From support \(\tau \)-tilting posets to algebras. I
type: Item
instance of: scholarly article
title: From support \(\tau \)-tilting posets to algebras. I
author: Ryoichi Kase
published in: Transactions of the American Mathematical Society
publication date: 2024-07-04T00:00:00Z
review text: Support \(\tau\)-tilting modules were introduced by \textit{T. Adachi} et al. [Compos. Math. 150, No. 3, 415--452 (2014; Zbl 1330.16004)]. \(\tau\)-tilting theory has great importance in representation theory and has a close relationship to torsion theory, silting theory and cluster-tilting theory.\N\NIn 2009, \textit{D. Happel} and \textit{L. Unger} proved an interesting reconstruction theorem [Trans. Am. Math. Soc. 361, No. 7, 3633--3660 (2009; Zbl 1243.16014)]. More precisely, they can reconstruct a quiver \(Q\) up to multiple arrows from the tilting poset of \(KQ\). The aim of the paper under review is to extend Happel-Unger's reconstruction theorem to support \(\tau\)-tilting posets.\N\NThe author first studies general properties of a poset isomorphism between support \(\tau\)-tilting posets. Assume \(\rho:\) \(s\tau\)-tilt\(\Lambda\to s\tau\)-tilt\(\Gamma\) is a poset isomorphism, then (1) \(\rho\) preserves supports of basic support \(\tau\)-tilting modules, (2) if \(s\tau\)-tilt\(\Lambda\) is a lattice, then \(\rho\) induces a natural bijection between isomorphism classes of basic \(\tau\)-rigid pairs of \(\Lambda\) and \(\Gamma\), (3) \(\rho\) preserves supports of indecomposable projective modules. Secondly, the author gives an analogue of Happel-Unger's reconstruction theorem. He proves that the support \(\tau\)-tilting poset of \(\Lambda\) determines the quiver of \(\Lambda\) up to multiple arrows and loops. If \(\Lambda=kQ/I\) is a \(\tau\)-tilting finite algebra, then \(Q\) has no multiple arrows and the group of poset automorphism of the support \(\tau\)-tilting poset of \(\Lambda\) is realized as a subgroup of the group of quiver automorphisms of \(Q\backslash\{\mathrm{loops}\}\)
reviewed by: Yongliang Sun
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 16D80
zbMATH DE Number: 7876061
zbMATH Keywords: representation of quivers
zbMATH Keywords: silting complex
zbMATH Keywords: support \( \tau \)-tilting poset
zbMATH Keywords: support \( \tau \)-tilting module
MaRDI profile type: Publication
cites work: -tilting theory
cites work: Silting mutation in triangulated categories
cites work: Semibricks
cites work: Wide subcategories and lattices of torsion classes
cites work: Q5704250
cites work: Preprojective modules over Artin algebras
cites work: $\boldsymbol{\tau}$ -Tilting Finite Algebras, Bricks, and $\boldsymbol{g}$-Vectors
cites work: Lattice theory of torsion classes: Beyond 𝜏-tilting theory
cites work: ICE-closed subcategories and wide \(\tau \)-tilting modules
cites work: On the quiver of tilting modules.
cites work: Reconstruction of path algebras from their posets of tilting modules
cites work: Lattice structure of Weyl groups via representation theory of preprojective algebras
cites work: Lattice structure of torsion classes for path algebras
cites work: Reduction of τ-Tilting Modules and Torsion Pairs
cites work: Taking tilting modules from the poset of support tilting modules.
cites work: Silting objects, simple-minded collections, \(t\)-structures and co-\(t\)-structures for finite-dimensional algebras.
cites work: Generalized associahedra via quiver representations
cites work: Classifying \(\tau\)-tilting modules over preprojective algebras of Dynkin type
DOI: 10.1090/TRAN/9137
description: scientific article; zbMATH DE number 7876061