Tilting modules over duplicated algebras

arXiv1105.2994MaRDI QIDQ6225418

Shun-Hua Zhang, Guopeng Wang

Publication date: 15 May 2011

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 $mathscrK(A)$ of $A$ into tilting quiver $mathscrK(A^{(1)})$ 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 $mathscrK(A)$ 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 $mathscrK(A)$ is a constant which does not depend on the orientation of $Q$ if $Q$ is Dynkin type.

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