On some Ringel-Hall numbers in tame cases

Szántó, Csaba (2014) On some Ringel-Hall numbers in tame cases. Acta Universitatis Sapientiae Mathematica, 6 (1). pp. 61-72. ISSN 1844-6094

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Abstract

Let $k$ be a finite field and consider the finite dimensional path algebra $kQ$ where $Q$ is a quiver of tame type i.e. of type $\tilde A_n,\tilde D_n,\tilde E_6, \tilde E_7,\tilde E_8$. Let $\mathcal{H}(kQ)$ be the corresponding Ringel-Hall algebra. We are going to determine the Ringel-Hall numbers of the form $F^{P'}_{XP}$ with $P,P'$ preprojective indecomposables of defect -1 and $F^{I'}_{IX}$ with $I,I'$ preinjective indecompo\-sables of defect 1. It turns out that these numbers are either 1 or 0.

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