On Finiteness of Critical Tits Forms of Posets

The quadratic Tits form, introduced by P. Gabriel [1] for quivers, Yu.A. Drozd [2] for posets, and M.M. Kleiner, A.V. Roiter [3] and Yu.A. Drozd [4] for a wide class of classification problems, plays an important role in representation theory. In particular, there are many results on connections between representation types of various objects and properties of the Tits forms. The reader interested in this topic is refereed to the papers of [5, 6], the monographs [7, 8] and, e.g., [9–15] (with the bibliographies therein). Above all one must mention the well known result that a quiver is of finite type if and only if its Tits form is positive [1]; in the case of posets the Tits form must be weakly positive [2] (recall that representations of posets were introduced in [16]). It follows from the results of [2] that the Tits form of a poset S is weakly positive if and only if S contains no subposet isomorphic to a critical, with respect to finiteness of type, poset (critical posets are indicated in [17]; their number is 5). Our paper is devoted to study critical, with respect to positivity, Tits forms of posets. Formulate first the main result. Let S be a (finite or infinite) poset and Z the integer numbers. Denote by ZS∪0 0 the subset of the cartesian product ZS∪0 consisting of all vectors z = (zi), i ∈ S ∪ 0 with only finitely many non-zero coordinates. The quadratic Tits form of S is by definition the form qS : ZS∪0 0 → Z defined by the equality