|
Extension-orthogonal components of preprojective varieties
Author(s):
Christof
Geiß;
Jan
Schröer
Journal:
Trans. Amer. Math. Soc.
357
(2005),
1953-1962.
MSC (2000):
Primary 14M99, 16D70, 16G20, 17B37
Posted:
August 11, 2004
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
Let be a Dynkin quiver, and let be the corresponding preprojective algebra. Let be a set of pairwise different indecomposable irreducible components of varieties of -modules such that generically there are no extensions between and for all . We show that the number of elements in is at most the number of positive roots of . Furthermore, we give a module-theoretic interpretation of Leclerc's counterexample to a conjecture of Berenstein and Zelevinsky.
References:
-
- 1.
- M. Auslander, I. Reiten, S. Smalø, Representation theory of Artin algebras. Corrected reprint of the 1995 original. Cambridge Studies in Advanced Mathematics, 36. Cambridge University Press, Cambridge (1997), xiv+425pp. MR 98e:16011
- 2.
- A. Berenstein, A. Zelevinsky, String bases for quantum groups of type
. I.M. Gelfand Seminar, 51-89, Adv. Soviet Math. 16, Part 1, Amer. Math. Soc., Providence, RI (1993). MR 94g:17019 - 3.
- K. Bongartz, A geometric version of the Morita equivalence. J. Algebra 139 (1991), no. 1, 159-171. MR 92f:16008
- 4.
- W. Crawley-Boevey, On tame algebras and bocses. Proc. London Math. Soc. (3) 56 (1988), no. 3, 451-483. MR 89c:16028
- 5.
- W. Crawley-Boevey, J. Schröer, Irreducible components of varieties of modules. J. Reine Angew. Math. 553 (2002), 201-220. MR 2004a:16020
- 6.
- V. Dlab, C.M. Ringel, The module theoretical approach to quasi-hereditary algebras. In: Representations of algebras and related topics (Kyoto, 1990), 200-224, Cambridge Univ. Press, Cambridge (1992). MR 94f:16026
- 7.
- P. Dowbor, A. Skowronski, Galois coverings of representation-infinite algebras. Comment. Math. Helv. 62 (1987), 311-337. MR 88m:16020
- 8.
- P. Gabriel, Auslander-Reiten sequences and representation-finite algebras. In: Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), pp. 1-71, Lecture Notes in Mathematics 831, Springer-Verlag, Berlin (1980). MR 82i:16030
- 9.
- P. Gabriel, Algèbres auto-injectives de reprèsentation finie (d'après Christine Riedtmann). (French) [Self-injective algebras of finite representation (according to Christine Riedtmann)] Bourbaki Seminar, Vol. 1979/80, pp. 20-39, Lecture Notes in Mathematics 842, Springer-Verlag, Berlin-New York (1981). MR 84j:16018
- 10.
- P. Gabriel, The universal cover of a representation-finite algebra. In: Representations of algebras (Puebla, 1980), 68-105, Lecture Notes in Mathematics 903, Springer-Verlag, Berlin (1981). MR 83f:16036
- 11.
- C. Geiß, J. Schröer, Varieties of modules over tubular algebras. Colloq. Math. 95 (2003), no. 2, 163-183. MR 2004d:16026
- 12.
- M. Kashiwara, Y. Saito, Geometric construction of crystal bases. Duke Math. J. 89 (1997), 9-36. MR 99e:17025
- 13.
- B. Leclerc, Imaginary vectors in the dual canonical basis of
. Transform. Groups 8 (2003), no. 1, 95-104. MR 2004d:17020 - 14.
- B. Leclerc, M. Nazarov, J.-Y. Thibon, Induced representations of affine Hecke algebras and canonical bases of quantum groups. In: Studies in Memory of Issai Schur, Progress in Mathematics 210, 115-153, Birkhauser (2003). MR 2004d:17007
- 15.
- G. Lusztig, Canonical bases arising from quantized enveloping algebras. II. Common trends in mathematics and quantum field theories (Kyoto, 1990). Progr. Theoret. Phys. Suppl. No. 102 (1990), 175-201 (1991). MR 93g:17019
- 16.
- G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras. J. Amer. Math. Soc. 4 (1991), no. 2, 365-421. MR 91m:17018
- 17.
- R. Marsh, M. Reineke, Private communication. Bielefeld, February 2002.
- 18.
- M. Reineke, Multiplicative properties of dual canonical bases of quantum groups. J. Algebra 211 (1999), 134-149. MR 99k:17034
- 19.
- C.M. Ringel, Tame algebras and integral quadratic forms. Lecture Notes in Mathematics 1099, Springer-Verlag, Berlin (1984), xiii+376pp. MR 87f:16027
- 20.
- C.M. Ringel, The preprojective algebra of a quiver. Algebras and modules II (Geiranger, 1996), 467-480, CMS Conf. Proc. 24, Amer. Math. Soc., Providence, RI (1998). MR 99i:16031
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
14M99, 16D70, 16G20, 17B37
Retrieve articles in all Journals with MSC
(2000):
14M99, 16D70, 16G20, 17B37
Additional Information:
Christof
Geiß
Affiliation:
Instituto de Matemáticas, UNAM, Ciudad Universitaria, 04510 Mexico D.F., Mexico
Email:
christof@math.unam.mx
Jan
Schröer
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
Email:
jschroer@maths.leeds.ac.uk
DOI:
10.1090/S0002-9947-04-03555-X
PII:
S 0002-9947(04)03555-X
Received by editor(s):
September 5, 2002
Received by editor(s) in revised form:
October 7, 2003
Posted:
August 11, 2004
Additional Notes:
The second author thanks the Nuffield Foundation (Grant Number NAL/00270/G) for financial support, and the IM UNAM, Mexico City, where most of this work was done
Copyright of article:
Copyright
2004,
American Mathematical Society
|