Classifying $$\tau $$τ-tilting modules over preprojective algebras of Dynkin type

By Yuya Mizuno. 2013

136 citations recorded by the source corpus.

We study support $$\tau $$τ-tilting modules over preprojective algebras of Dynkin type. We classify basic support $$\tau $$τ-tilting modules by giving a bijection with elements in the corresponding Weyl groups. Moreover we show that they are in bijection with the set of torsion classes, the set of torsion-free classes and many other important objects in representation theory. We also study $$g$$g-matrices of support $$\tau $$τ-tilting modules, which show terms of minimal projective presentations of indecomposable direct summands. We give an explicit description of $$g$$g-matrices and prove that cones given by $$g$$g-matrices coincide with chambers of the associated root systems.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/75281df080a27ac1a68fa1cab669f3882924c2eb

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In Algebra and Number Theory

1. I. Reiten
2. Osamu Iyama
3. Sotaro Asai
4. T. Adachi
5. B. Keller
6. Takuma Aihara
7. Hugh Thomas
8. J. Št’ovíček

In Geometry and Topology

1. Sotaro Asai
2. I. Reiten
3. Osamu Iyama
4. T. Adachi
5. B. Keller
6. Yuya Mizuno
7. Hugh Thomas
8. Takuma Aihara

In Mathematical Physics

1. Sotaro Asai
2. I. Reiten
3. Takuma Aihara
4. T. Adachi
5. Osamu Iyama
6. J. Št’ovíček
7. Yuya Mizuno
8. F. Marks

Citations in the corpus

• The classification of \tau-tilting modules over Nakayama algebras (2013, methodology): “ the elements in N+ are marked by rectangles. One can easily check that the relationship between sτ-tiltΛ and sτ-tiltΛ is given by Theorem 3.3. τ-TILTING MODULES OVER NAKAYAMA ALGEBRAS 23 We refer to [Mi2] for more information on H(Λ) for a preprojective algebra Λ of Dynkin type. In the rest of this”
• Lattice structure of torsion classes for path algebras (2013, background): “ τ-tilting modules. A related partial order has been studied in classical tilting theory by many authors (e.g. [RS, HU, AI, K]). There is also a connection with the weak order on finite Coxeter groups [M]. The aim of this paper is to study the following questions. Question 0.1. Let Λ be a finite dimen”
• Wide Subcategories and Lattices of Torsion Classes (2019, background): “The crucial ingredient is the bijection between the Coxeter group W associated to and tors established by Mizuno [24].”
• A reduction theorem for τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}-rigid modules (2016, unknown): “and B, which is the preprojective algebra of type An, have isomorphic posets of 2-term silting complexes. In fact, Theorem 4.1 immediately recovers all of [20, Theorem 5.3]. Now by a result of Mizuno [24], the poset of 2-term silting complexes over Bis isomorphic to the group Sn+1 with the generatio”
• Classifying tilting complexes over preprojective algebras of Dynkin type (2015, methodology): “bject P is minimal in ∇A(T) if it is a minimal element in the partially ordered set ∇A(T). To keep this notation, we will make the following assumption. 1 The convention of µ+ and µ− is different from [M1] in which we use the converse notation 6 TAKUMA AIHARA AND YUYA MIZUNO Assumption 2.7. In the re”
• Torsion classes, wide subcategories and localisations (2015, background): “8. This corollary, in particular, applies to representation fi nite K-algebras. Furthermore, examples of representation infinite algebras with only finitely many t orsion classes can be found in [A] and [Mi], namely certain Brauer graph algebras and preprojective algebras of Dynkin type. In some specia”