One point extensions of linear quivers and quadratic forms (Q1112152)

Let A be a basic and connected finite dimensional algebra over an algebraically closed field k. If A has no oriented cycles in its ordinary quiver, we let \(q_ A\) denote its Tits quadratic form. Recall that \(q_ A\) is said to be weakly positive if \(q_ A(x)>0\) for every positive integral vector x, and weakly non-negative if \(q_ A(x)\geq 0\) for every positive integral vector x, and moreover there is at least one positive integral vector z with \(q_ A(z)=0\). It was shown by \textit{K. Bongartz} [J. Lond. Math. Soc., II. Ser. 28, 461-469 (1983; Zbl 0532.16020)] that, if A has a preprojective component, then A is representation-finite if and only if \(q_ A\) is weakly positive. He also asked whether there exist a similar result for tame algebras. In this article, the author considers algebras A of the following form. Let \(kA_ n\) be the path algebra of a quiver with underlying graph \({\mathbb{A}}_ n\) (with an arbitrary orientation) and A be the one-point extension \(A=\left[ \begin{matrix} k&M\\0&kA_ n \end{matrix} \right]\) where M is an indecomposable \(kA_ n\)-module. He shows that A is tame representation-infinite if and only if \(q_ A\) is weakly non-negative. The proof is based on a reduction procedure to some limiting cases, and a study of the vector-space category associated to A. It also uses the fact that an algebra A as above is of global dimension at most two, and thus the Tits form of A coincides with its homological quadratic form (see \textit{C. M. Ringel} [``Tame algebras and integral quadratic forms'' (Lect. Notes Math. 1099, 1984; Zbl 0546.16013)]). A similar result has been recently obtained for one-point extensions of tame concealed algebras by \textit{J. A. de la Peña} [Manuscr. Math. 61, 183-194 (1988; Zbl 0647.16021)].