arXiv subjects

Wassilij Gnedin

Publications and source records attributed to Wassilij Gnedin.

6 recordsLinked to original sources

arXiv2026

Representation theory of the Gelfand quiver and Harish-Chandra modules for $\mathsf{SL}_2(\mathbb{R})$

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.

math.RT↗

arXiv2025

Derived invariants of gentle orders

This article is concerned with the derived representation theory of certain infinite-dimensional gentle algebras called gentle orders. For a gentle order, we provide a factorization of the derived Nakayama functor, study its fractionally Calabi-Yau objects and exceptional cycles, and establish that certain combinatorial invariants of its underlying quiver are derived invariants, analogous to results for finite-dimensional gentle algebras.

math.RT↗

arXiv2023

A class of Gorenstein algebras and their dualities

In the recent paper "The Nakayama functor and its completion for Gorenstein algebras", a class of Gorenstein algebras over commutative noetherian rings was introduced, and duality theorems for various categories of representations were established. The manuscript on hand provides more context to the results presented in the aforementioned work, identifies new classes of Gorenstein algebras, and explores their behaviour under standard operations like taking tensor products and tilting.

math.RT↗

arXiv2022

Silting theory under change of rings

The main goal of this paper is to compare the silting theory of an $R$-algebra $Λ$ over a Noetherian ring $R$ with that of its tensor product $Λ\otimes Γ$ with another $R$-algebra $Γ$. In the case that the $R$-algebra $Λ$ is Noetherian, $R$ a complete local ring and $\mathfrak{a}$ a certain ideal of the ring $R$, we obtain an isomorphism between the silting poset of $Λ$ and that of its quotient $Λ\otimes R/ \mathfrak{a}$. Furthermore, we study the restrictions of such a bijection to tilting complexes and deduce silting embedding and descent results for the algebra $Λ$ and a certain family of algebras $(Λ\otimes Γ_i)_{i \in I}$.

math.RT↗

arXiv2019

Calabi-Yau properties of ribbon graph orders

We pursue the order-theoretic approach to ribbon graphs initiated by Kauer and Roggenkamp. We show that any ribbon graph order is twisted $1$-Calabi-Yau in general and $1$-Calabi-Yau if the ribbon graph is bipartite. We derive analogous results for anti-commutative versions of Brauer graph algebras.

math.RT↗

arXiv2013

Cohen-Macaulay modules over some non-reduced curve singularities

In this article, we study Cohen-Macaulay modules over non-reduced curve singularities. We prove that the rings $k[[x,y,z]]/(xy, y^q -z^2)$ have tame Cohen-Macaulay representation type. For the singularity $k[[x,y,z]]/(xy, z^2)$ we give an explicit description of all indecomposable Cohen--Macaulay modules and apply the obtained classification to construct explicit families of indecomposable matrix factorizations of $(xy)^2 \in k[[x,y]]$.

math.AG↗