On Weakly Non-Negative Unit forms and Tame Algebras

Abstract

In this paper we consider integral quadratic forms q: IP —>Z of the form q(x) = 2i x] + ^i<jqijX,Xj and call them unit forms. Such a unit form q is said to be weakly non-negative if q(x)^Q whenever Jt2*0, that is, whenever x^O for i = 1,..., n. Our main purpose is to formulate a criterion for weak non-negativity of unit forms. This question arises in various areas of representation theory in connection with the characterization of tame representation type: in the representation theory of quivers [17, 6], partially ordered sets [18, 19], certain classes of bimodules [8, 23, 1.3] and certain classes of algebras [20, 14, 24, 16]. For each of these structures Q, there is a unit form qn, naturally attached to Q, such that the weak non-negativity of qn is an equivalent (or at least a necessary) condition for Q to be of tame representation type. A class of algebras for which tame representation type is equivalent to the weak non-negativity of the associated unit form is described in the last section of this paper. A general criterion for weak non-negativity using determinants was given by Zeldich [28] (see also [21]), another criterion was obtained by de la Pena [21]. Our approach is similar to that in [11], where we gave a characterization of weakly positive unit forms based on the classification of the critical forms. Let us recall some notions. Given a unit form q: IP-*TL and a non-empty subset / = {ix < ... < im) of {1,..., n), we denote by qt the quadratic form qd, where d, Z —*• IT maps the /cth natural basis vector ek onto eik. Such a unit form q, is called the restriction of q to /. We call a unit form q on IP a hypercritical form if q, is weakly non-negative for every proper subset /<={1,..., n), though q itself is not. In these terms, a unit form q is weakly non-negative if and only if q admits no hypercritical restriction. So we have to classify the hypercritical forms. In § 1 we show that they can be obtained by certain elementary transformations from the unit forms naturally attached to the hyperbolic graphs of Fig. 1 on p. 50. In particular, there are only finitely many hypercritical forms. However, their number is large. By a simple process we reduce the problem to the classification of the stretched hypercritical forms. In § 2 we show that, except for a few of them, these are of the shape q(x) = q'(x') + xn -xn-iXn where q' = q{\,...,n-D is critical. (In a similar way, the unit forms attached to the hyperbolic graphs Am, Dm, Em correspond to those