The Tits and Euler forms of a tame algebra
The use of quadratic forms in representation theory of algebras has lead to important results in the area, in particular in the study of the representation type of finite dimensional algebras. In the paper under review, the authors study some properties of the Euler and the Tits forms associated to a tame (strongly) simply connected algebra. Recall that the Tits form of an algebra gives informations related to geometrical aspects while the Euler form reflects the homological behavior of the module category. Let \(k\) be an algebraically closed field and \(A=kQ/I\) be a strongly simply connected algebra, where \(Q\) a finite quiver and \(I\) is an admissible ideal of the path algebra \(kQ\). Denote by \(q_A\) and by \(\chi_A\) the Tits and the Euler forms of \(A\), respectively. The authors show that if \(A\) is of polynomial growth and \(X\) is an indecomposable \(A\)-module, then: (i) \(0\leq q_A(\mathbf{dim} X)\leq 18n\), where \(n\) indicates the number of vertices of \(Q\); (ii) \(0\leq\chi_A(\mathbf{dim} X)\leq 2+e\), where \(e\) indicates the number of projective-injective indecomposable modules; and (iii) if \(X\) is faithful, then \(q_A(\mathbf{dim} X),\chi_A(\mathbf{dim} X)\leq 2\).
scientific article; zbMATH DE number 1347388