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)].
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 |
Keywords:
admissible grading; simply connected translation quiver; mesh category; critical grading; extended Dynkin diagram; minimal algebras of infinite representation typeReferences:
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