Critical simply connected algebras.(English)Zbl 0537.16024

For the definitions, the reader is referred to [K. Bongartz and P. Gabriel, Invent. Math. 65, 331-378 (1982; Zbl 0482.16026)] and [O. Bretscher and P. Gabriel, Bull. Math. Soc. Fr. 111, 21-40 (1983; Zbl 0527.16021)]. Let g be an admissible grading of a tree T, $$R_ g$$ the corresponding simply connected translation quiver, and $$A_ g$$ the full subcategory of the mesh category $$k(R_ g)$$ (over an algebraically closed field k) supported by the projective points. An algebra A is called critical if it is of infinite representation type, but every proper convex full subalgebra is of finite representation type. The grading g is called critical if $$A_ g$$ is critical. The author obtains the following results.
Theorem 1. The tree T bears a critical grading if and only if T is an extended Dynkin diagram, i.e. T is of the form $$\tilde D_ n$$ for $$n\geq 4$$ or $$\tilde E_ m$$ for 6$$\leq m\leq 8.$$
Theorem 2. The algebras $$A_ g$$ obtained by critical gradings coincide with the minimal algebras of infinite representation type with preprojective component of frames $$\tilde D_ n$$ and $$\tilde E_ m$$ as described by D. Happel and D. Vossieck [Manuscr. Math. 42, 221-243 (1983; Zbl 0516.16023)].
Reviewer: M.M.Kleiner

MSC:

 16Gxx Representation theory of associative rings and algebras 16P10 Finite rings and finite-dimensional associative algebras
Full Text:

References:

 [1] BAUTISTA,R.,BRENNER,S.:On the number of terms in the middle of an almost split sequence,Proc. of ICRA III,Puebla 1980,LNM 903,pp.1-9 [2] BONGARTZ,K.: Treue einfach zusammenh?ngende Algebren I, Comment.Math.Helv. 57 (1982),282-330 · Zbl 0502.16022 [3] BONGARTZ,K.: Algebras and quadratic forms,to appear in JLMS · Zbl 0532.16020 [4] BONGARTZ,K.:Ein Kriterium f?r endlichen Darstellungstyp,preprint January 1983,22 pages [5] BONGARTZ,K.,GABRIEL,P.:Covering spaces in representation theory, Invent.math. 65 (1982),331-378 · Zbl 0482.16026 [6] BRETSCHER,O.,GABRIEL,P.:The standard form of a representation-finite algebra, Bull.Soc.math.France, 111 (1983),pp.21-40 · Zbl 0527.16021 [7] DLAB,V.,RINGEL,C.M.: Indecomposable representations of graphs and algebras,Memoirs Amer.Math.Soc. 173 (1976) · Zbl 0332.16015 [8] HAPPEL,D.,VOSSIECK,D.:Minimal algebras of infinite representation type with preprojective component, Man.math.,Vol.42 (1983),221-243 · Zbl 0516.16023 [9] KLEINER,M.M.: Partially ordered sets of finite type, Zap.Naucn.Sem. LOMI 28 (1972), 32-41 · Zbl 0345.06001 [10] NAZAROVA,L.A.,ROITER,A.V.:Representations of partially ordered sets,Zap.Naucn.Sem. LOMI 28 (1972) · Zbl 0336.16031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.