Algebras whose Tits form accepts a maximal omnipresent root

de la Peña, José; Skowroński, Andrzej

doi:10.1090/S0002-9947-2013-05841-2

Algebras whose Tits form accepts a maximal omnipresent root
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by José A. de la Peña and Andrzej Skowroński PDF

Trans. Amer. Math. Soc. 366 (2014), 395-417 Request permission

Abstract:

Let $k$ be an algebraically closed field and $A$ be a finite-dimensional associative basic $k$-algebra of the form $A=kQ/I$ where $Q$ is a quiver without oriented cycles or double arrows and $I$ is an admissible ideal of $kQ$. We consider roots of the Tits form $q_A$, in particular in the case where $q_A$ is weakly non-negative. We prove that for any maximal omnipresent root $v$ of $q_A$, there exists an indecomposable $A$-module $X$ such that v=dim X. Moreover, if $A$ is strongly simply connected, the existence of a maximal omnipresent root of $q_A$ implies that $A$ is tame of tilted type.

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