Representation theory of orders (Q2759660)

A survey on important developments concerning classical orders, which have been obtained recently by the author, with results taken from several of his papers, is presented here. Main topics are Bass orders, lattice-finite orders, and the structure of their Auslander-Reiten quivers. The classification of Bass orders is put into the wider frame of quadratic extensions of algebras which play an important rôle in the theory of orders. They have been studied recently in a series of papers by the author. Following Hijikata and Nishida, Bass orders in non-semisimple algebras are included into the consideration. The Auslander-Reiten quivers of the different types of Bass orders are listed, and the relationship between infinite descending chains of Bass orders and subskewfields of index two is mentioned. In this context, Hijikata's description of almost Bass orders, i.e. Gorenstein orders such that the left multiplier of the radical is again Gorenstein, is given.NEWLINENEWLINENEWLINEThe next main topic is the author's theory of \(\tau\)-categories. Roughly speaking, a \(\tau\)-category is a Krull-Schmidt category with (a generalized version of) almost split sequences. Generalizing the work of Igusa and Todorov, the \(\tau\)-categories \(\Lambda\text{-lat}\) of \(\Lambda\)-lattices over a lattice-finite order \(\Lambda\) are characterized. The proof of this characterization was obtained recently in a triad of papers by the author. A brief outline is given here. The main idea consists in an induction which passes to subcategories of \(\Lambda\text{-lat}\) corresponding to an overorder \(\Gamma\) of \(\Lambda\). For the purpose of that proof the relationship between \(\Lambda\text{-lat}\) and \(\Gamma\text{-lat}\) has to be described in terms of categories. It turns out that the passage from \(\Lambda\text{-lat}\) to \(\Gamma\text{-lat}\) rests on a generalization of the Drozd-Kirichenko rejection lemma. Remarkably, the minimal non-empty difference sets \((\Lambda\text{-lat})\setminus(\Gamma\text{-lat})\) can be detected within \(\Lambda\text{-lat}\), solely by means of their Auslander-Reiten quiver which always starts at a projective and ends at an injective \(\Lambda\)-lattice. A reinspection and generalization of Igusa and Todorov's result on Artinian algebras in the light of \(\tau\)-categories illustrates the power of the author's method.NEWLINENEWLINENEWLINEAlmost Bass orders reappear in the next topic which deals with tame orders in non-semisimple algebras. Here the dichotomy for tame and wild [\textit{Yu. A. Drozd, G.-M. Greuel}, Math. Ann. 294, No. 3, 387-394 (1992; Zbl 0760.16005)] is open. Nevertheless, the author makes a convincing proposal for both tame and wild and provides a criterion for tameness in this setting. As a special case, he characterizes orders for which the rational rank of indecomposables is bounded.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00049].