@unpublished{66293,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@unpublished{66292,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@unpublished{63620,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}},
}

@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}},
}

@unpublished{44537,
  author       = {{Burban, Igor and Alfes-Neumann, C. and Raum, M.}},
  title        = {{{A classification of polyharmonic Maaß forms via quiver representations}}},
  year         = {{2022}},
}

@unpublished{44538,
  author       = {{Burban, Igor and Drozd, Yu.}},
  title        = {{{Non-commutative nodal curves and derived tame algebras}}},
  year         = {{2018}},
}

@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}},
}