Semicanonical bases and preprojective algebras

Geiss, Christof; Leclerc, Bernard; Schröer, Jan

doi:10.1016/j.ansens.2004.12.001

Geiss, Christof ; Leclerc, Bernard ; Schröer, Jan

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 38 (2005) no. 2, pp. 193-253.

MR Zbl | 3 citations dans Numdam

DOI : 10.1016/j.ansens.2004.12.001

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     title = {Semicanonical bases and preprojective algebras},
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Geiss, Christof; Leclerc, Bernard; Schröer, Jan. Semicanonical bases and preprojective algebras. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 38 (2005) no. 2, pp. 193-253. doi : 10.1016/j.ansens.2004.12.001. http://archive.numdam.org/articles/10.1016/j.ansens.2004.12.001/

Bibliographie
Cité par

[1] Auslander M., Reiten I., Smalø S., Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1997, Corrected reprint of the 1995 original. xiv+425pp. | MR | Zbl

[2] Bautista R., On algebras of strongly unbounded representation type, Comment. Math. Helv. 60 (3) (1985) 392-399. | MR | Zbl

[3] Berenstein A., Fomin S., Zelevinsky A., Cluster algebras III: Upper bounds and double Bruhat cells, Duke Math. J. 126 (1) (2005) 1-52. | MR | Zbl

[4] Berenstein A., Zelevinsky A., String bases for quantum groups of type $A_{r}$ , in: I.M. Gelfand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 51-89. | MR | Zbl

[5] Bobiński G., Skowroński A., Geometry of modules over tame quasi-tilted algebras, Colloq. Math. 79 (1) (1999) 85-118. | MR | Zbl

[6] Bongartz K., Algebras and quadratic forms, J. London Math. Soc. 28 (3) (1983) 461-469. | MR | Zbl

[7] Bongartz K., A geometric version of the Morita equivalence, J. Algebra 139 (1) (1991) 159-171. | MR | Zbl

[8] Caldero P., Adapted algebras for the Berenstein Zelevinsky conjecture, Transform. Groups 8 (1) (2003) 37-50. | MR | Zbl

[9] Caldero P., A multiplicative property of quantum flag minors, Representation Theory 7 (2003) 164-176. | MR | Zbl

[10] Caldero P., Marsh R., A multiplicative property of quantum flag minors II, J. London Math. Soc. 69 (3) (2004) 608-622. | MR | Zbl

[11] Chapoton F., Fomin S., Zelevinsky A., Polytopal realizations of generalized associahedra, Canad. Math. Bull. 45 (2002) 537-566. | MR | Zbl

[12] Crawley-Boevey W., On the exceptional fibres of Kleinian singularities, Amer. J. Math. 122 (2000) 1027-1037. | MR | Zbl

[13] Crawley-Boevey W., Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001) 257-293. | MR | Zbl

[14] Crawley-Boevey W., Schröer J., Irreducible components of varieties of modules, J. Reine Angew. Math. 553 (2002) 201-220. | MR | Zbl

[15] Dlab V., Ringel C.M., The module theoretical approach to quasi-hereditary algebras, in: Representations of Algebras and Related Topics, Kyoto, 1990, Cambridge Univ. Press, Cambridge, 1992, pp. 200-224. | MR | Zbl

[16] Dowbor P., Skowroński A., On Galois coverings of tame algebras, Arch. Math. 44 (1985) 522-529. | MR | Zbl

[17] Dowbor P., Skowroński A., Galois coverings of representation-infinite algebras, Comment. Math. Helv. 62 (1987) 311-337. | MR | Zbl

[18] Fomin S., Zelevinsky A., Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002) 497-529. | MR | Zbl

[19] Fomin S., Zelevinsky A., Y-systems and generalized associahedra, Annals of Math. 158 (3) (2003) 977-1018. | MR | Zbl

[20] Fomin S., Zelevinsky A., Cluster algebras II: Finite type classification, Invent. Math. 154 (2003) 63-121. | MR | Zbl

[21] Gabriel P., Auslander-Reiten sequences and representation-finite algebras, in: Representation Theory I, Carleton, 1979, Lecture Notes in Math., vol. 831, Springer-Verlag, Berlin, 1980, pp. 1-71. | MR | Zbl

[22] Gabriel P., The universal cover of a representation-finite algebra, in: Representations of Algebras, Puebla, 1980, Lecture Notes in Math., vol. 903, Springer-Verlag, Berlin, 1981, pp. 68-105. | MR | Zbl

[23] Geigle W., Lenzing H., A class of weighted projective curves arising in representation theory of finite-dimensional algebras, in: Singularities, Representation of Algebras, and Vector Bundles, Lambrecht, 1985, Lecture Notes in Math., vol. 1273, Springer-Verlag, Berlin, 1987, pp. 265-297. | MR | Zbl

[24] Geiß C., Schröer J., Varieties of modules over tubular algebras, Colloq. Math. 95 (2003) 163-183. | MR | Zbl

[25] Geiss C., Schröer J., Extension-orthogonal components of nilpotent varieties, Trans. Amer. Math. Soc. 357 (2005) 1953-1962. | MR | Zbl

[26] Happel D., Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988, x+208pp. | MR | Zbl

[27] Happel D., Ringel C.M., The derived category of a tubular algebra, in: Representation Theory, I, Ottawa, Ont., 1984, Lecture Notes in Math., vol. 1177, Springer-Verlag, Berlin, 1986, pp. 156-180. | MR | Zbl

[28] Happel D., Vossieck D., Minimal algebras of infinite representation type with preprojective component, Manuscripta Math. 42 (1983) 221-243. | MR | Zbl

[29] Kashiwara M., On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991) 465-516. | MR | Zbl

[30] Kashiwara M., Saito Y., Geometric construction of crystal bases, Duke Math. J. 89 (1997) 9-36. | MR | Zbl

[31] Leclerc B., Imaginary vectors in the dual canonical basis of $U_{q} (n)$ , Transform. Groups 8 (2003) 95-104. | MR | Zbl

[32] Leclerc B., Dual canonical bases, quantum shuffles and q-characters, Math. Z. 246 (2004) 691-732. | MR | Zbl

[33] Leclerc B., Nazarov M., Thibon J.-Y., Induced representations of affine Hecke algebras and canonical bases of quantum groups, in: Studies in Memory of Issai Schur, Progress in Mathematics, vol. 210, Birkhäuser, Basel, 2003. | MR | Zbl

[34] Lenzing H., A K-theoretic study of canonical algebras, in: CMS Conf. Proc., vol. 18, 1996, pp. 433-454. | MR | Zbl

[35] Lenzing H., Coxeter transformations associated with finite dimensional algebras, in: Computational Methods for Representations of Groups and Algebras, Essen, 1997, Progress in Math., vol. 173, Birkhäuser, Basel, 1999. | MR | Zbl

[36] Lenzing H., Meltzer H., Sheaves on a weighted projective line of genus one, and representations of a tubular algebra, in: Representations of Algebras, Ottawa, ON, 1992, CMS Conf. Proc., vol. 14, 1993, pp. 313-337. | Zbl

[37] Lusztig G., Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990) 447-498. | MR | Zbl

[38] Lusztig G., Quivers, perverse sheaves and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991) 365-421. | MR | Zbl

[39] Lusztig G., Affine quivers and canonical bases, Publ. Math. IHES 76 (1994) 365-416. | Numdam | Zbl

[40] Lusztig G., Constructible functions on the Steinberg variety, Adv. Math. 130 (1997) 365-421. | MR | Zbl

[41] Lusztig G., Semicanonical bases arising from enveloping algebras, Adv. Math. 151 (2000) 129-139. | MR | Zbl

[42] Marsh R., Reineke M., Personal communication.

[43] Reineke M., Multiplicative properties of dual canonical bases of quantum groups, J. Algebra 211 (1999) 134-149. | MR | Zbl

[44] Reiten I., Dynkin diagrams and the representation theory of algebras, Notices AMS 44 (1997) 546-556. | MR | Zbl

[45] Ringel C.M., Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math., vol. 1099, Springer-Verlag, Berlin, 1984, xiii+376pp. | MR | Zbl

[46] Ringel C.M., The preprojective algebra of a quiver, in: Algebras and Modules II, Geiranger, 1966, CMS Conf. Proc., vol. 24, AMS, 1998, pp. 467-480. | MR | Zbl

[47] Ringel C.M., The multisegment duality and the preprojective algebras of type A, Algebra Montpellier Announcements 1.1 (1999) (6 pages). | MR | Zbl

[48] Ringel C.M., Representation theory of finite-dimensional algebras, in: Representations of Algebras, Proceedings of the Durham Symposium 1985, Lecture Note Series, vol. 116, LMS, 1986, pp. 9-79. | MR | Zbl

[49] Schröer J., Module theoretic interpretation of quantum minors, in preparation.

[50] Saito K., Extended affine root systems I, Publ. Res. Inst. Math. Sci. 21 (1985) 75-179. | MR | Zbl

[51] Zelevinsky A., Personal communication.

[52] Zelevinsky A., The multisegment duality, in: Documenta Mathematica, Extra Volume, ICM, Berlin, 1998, III, pp. 409-417. | MR | Zbl

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