Abstract :
To each finite dimensional algebra A one can associate an integral quadratic form qA, called the Tits form of A. We show that for some pg-critical algebra B, which is given below, the value of its Tits form on all indecomposable modules is bounded by 2. This gives a negative answer to the following problem, posed by J.A. de la Peña [Representation Theory of Algebras and Related Topics (Mexico City, 1994), CMS Conference Proceedings, vol. 19, AMS, Providence, RI, 1996, p. 159]: Can one characterize the polynomial growth of a tame stronly simply connected algebra by the boundedness of its Tits form on the indecomposable modules? In order to compute the value of qB on indecomposables, we first translate the problem of classifying isomorphism classes of A-modules into a linear matrix problem. Then we show that the matrix problem obtained this way belongs to a class of repetitive matrix problems whose dimension vectors are easy to calculate.
