A characterization of representation-finite algebras

• Autorzy: Andrzej Skowroński , M. Wenderlich
• Języki publikacji: EN

Abstrakty

EN

Let A be a finite-dimensional, basic, connected algebra over an algebraically closed field. Denote by Γ(A) the Auslander-Reiten quiver of A. We show that A is representation-finite if and only if Γ(A) has at most finitely many vertices lying on oriented cycles and finitely many orbits with respect to the action of the Auslander-Reiten translation.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1991-1992
• Tom: 140
• Numer: 1
• Strony: 31-34

Daty

wydano

1991

otrzymano

1990-04-20

Twórcy

autor

Andrzej Skowroński

• Institute of Mathematics, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

autor

M. Wenderlich

Bibliografia

• [1] W. W. Crawley-Boevey, On tame algebras and bocses, Proc. London Math. Soc. 56 (1988), 451-483.
• [2] U. Fischbacher, Une nouvelle preuve d'un théorème de Nazarova et Roiter, C. R. Acad. Sci. Paris Sér. I 300 (9) (1985), 259-262.
• [3] M. Harada and Y. Sai, On categories of indecomposable modules I, Osaka J. Math. 7 (1970), 323-344.
• [4] O. Kerner and A. Skowroński, On module categories with nilpotent infinite radical, Compositio Math. 77 (1991), 313-333.
• [5] C. M. Ringel, Finite-dimensional hereditary algebras of wild representation type, Math. Z. 161 (1978), 235-255.
• [6] C. M. Ringel, Report on the Brauer-Thrall conjectures, in: Proc. ICRA II (Ottawa 1979), Lecture Notes in Math. 831, Springer, 1980, 104-136.
• [7] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, 1984.
• [8] A. Skowroński, Algebras of polynomial growth, in: Topics in Algebra, Banach Center Publ. 26, Part 1, PWN, Warszawa 1990, 535-568.
• [9] A. Skowroński and S. O. Smalø, Directing modules, J. Algebra, to appear.