From support \(\tau \)-tilting posets to algebras. I (Q6567178)

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 \(Q\backslash\{\mathrm{loops}\}\)