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Towards the classification of sincere weakly positive unit forms. (English) Zbl 0830.16013

Unit forms \(\chi:\mathbb{Z}^n \to \mathbb{Z}\) plays an important role in the representation theory of finite-dimensional algebras, quivers, posets and other algebraic structures. In many cases they characterize the representation type of the considered object: It is of finite, resp. tame representation type iff the associated unit form is weakly positive, resp. weakly non negative [K. Bongartz, J. Lond. Math. Soc., II. Ser. 28, 461-469 (1983; Zbl 0532.16020); Yu. A. Drozd, Funkts. Anal. Prilozh. 8, No. 3, 34-42 (1974; Zbl 0356.06003); P. Gabriel, Manuscr. Math. 6, 71-103 (1972; Zbl 0232.08001); H.-J. von Höhne, Comment. Math. Helv. 63, 312-336 (1988; Zbl 0662.15014); and On weakly nonnegative unit forms and tame algebras, Proc. Lond. Math. Soc., III. Ser. (to appear); J. A. de la Peña, Quadratic forms and the representation type of an algebra, Ergänzungsreihe 90-003, SFB 343, Bielefeld; C. M. Ringel, Tame algebras and integral quadratic forms (Lect. Notes Math. 1099, 1984; Zbl 0546.16013)].
The paper is aimed at a first step to the classification of all sincere weakly positive unit forms. The authors consider a subclass of these forms, namely “good thin forms”. Here, a weakly positive unit form \(\chi\) in \(n\) variables is thin if \(\varepsilon^{(n)}=(1,1,\dots,1)\) is a root and good if there is no radical vector \(\mu \neq 0\) of \(\chi\) such that both \(\varepsilon^{(n)}+\mu\) and \(\varepsilon^{(n)}-\mu\) are positive roots.
The main result of the paper is a complete classification of all good thin weakly positive unit forms. This consists of finitely many series containing all forms in a sufficiently large number of variables, together with huge lists of exceptional forms in few variables \((n \leq 20)\) produced by a computer (lists which upon request will be sent to the interested reader by the authors).
In more detail: a list of 63 good thin forms in at most 11 variables is presented, given by their bigraphs in which certain sets \(\Lambda\) of “linking” vertices are distinguished. Each of these forms \(\chi\) and sets \(\Lambda\) define a series \(S(\chi, \Lambda)\) by blowing up linking points to “chains”. Let \(S(n)\) denote the set of all unit forms in \(n\) variables which, up to renumbering of the vertices, occur in one of these series. Using the computer lists as base of induction it is proved that for \(n \geq 15\) the set of good thin weakly positive unit forms in \(n\) variables coincides with \(S(n)\). Essential tools for the paper are Gabrielov transformations and reflection extensions.
Reviewer: H.Kupisch (Berlin)

MSC:

16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16G10 Representations of associative Artinian rings
15A63 Quadratic and bilinear forms, inner products
16G20 Representations of quivers and partially ordered sets
16G30 Representations of orders, lattices, algebras over commutative rings
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References:

[1] Bongartz, K., Algebras and quadratic forms, J. Lond. Math. Soc., 28, 2, 461-469 (1983) · Zbl 0532.16020
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[10] Ringel, C. M., Tame Algebras and Integral Quadratic Forms, (Lecture Notes in Mathematics 1099 (1984), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0546.16013
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