathematical Physics, 364(1), 123–169. https://doi.org/10.1007/s00220-018-3172-2 bibtex: '@article{Burban_Galinat_2018, title={Torsion free sheaves on Weierstraß cubic curves and the classical Yang-Baxter equation}, volume={364}, DOI={10.1007/s00220-018-3172-2}, number={1}, journal={Communications in Mathematical Physics}, author={Burban, Igor and Galinat, L.}, year={2018}, pages={123–169} }' chicago: 'Burban, Igor, and L. Galinat. “Torsion Free Sheaves on Weierstraß Cubic Curves and the Classical Yang-Baxter Equation.” Communications in Mathematical Physics 364, no. 1 (2018): 123–169. https://doi.org/10.1007/s00220-018-3172-2.' ieee: 'I. Burban and L. Galinat, “Torsion free sheaves on Weierstraß cubic curves and the classical Yang-Baxter equation,” Communications in Mathematical Physics, vol. 364, no. 1, pp. 123–169, 2018, doi: 10.1007/s00220-018-3172-2.' mla: Burban, Igor, and L. Galinat. “Torsion Free Sheaves on Weierstraß Cubic Curves and the Classical Yang-Baxter Equation.” Communications in Mathematical Physics, vol. 364, no. 1, 2018, pp. 123–169, doi:10.1007/s00220-018-3172-2. short: I. Burban, L. Galinat, Communications in Mathematical Physics 364 (2018) 123–169. date_created: 2023-05-02T18:50:35Z date_updated: 2023-05-07T01:34:43Z department: - _id: '602' doi: 10.1007/s00220-018-3172-2 intvolume: ' 364' issue: '1' language: - iso: eng page: 123–169 publication: Communications in Mathematical Physics publication_status: published status: public title: Torsion free sheaves on Weierstraß cubic curves and the classical Yang-Baxter equation type: journal_article user_id: '49063' volume: 364 year: '2018' ... --- _id: '44538' author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: Yu. full_name: Drozd, Yu. last_name: Drozd citation: ama: Burban I, Drozd Yu. Non-commutative nodal curves and derived tame algebras. Published online 2018. apa: Burban, I., & Drozd, Yu. (2018). Non-commutative nodal curves and derived tame algebras. bibtex: '@article{Burban_Drozd_2018, title={Non-commutative nodal curves and derived tame algebras}, author={Burban, Igor and Drozd, Yu.}, year={2018} }' chicago: Burban, Igor, and Yu. Drozd. “Non-Commutative Nodal Curves and Derived Tame Algebras,” 2018. ieee: I. Burban and Yu. Drozd, “Non-commutative nodal curves and derived tame algebras.” 2018. mla: Burban, Igor, and Yu. Drozd. Non-Commutative Nodal Curves and Derived Tame Algebras. 2018. short: I. Burban, Yu. Drozd, (2018). date_created: 2023-05-07T00:56:31Z date_updated: 2023-05-07T01:36:42Z department: - _id: '602' language: - iso: eng main_file_link: - url: https://doi.org/10.48550/arXiv.1805.05174 publication_status: published status: public title: Non-commutative nodal curves and derived tame algebras type: preprint user_id: '49063' year: '2018' ... --- _id: '44332' article_number: 1850064-46 pp author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: A. full_name: Zheglov, A. last_name: Zheglov citation: ama: Burban I, Zheglov A. Fourier-Mukai transform on Weierstraß cubics and commuting differential operators. International Journal of Mathematics. 2018;29(10). apa: Burban, I., & Zheglov, A. (2018). Fourier-Mukai transform on Weierstraß cubics and commuting differential operators. International Journal of Mathematics, 29(10), Article 1850064- 46 pp. bibtex: '@article{Burban_Zheglov_2018, title={Fourier-Mukai transform on Weierstraß cubics and commuting differential operators}, volume={29}, number={101850064–46 pp}, journal={International Journal of Mathematics}, author={Burban, Igor and Zheglov, A.}, year={2018} }' chicago: Burban, Igor, and A. Zheglov. “Fourier-Mukai Transform on Weierstraß Cubics and Commuting Differential Operators.” International Journal of Mathematics 29, no. 10 (2018). ieee: I. Burban and A. Zheglov, “Fourier-Mukai transform on Weierstraß cubics and commuting differential operators,” International Journal of Mathematics, vol. 29, no. 10, Art. no. 1850064–46 pp, 2018. mla: Burban, Igor, and A. Zheglov. “Fourier-Mukai Transform on Weierstraß Cubics and Commuting Differential Operators.” International Journal of Mathematics, vol. 29, no. 10, 1850064-46 pp, 2018. short: I. Burban, A. Zheglov, International Journal of Mathematics 29 (2018). date_created: 2023-05-02T18:49:12Z date_updated: 2023-05-07T01:41:27Z department: - _id: '602' intvolume: ' 29' issue: '10' language: - iso: eng publication: International Journal of Mathematics publication_status: published status: public title: Fourier-Mukai transform on Weierstraß cubics and commuting differential operators type: journal_article user_id: '49063' volume: 29 year: '2018' ... --- _id: '44337' author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: Yu. full_name: Drozd, Yu. last_name: Drozd citation: ama: Burban I, Drozd Yu. Maximal Cohen-Macaulay Modules over Non-Isolated Surface Singularities and Matrix Problems. Vol 248. 1178th ed.; 2017. doi:10.1090/memo/1178 apa: Burban, I., & Drozd, Yu. (2017). Maximal Cohen-Macaulay modules over non-isolated surface singularities and matrix problems (1178th ed., Vol. 248). https://doi.org/10.1090/memo/1178 bibtex: '@book{Burban_Drozd_2017, edition={1178}, series={Memoirs of the American Mathematical Society}, title={Maximal Cohen-Macaulay modules over non-isolated surface singularities and matrix problems}, volume={248}, DOI={10.1090/memo/1178}, author={Burban, Igor and Drozd, Yu.}, year={2017}, collection={Memoirs of the American Mathematical Society} }' chicago: Burban, Igor, and Yu. Drozd. Maximal Cohen-Macaulay Modules over Non-Isolated Surface Singularities and Matrix Problems. 1178th ed. Vol. 248. Memoirs of the American Mathematical Society, 2017. https://doi.org/10.1090/memo/1178. ieee: I. Burban and Yu. Drozd, Maximal Cohen-Macaulay modules over non-isolated surface singularities and matrix problems, 1178th ed., vol. 248. 2017. mla: Burban, Igor, and Yu. Drozd. Maximal Cohen-Macaulay Modules over Non-Isolated Surface Singularities and Matrix Problems. 1178th ed., vol. 248, 2017, doi:10.1090/memo/1178. short: I. Burban, Yu. Drozd, Maximal Cohen-Macaulay Modules over Non-Isolated Surface Singularities and Matrix Problems, 1178th ed., 2017. date_created: 2023-05-02T18:59:05Z date_updated: 2023-05-07T01:33:37Z department: - _id: '602' doi: 10.1090/memo/1178 edition: '1178' extern: '1' intvolume: ' 248' language: - iso: eng publication_identifier: isbn: - 978-1-4704-2537-1 publication_status: published series_title: Memoirs of the American Mathematical Society status: public title: Maximal Cohen-Macaulay modules over non-isolated surface singularities and matrix problems type: book user_id: '49063' volume: 248 year: '2017' ... --- _id: '44539' author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: Yu. full_name: Drozd, Yu. last_name: Drozd citation: ama: 'Burban I, Drozd Yu. On the derived categories of gentle and skew-gentle algebras: homological algebra and matrix problems. Published online 2017.' apa: 'Burban, I., & Drozd, Yu. (2017). On the derived categories of gentle and skew-gentle algebras: homological algebra and matrix problems.' bibtex: '@article{Burban_Drozd_2017, title={On the derived categories of gentle and skew-gentle algebras: homological algebra and matrix problems}, author={Burban, Igor and Drozd, Yu.}, year={2017} }' chicago: 'Burban, Igor, and Yu. Drozd. “On the Derived Categories of Gentle and Skew-Gentle Algebras: Homological Algebra and Matrix Problems,” 2017.' ieee: 'I. Burban and Yu. Drozd, “On the derived categories of gentle and skew-gentle algebras: homological algebra and matrix problems.” 2017.' mla: 'Burban, Igor, and Yu. Drozd. On the Derived Categories of Gentle and Skew-Gentle Algebras: Homological Algebra and Matrix Problems. 2017.' short: I. Burban, Yu. Drozd, (2017). date_created: 2023-05-07T00:57:34Z date_updated: 2023-05-07T01:36:54Z department: - _id: '602' extern: '1' language: - iso: eng main_file_link: - url: https://doi.org/10.48550/arXiv.1706.08358 publication_status: published status: public title: 'On the derived categories of gentle and skew-gentle algebras: homological algebra and matrix problems' type: preprint user_id: '49063' year: '2017' ... --- _id: '44334' article_number: '454002' author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: L. full_name: Galinat, L. last_name: Galinat citation: ama: 'Burban I, Galinat L. Simple vector bundles on a nodal Weierstraß cubic and quasi-trigonometric solutions of CYBE. Journal of Physics A: Mathematical and Theoretical. 2017;50.' apa: 'Burban, I., & Galinat, L. (2017). Simple vector bundles on a nodal Weierstraß cubic and quasi-trigonometric solutions of CYBE. Journal of Physics A: Mathematical and Theoretical, 50, Article 454002.' bibtex: '@article{Burban_Galinat_2017, title={Simple vector bundles on a nodal Weierstraß cubic and quasi-trigonometric solutions of CYBE}, volume={50}, number={454002}, journal={Journal of Physics A: Mathematical and Theoretical}, author={Burban, Igor and Galinat, L.}, year={2017} }' chicago: 'Burban, Igor, and L. Galinat. “Simple Vector Bundles on a Nodal Weierstraß Cubic and Quasi-Trigonometric Solutions of CYBE.” Journal of Physics A: Mathematical and Theoretical 50 (2017).' ieee: 'I. Burban and L. Galinat, “Simple vector bundles on a nodal Weierstraß cubic and quasi-trigonometric solutions of CYBE,” Journal of Physics A: Mathematical and Theoretical, vol. 50, Art. no. 454002, 2017.' mla: 'Burban, Igor, and L. Galinat. “Simple Vector Bundles on a Nodal Weierstraß Cubic and Quasi-Trigonometric Solutions of CYBE.” Journal of Physics A: Mathematical and Theoretical, vol. 50, 454002, 2017.' short: 'I. Burban, L. Galinat, Journal of Physics A: Mathematical and Theoretical 50 (2017).' date_created: 2023-05-02T18:51:44Z date_updated: 2023-05-07T01:40:53Z department: - _id: '602' extern: '1' intvolume: ' 50' language: - iso: eng publication: 'Journal of Physics A: Mathematical and Theoretical' publication_status: published status: public title: Simple vector bundles on a nodal Weierstraß cubic and quasi-trigonometric solutions of CYBE type: journal_article user_id: '49063' volume: 50 year: '2017' ... --- _id: '44335' author: - first_name: Igor full_name: Burban, Igor id: '72064' last_name: Burban - first_name: Yu. full_name: Drozd, Yu. last_name: Drozd - first_name: V. full_name: Gavran, V. last_name: Gavran citation: ama: Burban I, Drozd Yu, Gavran V. Minors of non-commutative schemes. European Journal of Mathematics. 2017;3(2):311–341. apa: Burban, I., Drozd, Yu., & Gavran, V. (2017). Minors of non-commutative schemes. European Journal of Mathematics, 3(2), 311–341. bibtex: '@article{Burban_Drozd_Gavran_2017, title={Minors of non-commutative schemes}, volume={3}, number={2}, journal={European Journal of Mathematics}, author={Burban, Igor and Drozd, Yu. and Gavran, V.}, year={2017}, pages={311–341} }' chicago: 'Burban, Igor, Yu. Drozd, and V. Gavran. “Minors of Non-Commutative Schemes.” European Journal of Mathematics 3, no. 2 (2017): 311–341.' ieee: I. Burban, Yu. Drozd, and V. Gavran, “Minors of non-commutative schemes,” European Journal of Mathematics, vol. 3, no. 2, pp. 311–341, 2017. mla: Burban, Igor, et al. “Minors of Non-Commutative Schemes.” European Journal of Mathematics, vol. 3, no. 2, 2017, pp. 311–341. short: I. Burban, Yu. Drozd, V. Gavran, European Journal of Mathematics 3 (2017) 311–341. date_created: 2023-05-02T18:53:20Z date_updated: 2023-05-07T01:41:05Z department: - _id: '602' extern: '1' intvolume: ' 3' issue: '2' language: - iso: eng page: 311–341 publication: European Journal of Mathematics publication_status: published status: public title: Minors of non-commutative schemes type: journal_article user_id: '49063' volume: 3 year: '2017' ...