Tilting up iterated tilted algebras
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- by Ibrahim Assem, Dieter Happel and Sonia Trepode PDF
- Proc. Amer. Math. Soc. 128 (2000), 2223-2232 Request permission
Abstract:
We show that, if $A$ is a representation-finite iterated tilted algebra of euclidean type $Q$, then there exist a sequence of algebras $A=A_{0},A_{1},A_{2},\dots , A_{m}$, and a sequence of modules $T^{(i)}_{A_{i}}$, where $0\leq i<m$, such that each $T^{(i)}_{A_{i}}$ is an APR-tilting $A_{i}$-module, or an APR-cotilting $A_{i}$-module, $\operatorname {End} T^{(i)}_{A_{i}}=A_{i+1}$ and $A_{m}$ is tilted representation-finite.References
- Ibrahim Assem, Tilting theory—an introduction, Topics in algebra, Part 1 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 127–180. MR 1171230
- 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
- Ibrahim Assem and Andrzej Skowroński, Iterated tilted algebras of type $\tilde \textbf {A}_n$, Math. Z. 195 (1987), no. 2, 269–290. MR 892057, DOI 10.1007/BF01166463
- Assem, I. and Zhang, Y., Endomorphism algebras of exceptional sequences over path algebras of type $\tilde {\mathbb {A}}_{n}$, Colloq. Math. 77 (1998), 271–292.
- Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR 1314422, DOI 10.1017/CBO9780511623608
- 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
- Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. MR 935124, DOI 10.1017/CBO9780511629228
- Richard, J., A Morita theory for derived categories, vol. 2, 39, J. London Math. Soc., 1989, p. 436–456.
- Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589, DOI 10.1007/BFb0072870
- Roldán, O., Tilted algebras of types $\tilde {\mathbb {A}}_{n}$, $\tilde {\mathbb {B}}_{n}$, $\tilde {\mathbb {C}}_{n}$ and $\widetilde {{\mathbb {B}}{\mathbb {C}}}_{n}$, Ph. D. Thesis, Carleton University (1983).
- Andrzej Skowroński, Selfinjective algebras of polynomial growth, Math. Ann. 285 (1989), no. 2, 177–199. MR 1016089, DOI 10.1007/BF01443513
- Trepode, S.E., A conjectura de Roldán para álgebras inclinadas iteradas de tipo euclideano, Ph. D. Thesis, Universidade de São Paulo, 1995.
- Sonia Elisabet Trepode, Roldán’s conjecture in the case $\widetilde A_n$, Proceedings of the Third “Dr. Antonio A. R. Monteiro” Congress on Mathematics (Spanish) (Bahía Blanca, 1995) Univ. Nac. del Sur, Bahía Blanca, 1996, pp. 51–68 (Spanish). MR 1403804
Additional Information
- Ibrahim Assem
- Affiliation: Département de mathématiques et d’informatique, Faculté des sciences, Université de Sherbrooke, Québec, Canada J1K 2R1
- MR Author ID: 27850
- ORCID: 0000-0001-6217-9876
- Email: ibrahim.assem@dmi.usherb.ca
- Dieter Happel
- Affiliation: Fakultät für Mathematik, TU Chemmitz, PSF 964, D-09107 Chemnitz, Federal Republic of Germany
- Email: happel@mathematik.tu-chemnitz.de
- Sonia Trepode
- Affiliation: Departamento de Matemáticas, Facultad de Ciencias Exactas y Naturales, Universidad nacional de Mar del Plata, Funes 3350, 7600 Mar del Plata, Argentina
- Address at time of publication: Instituto de Matemáticas, UNAM, Circuito exterior, Cd. Universitaria, México, 04510 D.F., Mexico
- Email: strepode@ mdp.edu.ar, sonia@math.unam.mx
- Received by editor(s): December 15, 1997
- Received by editor(s) in revised form: September 10, 1998
- Published electronically: November 29, 1999
- Communicated by: Ken Goodearl
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 2223-2232
- MSC (2000): Primary 16G60, 16G20
- DOI: https://doi.org/10.1090/S0002-9939-99-05230-2
- MathSciNet review: 1653413