The Tits and Euler forms of a tame algebra

Abstract

Since the introduction by Gabriel in 1972 of the Tits quadratic form associated to finite dimensional hereditary algebras, quadratic forms have played an important role in the representation theory of algebras and related categories. There have been quadratic forms associated to finite dimensional algebras, partially ordered sets and other objects in order to determine their representation type and the classes of indecomposable representations in the corresponding Grothendieck group (see for example [3,9,14,21]). The class of finite dimensional algebras (associative, with in identity) over an algebraically closed field k is divided into two disjoint classes. One class consists of tame algebras for which the indecomposable modules occur, in each dimension, in a finite number of discrete and one-parameter families. The second class consist of wild algebras whose representation theory is as complicated as the classification of pairs of matrices up to simultaneous conjugation. In this paper we establish properties of the Tits and Euler forms associated to tame simply connected algebras. Let A be a basic connected finite dimensional algebra over k. Then A has a presentation A = kQA/I , where QA = (Q0, Q1) is the ordinary quiver of A with set of vertices Q0 and set of arrows Q1 and I is an admissible ideal of the path algebra kQA (see [9]). We shall assume that QA has no oriented cycles (and hence g`dim A < ∞). Then the Euler form χA : ZQ0 → Z is defined by