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Socle projective representations of partially ordered sets and Tits quadratic forms with applications to lattices over orders. (English) Zbl 0869.16008

Arnold, David M. (ed.) et al., Abelian groups and modules. Proceedings of the international conference at Colorado Springs, CO, USA, August 7–12, 1995. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 182, 73-111 (1996).
Some concepts arising in the theory of poset representations and applications to lattices over orders (over a complete discrete valuation domain) are presented. Mostly by means of examples, the author explains the particular importance of peak \(I\)-spaces for a poset \(I\), their connection with socle projective representations, and their occurrence in the representation theory of Bäckström orders. A Tits form \(q_I\) is associated with every poset \(I\) with respect to its maximal elements, generalizing the one-peak Tits form of Drozd. This allows to characterize finite as well as fully wild representation type. By an example of Kasjan, however, it is shown that wildness does not imply fully wildness, in general. For a special class of two-peak posets, a characterization of tameness is also given by means of the Tits form, or equivalently, by exclusion of Nazarova’s and 41 additional hypercritical posets [cf. H. J. von Höhne, Proc. Lond. Math. Soc., III. Ser. 73, No. 1, 47-67 (1996; see Zbl 0869.16011 below)]. Together with the minimal posets of infinite prinjective type, these posets are listed in the article. In the finite type case, the indecomposable peak \(I\)-spaces are characterized by their coordinate vector which has to be a positive root of the Tits form.
For an \(n\times n\) tiled order \(\Lambda_0\) containing the radical of a maximal order, two constructions of orders are considered, firstly, an amalgamation of \(\Lambda_0\) with itself along the radical, and secondly, a related order in a matrix algebra of size \(2n\). For these orders, a reduced Tits form is introduced, which is weakly positive if and only if \(\Lambda_0\) is hereditary, and this happens exactly if either of the derived orders is representation-finite. A similar result is presented for the tame case. Moreover, for one of the constructed orders, a description of the Auslander-Reiten quiver is given.
A brief review of tiled orders is included, and Plakhotnik’s result on representations of a poset \(I\) over an artinian p.i.d. \(F\) is treated in a more modern fashion. To this end, the notion of a filtered subprojective poset representation is introduced. By means of an enlarged (infinite) poset \(I^\wedge_m\), depending on the Loewy length \(m\) of \(F\), representation-finiteness of \(I\) over \(F\) is characterized by exclusion of Kleiner’s critical posets.
For the entire collection see [Zbl 0853.00035].

MSC:

16G20 Representations of quivers and partially ordered sets
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
11E04 Quadratic forms over general fields
15A63 Quadratic and bilinear forms, inner products
11H55 Quadratic forms (reduction theory, extreme forms, etc.)

Citations:

Zbl 0869.16011
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