Quivers with relations for symmetrizable Cartan matrices I: Foundations
International audience
SYMMETRIES OF INTEGRABLE DEFORMATIONS (Combinatorics of Lie Type)
In this note, we will explain symmetries of isomonodromic deformations as Weyl groups of some quivers and give classifications of isomondromic deformations of linear ordinary differential equations with at most unramified irregular singularities and 2 or 4 accessory parameters.
Representations of Regular Trees and Invariants of AR-Components for Generalized Kronecker Quivers
Published in Algebras and Representation Theory
We investigate the generalized Kronecker algebra 𝒦r = kΓr with r ≥ 3 arrows. Given a regular component 𝒞 of the Auslander-Reiten quiver of 𝒦r, we show that the quasi-rank rk(𝒞) ∈ ℤ≤1 can be described almost exactly as the distance 𝒲(𝒞) ∈ ℕ0 between two non-intersecting cones in 𝒞, given by modules with the equal images and the equal kernels propert...
Decomposition theory of modules: the case of Kronecker algebra
Published in Japan Journal of Industrial and Applied Mathematics
Let A be a finite-dimensional algebra over an algebraically closed field k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Bbbk $$\end{document}. For any finite-dimensi...
Relative stable equivalences of Morita type
Published in Mathematische Zeitschrift
In the representation theory of finite groups, the stable equivalence of Morita type plays a prominent role. However, except for self-injective algebras, one does not know much on existence of such equivalences between arbitrary algebras. Moreover, this notion seems to be not general enough, to be preserved by classical algebraic constructions. In ...
Degenerations of submodules and composition series
Let $M$ and $N$ be modules over an artin algebra such that $M$ degenerates to $N$. We show that any submodule of $M$ degenerates to a submodule of $N$. This suggests that a composition series of $M$ will in some sense degenerate to a composition series of $N$. We then study a subvariety of the module variety, consisting of those representations whe...
Pro-Species of Algebras I: Basic Properties
Published in Algebras and Representation Theory
In this paper, we generalise part of the theory of hereditary algebras to the context of pro-species of algebras. Here, a pro-species is a generalisation of Gabriel’s concept of species gluing algebras via projective bimodules along a quiver to obtain a new algebra. This provides a categorical perspective on a recent paper by Geiß et al. (2016). In...
Differential Modules over Quadratic Monomial Algebras
Published in Algebras and Representation Theory
We compare the so-called clock condition to the gradability of certain differential modules over quadratic monomial algebras. These considerations show that a stably hereditary or gentle one-cycle algebra is piecewise hereditary if and only if the orbit category of its bounded derived category with respect to a positive power of the shift functor i...
Geometric Realizations of Lusztig’s Symmetries of Symmetrizable Quantum Groups
Published in Algebras and Representation Theory
Let U be the quantum group and f be the Lusztig’s algebra associated with a symmetrizable generalized Cartan matrix. The algebra f can be viewed as the positive part of U. Lusztig introduced some symmetries Ti on U for all i ∈ I. Since Ti(f) is not contained in f, Lusztig considered two subalgebras if and if of f for any i ∈ I, where if={x ∈ f | Ti...
Poisson-Nijenhuis structures on quiver path algebras
We introduce a notion of noncommutative Poisson-Nijenhuis structure on the path algebra of a quiver. In particular, we focus on the case when the Poisson bracket arises from a noncommutative symplectic form. The formalism is then applied to the study of the Calogero-Moser and Gibbons-Hermsen integrable systems. In the former case, we give a new int...
