---
_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). <i>arXiv:260400274</i>. Published online 2026.
  apa: Burban, I., &#38; Gnedin, W. (2026). Representation theory of the Gelfand quiver
    and Harish-Chandra modules for SL_2(R). In <i>arXiv:2604.00274</i>.
  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).” <i>ArXiv:2604.00274</i>, 2026.
  ieee: I. Burban and W. Gnedin, “Representation theory of the Gelfand quiver and
    Harish-Chandra modules for SL_2(R),” <i>arXiv:2604.00274</i>. 2026.
  mla: Burban, Igor, and Wassilij Gnedin. “Representation Theory of the Gelfand Quiver
    and Harish-Chandra Modules for SL_2(R).” <i>ArXiv:2604.00274</i>, 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). <i>arXiv:260518000</i>. Published online 2026.
  apa: Burban, I., &#38; Drozd, Y. (2026). Representation theory of the real Gelfand
    order and real Harish-Chandra modules for SL_2(R). In <i>arXiv:2605.18000</i>.
  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).” <i>ArXiv:2605.18000</i>, 2026.
  ieee: I. Burban and Y. Drozd, “Representation theory of the real Gelfand order and
    real Harish-Chandra modules for SL_2(R),” <i>arXiv:2605.18000</i>. 2026.
  mla: Burban, Igor, and Yuriy Drozd. “Representation Theory of the Real Gelfand Order
    and Real Harish-Chandra Modules for SL_2(R).” <i>ArXiv:2605.18000</i>, 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. <i>arXiv:251201729</i>. Published online 2025.
  apa: Baumeister, B., Burban, I., Neaime, G., &#38; Schwabe, C. M. (2025). Non-crossing
    partitions for exceptional hereditary curves. In <i>arXiv:2512.01729</i>.
  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.” <i>ArXiv:2512.01729</i>,
    2025.
  ieee: B. Baumeister, I. Burban, G. Neaime, and C. M. Schwabe, “Non-crossing partitions
    for exceptional hereditary curves,” <i>arXiv:2512.01729</i>. 2025.
  mla: Baumeister, Barbara, et al. “Non-Crossing Partitions for Exceptional Hereditary
    Curves.” <i>ArXiv:2512.01729</i>, 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: '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. <i>arXiv:241005792</i>.
    Published online 2024.
  apa: Burban, I., &#38; Drozd, Y. (2024). Classification of real nodal orders. In
    <i>arXiv:2410.05792</i>.
  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.” <i>ArXiv:2410.05792</i>,
    2024.
  ieee: I. Burban and Y. Drozd, “Classification of real nodal orders,” <i>arXiv:2410.05792</i>.
    2024.
  mla: Burban, Igor, and Yuriy Drozd. “Classification of Real Nodal Orders.” <i>ArXiv:2410.05792</i>,
    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: '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., &#38; Raum, M. (2022). <i>A classification of
    polyharmonic Maaß forms via quiver representations</i>.
  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. <i>A Classification of Polyharmonic Maaß Forms via Quiver
    Representations</i>. 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: '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., &#38; Drozd, Yu. (2018). <i>Non-commutative nodal curves and derived
    tame algebras</i>.
  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. <i>Non-Commutative Nodal Curves and Derived Tame
    Algebras</i>. 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: '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., &#38; Drozd, Yu. (2017). <i>On the derived categories of gentle
    and skew-gentle algebras: homological algebra and matrix problems</i>.'
  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. <i>On the Derived Categories of Gentle and Skew-Gentle
    Algebras: Homological Algebra and Matrix Problems</i>. 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'
...