Tilting Quivers for Hereditary Algebras

Shen Li ()
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Shen Li: School of Science, Shandong Jianzhu University, Jinan 250101, China

Mathematics, 2024, vol. 12, issue 2, 1-9

Abstract: Let A be a finite dimensional hereditary algebra over an algebraically closed field k . In this paper, we study the tilting quiver of A from the viewpoint of τ -tilting theory. First, we prove that there exists an isomorphism between the support τ -tilting quiver Q (s τ -tilt A ) of A and the tilting quiver Q (tilt A ¯ ) of the duplicated algebra A ¯ . Then, we give a new method to calculate the number of arrows in the tilting quiver Q (tilt A ) when A is representation-finite. Finally, we study the conjecture given by Happel and Unger, which claims that each connected component of Q (tilt A ) 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.

Keywords: tilting module; support ? -tilting module; tilting quiver; support ? -tilting quiver (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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