Gradings of 𝐵_{𝑛} and 𝐶_{𝑛} of finite representation type

Assem, Ibrahim; Roldán, Oscar

doi:10.1090/S0002-9947-1983-0709570-1

Gradings of $\textbf {B}_{n}$ and $\textbf {C}_{n}$ of finite representation type
HTML articles powered by AMS MathViewer

by Ibrahim Assem and Oscar Roldán PDF

Trans. Amer. Math. Soc. 279 (1983), 589-609 Request permission

Abstract:

It was shown by Bongartz and Gabriel that the classification of simplyconnected algebras (i.e. finite-dimensional, basic, of finite representation type and with a simply-connected Auslander-Reiten graph) can be reduced to the study of certain numerical functions, called gradings, operating on a tree. Here, we classify in terms of their bounden species the simply-connected algebras arising from gradings of the Dynkin trees ${{\mathbf {B}}_n}$ and ${{\mathbf {C}}_n}$, and show that these are exactly the tilted algebras of types ${{\mathbf {B}}_n}$ and ${{\mathbf {C}}_n}$, respectively.

References

Ibrahim Assem, Tilted algebras of type $\textbf {A}_{n}$, Comm. Algebra 10 (1982), no. 19, 2121–2139. MR 675347, DOI 10.1080/00927878208822826
Ibrahim Assem, Iterated tilted algebras of types $B_{n}$ and $C_{n}$, J. Algebra 84 (1983), no. 2, 361–390. MR 723397, DOI 10.1016/0021-8693(83)90083-2
Ibrahim Assem and Dieter Happel, Generalized tilted algebras of type $A_{n}$, Comm. Algebra 9 (1981), no. 20, 2101–2125. MR 640613, DOI 10.1080/00927878108822697
Maurice Auslander and Idun Reiten, Representation theory of Artin algebras. IV. Invariants given by almost split sequences, Comm. Algebra 5 (1977), no. 5, 443–518. MR 439881, DOI 10.1080/00927877708822180
K. Bongartz and P. Gabriel, Covering spaces in representation-theory, Invent. Math. 65 (1981/82), no. 3, 331–378. MR 643558, DOI 10.1007/BF01396624
Vlastimil Dlab, Representations of valued graphs, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 73, Presses de l’Université de Montréal, Montreal, Que., 1980. MR 586769
Vlastimil Dlab and Claus Michael Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 6 (1976), no. 173, v+57. MR 447344, DOI 10.1090/memo/0173
Vlastimil Dlab and Claus Michael Ringel, The preprojective algebra of a modulated graph, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) Lecture Notes in Math., vol. 832, Springer, Berlin-New York, 1980, pp. 216–231. MR 607155
Peter Gabriel, Indecomposable representations. II, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971) Academic Press, London, 1973, pp. 81–104. MR 0340377
Dieter Happel and Claus Michael Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 399–443. MR 675063, DOI 10.1090/S0002-9947-1982-0675063-2
C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), no. 2, 199–224 (German). MR 576602, DOI 10.1007/BF02566682
Claus Michael Ringel, On algorithms for solving vector space problems. I. Report on the Brauer-Thrall conjectures: Rojter’s theorem and the theorem of Nazarova and Rojter, Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 831, Springer, Berlin, 1980, pp. 104–136. MR 607142

Tame algebras

Trivial extensions of tilted algebras