Tilting modules over duplicated algebras

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

arXiv comment13 pages

References in corpus (1)

• The number of arrows in the quiver of tilting modules over a path algebra of type $A$ and $D$