@unpublished{66293, abstract = {{In 1970, Gelfand posed the problem of classifying the indecomposable objects in a representation category equivalent to the principal block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 = {{Burban, Igor and Gnedin, Wassilij}}, booktitle = {{arXiv:2604.00274}}, title = {{{Representation theory of the Gelfand quiver and Harish-Chandra modules for SL_2(R)}}}, year = {{2026}}, } @unpublished{66292, abstract = {{In this article we study the principal block of the category of real Harish-Chandra modules for the group $\mathsf{SL}_2(\RR)$ and relate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 = {{Burban, Igor and Drozd, Yuriy}}, booktitle = {{arXiv:2605.18000}}, title = {{{Representation theory of the real Gelfand order and real Harish-Chandra modules for SL_2(R)}}}, year = {{2026}}, } @unpublished{63620, abstract = {{We introduce a new class of reflection groups associated with the canonical bilinear lattices of Lenzing, which we call reflection groups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 = {{Baumeister, Barbara and Burban, Igor and Neaime, Georges and Schwabe, Charly Merlin}}, booktitle = {{arXiv:2512.01729}}, title = {{{Non-crossing partitions for exceptional hereditary curves}}}, year = {{2025}}, } @article{66291, abstract = {{Abstract 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 $$\delta $$ δ -dimensional, where $$\delta $$ δ is the determinant of K. We construct a hermitian holomorphic bundle of rank $$\delta $$ δ 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.}}, author = {{Burban, Igor and Klevtsov, Semyon}}, issn = {{0010-3616}}, journal = {{Communications in Mathematical Physics}}, number = {{5}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Algebraic Geometry of the Multilayer Model of the Fractional Quantum Hall Effect on a Torus}}}, doi = {{10.1007/s00220-025-05267-9}}, volume = {{406}}, year = {{2025}}, } @inbook{66296, abstract = {{

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 = {{Burban, Igor and Drozd, Yuriy}}, booktitle = {{Contemporary Mathematics}}, isbn = {{9781470481049}}, issn = {{0271-4132}}, publisher = {{American Mathematical Society}}, title = {{{Some aspects of the theory of nodal orders}}}, doi = {{10.1090/conm/829/16543}}, year = {{2025}}, } @article{66294, author = {{Burban, Igor}}, issn = {{1726-3255}}, journal = {{Algebra and Discrete Mathematics}}, number = {{2}}, pages = {{166--203}}, publisher = {{Luhansk Taras Shevchenko National University}}, title = {{{Exceptional hereditary curves and real curve orbifolds}}}, doi = {{10.12958/adm2365}}, volume = {{38}}, year = {{2025}}, } @inbook{66297, abstract = {{

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 = {{Burban, Igor and Klevtsov, Semyon}}, booktitle = {{Contemporary Mathematics}}, isbn = {{9781470481049}}, issn = {{0271-4132}}, publisher = {{American Mathematical Society}}, title = {{{Norms of wave functions for FQHE models on a torus}}}, doi = {{10.1090/conm/829/16544}}, year = {{2025}}, } @unpublished{66295, abstract = {{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 = {{Burban, Igor and Drozd, Yuriy}}, booktitle = {{arXiv:2410.05792}}, title = {{{Classification of real nodal orders}}}, year = {{2024}}, } @article{44328, abstract = {{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.}}, author = {{Burban, Igor and Drozd, Yu.}}, journal = {{Advances in Mathematics}}, title = {{{Morita theory for non-commutative noetherian schemes}}}, doi = {{10.1016/j.aim.2022.108273}}, volume = {{399}}, year = {{2022}}, } @article{44327, author = {{Burban, Igor and Peruzzi, A.}}, journal = {{Journal of Geometry and Physics}}, title = {{{On elliptic solutions of the associative Yang-Baxter equation}}}, volume = {{176}}, year = {{2022}}, } @unpublished{44537, author = {{Burban, Igor and Alfes-Neumann, C. and Raum, M.}}, title = {{{A classification of polyharmonic Maaß forms via quiver representations}}}, year = {{2022}}, } @article{44329, abstract = {{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 = {{Burban, Igor and Abedin, R.}}, journal = {{Communications in Mathematical Physics}}, number = {{2}}, pages = {{1051–1109}}, title = {{{Algebraic geometry of Lie bialgebras defined by solutions of the classical Yang-Baxter equation}}}, doi = {{10.1007/s00220-021-04188-7}}, volume = {{387}}, year = {{2021}}, } @article{44331, abstract = {{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 = {{Burban, Igor and Zheglov, A.}}, journal = {{Proceedings of the London Mathematical Society}}, number = {{4}}, pages = {{1033–1082}}, title = {{{Cohen-Macaulay modules over the algebra of planar quasi-invariants and Calogero-Moser systems}}}, doi = {{10.1112/plms.12341}}, volume = {{121}}, year = {{2020}}, } @article{44333, abstract = {{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 = {{Burban, Igor and Galinat, L.}}, journal = {{Communications in Mathematical Physics}}, number = {{1}}, pages = {{123–169}}, title = {{{Torsion free sheaves on Weierstraß cubic curves and the classical Yang-Baxter equation}}}, doi = {{10.1007/s00220-018-3172-2}}, volume = {{364}}, year = {{2018}}, } @unpublished{44538, author = {{Burban, Igor and Drozd, Yu.}}, title = {{{Non-commutative nodal curves and derived tame algebras}}}, year = {{2018}}, } @article{44332, author = {{Burban, Igor and Zheglov, A.}}, journal = {{International Journal of Mathematics}}, number = {{10}}, title = {{{Fourier-Mukai transform on Weierstraß cubics and commuting differential operators}}}, volume = {{29}}, year = {{2018}}, } @book{44337, author = {{Burban, Igor and Drozd, Yu.}}, isbn = {{978-1-4704-2537-1}}, title = {{{Maximal Cohen-Macaulay modules over non-isolated surface singularities and matrix problems}}}, doi = {{10.1090/memo/1178}}, volume = {{248}}, year = {{2017}}, } @unpublished{44539, author = {{Burban, Igor and Drozd, Yu.}}, title = {{{On the derived categories of gentle and skew-gentle algebras: homological algebra and matrix problems}}}, year = {{2017}}, } @article{44334, author = {{Burban, Igor and Galinat, L.}}, journal = {{Journal of Physics A: Mathematical and Theoretical}}, title = {{{Simple vector bundles on a nodal Weierstraß cubic and quasi-trigonometric solutions of CYBE}}}, volume = {{50}}, year = {{2017}}, } @article{44335, author = {{Burban, Igor and Drozd, Yu. and Gavran, V.}}, journal = {{European Journal of Mathematics}}, number = {{2}}, pages = {{311–341}}, title = {{{Minors of non-commutative schemes}}}, volume = {{3}}, year = {{2017}}, }