Tilting modules over duplicated algebras

TL;DR

This paper studies tilting modules over duplicated hereditary algebras, establishing bounds on endomorphism ring global dimension, embedding properties of tilting quivers, and implications for tame and Dynkin types.

Contribution

It proves the global dimension bound for endomorphism rings, explores tilting quiver embeddings, and addresses conjectures related to tame and Dynkin types.

Findings

Global dimension of endomorphism rings is at most 3.

Embedding of tilting quivers from A to A^{(1)}.

Number of arrows in tilting quivers is orientation-independent for Dynkin types.

Abstract

Let $A$ be a finite dimensional hereditary algebra over a field $k$ and $A^{(1)}$ the duplicated algebra of $A$. We first show that the global dimension of endomorphism ring of tilting modules of $A^{(1)}$ is at most 3. Then we investigate embedding tilting quiver $\mathscr{K}(A)$ of $A$ into tilting quiver $\mathscr{K}(A^{(1)})$ of $A^{(1)}$. As applications, we give new proofs for some results of D.Happel and L.Unger, and prove that every connected component in $\mathscr{K}({A})$ has finite non-saturated points if $A$ is tame type, which gives a partially positive answer to the conjecture of D.Happel and L.Unger in [10]. Finally, we also prove that the number of arrows in $\mathscr{K}({A})$ is a constant which does not depend on the orientation of $Q$ if $Q$ is Dynkin type.