Tame roots of wild quivers.
Let \(Q\) be a finite connected quiver. Thanks to \textit{V. G. Kac}'s theorem [Lect. Notes Math. 996, 74-108 (1983; Zbl 0534.14004)] the root system associated with \(Q\) plays a fundamental role in the representation theory of \(Q\). If \(\mathbf d\) is a positive (real or imaginary) root of \(Q\) then it is called a tame root if it is sincere (that is, every coordinate of \(\mathbf d\) is positive) and any positive subroot \(\mathbf d'\) of \(\mathbf d\) satisfies \(q({\mathbf d'})\geq 0\), where \(q\) is the Tits quadratic form of \(Q\). The definition is motivated by the behaviour of representations of Euclidian quivers. From now on assume that \(Q\) is a wild quiver, that is, neither Dynkin nor Euclidian. One of the main results is that a positive root is a tame root if and only if it is sincere and for any decomposition \({\mathbf d}={\mathbf d}^1+\cdots+{\mathbf d}^m\) into subroots we have \(q({\mathbf d}^i)\geq 0\) for \(i=1,\dots,m\). Moreover, \(q({\mathbf d}^i)=0\) for at most one \(i\). Another theorem asserts in particular that if \(\mathbf d\) is a tame root and \(q({\mathbf d})=0\) then \(\mathbf d\) is a Schur root (= the dimension vector of a representation \(M\) of \(Q\) having trivial endomorphism algebra). There are also results on existence of tame roots: it is proved that \(Q\) has tame roots if and only if \(Q\) has at most one subgraph of type \(\widetilde A_n\), and any wild quiver \(Q\) admits only finitely many tame roots.
scientific article; zbMATH DE number 2136737
scientific article