Algebras whose Tits form accepts a maximal omnipresent root. (Q2862134)

Let \(A=KQ\) be a hereditary algebra over an algebraically closed field \(K\). Then a fundamental theorem of \textit{V. G. Kac} [Invent. Math. 56, 57-92 (1980; Zbl 0427.17001)] states that the dimension vectors of the indecomposable representations are in one-to-one correspondence with the roots of an associated quadratic form \(q_A\) (i.e. those vectors \(v\in\mathbb Z^n\) with non-negative coordinates \(v(i)\) such that \(q_A(v)\leq 1\)).NEWLINENEWLINE If the algebra is no longer hereditary, one can still associate a quadratic form \(q_A\) to \(A=KQ/I\), the Tits form, but Kac's theorem is no longer true (see e.g. Example 1.5 (5) in the paper under review for a root \(v\) which is not realisable as the dimension vector of an indecomposable representation).NEWLINENEWLINE In the paper under review, the authors prove realisability for a certain class of algebras and dimension vectors: Namely under the assumptions that \(Q\) has neither oriented cycles nor double arrows and \(q_A\) admits a maximal omnipresent (real) root. Here real root means \(q_A(v)=1\), omnipresent that \(v(i)>0\) for all \(i\), and maximal that there is no other root \(w\) with \(w(i)\geq v(i)\) for all \(i\).NEWLINENEWLINE Furthermore, under the additional assumption that \(A\) is strongly simply connected they prove a weak form of Kac's theorem, namely that there is a one-to-one correspondence between the indecomposable sincere \(A\)-modules and the omnipresent roots of \(q_A\).