Tilting Quivers for Hereditary Algebras

Let <i>A</i> be a finite dimensional hereditary algebra over an algebraically closed field <i>k</i>. In this paper, we study the tilting quiver of <i>A</i> from the viewpoint of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" di...

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Bibliographic Details
Main Author: Shen Li
Format: Article
Language:English
Published: MDPI AG 2024-01-01
Series:Mathematics
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Online Access:https://www.mdpi.com/2227-7390/12/2/191
Description
Summary:Let <i>A</i> be a finite dimensional hereditary algebra over an algebraically closed field <i>k</i>. In this paper, we study the tilting quiver of <i>A</i> from the viewpoint of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>-tilting theory. First, we prove that there exists an isomorphism between the support <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>-tilting quiver <i>Q</i>(s<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>-tilt <i>A</i>) of <i>A</i> and the tilting quiver <i>Q</i>(tilt <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>A</mi><mo>¯</mo></mover></semantics></math></inline-formula>) of the duplicated algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>A</mi><mo>¯</mo></mover></semantics></math></inline-formula>. Then, we give a new method to calculate the number of arrows in the tilting quiver <i>Q</i>(tilt <i>A</i>) when <i>A</i> is representation-finite. Finally, we study the conjecture given by Happel and Unger, which claims that each connected component of <i>Q</i>(tilt <i>A</i>) contains only finitely many non-saturated vertices. We provide an example to show that this conjecture does not hold for some algebras whose quivers are wild with at least four vertices.
ISSN:2227-7390