Rigid modules over preprojective algebras
Tóm tắt
Let Λ be a preprojective algebra of simply laced Dynkin type Δ. We study maximal rigid Λ-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring ℂ[N] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of type Δ. As an application we obtain that all cluster monomials of ℂ[N] belong to the dual semicanonical basis.
Tài liệu tham khảo
Auslander, M.: Selected works of Maurice Auslander, part 1, edited and with a foreword by I. Reiten, S. Smalø, O. Solberg, xxii+895pp. Providence, RI: American Mathematical Society 1999
Auslander, M., Platzeck, M.I., Reiten, I.: Coxeter functors without diagrams. Trans. Am. Math. Soc. 250, 1–46 (1979)
Auslander, M., Reiten, I., Smalø, S.: Representation theory of Artin algebras. Corrected reprint of the 1995 original. Cambridge Studies in Advanced Mathematics, vol. 36, xiv+425pp. Cambridge: Cambridge University Press 1997
Berenstein, A., Fomin, S., Zelevinsky, A.: Cluster algebras III: Upper bounds and double Bruhat cells. Duke Math. J. 126, 1–52 (2005)
Bongartz, K.: Algebras and quadratic forms. J. Lond. Math. Soc. 28, 461–469 (1983)
Buan, A., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. 36 pp., accepted in Adv. Math., arXiv:math.RT/0402054
Buan, A., Marsh, R., Reiten, I.: Cluster-tilted algebras. Trans. Am. Math. Soc., to appear, 12 pp., arXiv:math.RT/0402075
Buan, A., Marsh, R., Reiten, I.: Cluster mutation via quiver representations, 23 pp., arXiv:math.RT/0412077
Butler, M.C.R., King, A.D.: Minimal resolutions of algebras. J. Algebra 212, 323–362 (1999)
Caldero, P., Chapoton, F.: Cluster algebras as Hall algebras of quiver representations, 17 pp., accepted in Comment. Math. Helv., arXiv:math.RT/0410187
Caldero, P., Keller, B.: From triangulated categories to cluster algebras, 31 pp., arXiv:math.RT/0506018
Crawley-Boevey, W.: On the exceptional fibres of Kleinian singularities. Am. J. Math. 122, 1027–1037 (2000)
Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15, 497–529 (2002)
Gabriel, P.: Unzerlegbare Darstellungen. I (German). Manuscr. Math. 6, 71–103 (1972); correction, ibid 6, 309 (1972)
Geiß, C., Leclerc, B., Schröer, J.: Semicanonical bases and preprojective algebras. Ann. Sci. Éc. Norm. Supér., IV. Sér. 38, 193–253 (2005)
Geiß, C., Leclerc, B., Schröer, J.: Auslander algebras and initial seeds for cluster algebras, preprint 2005, 1–23, arXiv:math.RT/0506405
Geiß, C., Leclerc, B., Schröer, J.: Semicanonical bases and preprojective algebras II: A multiplication formula, preprint 2005, 1–16, arXiv:math.RT/0509483
Geiß, C., Schröer, J.: Extension-orthogonal components of preprojective varieties. Trans. Am. Math. Soc. 357, 1953–1962 (2005)
Gelfand, I.M., Ponomarev, V.A.: Model algebras and representations of graphs (Russian). Funkts. Anal. Prilozh. 13, 1–12 (1979); English translation: Funct. Anal. Appl. 13 (1979), 157–166 (1979/1980)
Happel, D., Ringel, C.M.: Tilted algebras. Trans. Am. Math. Soc. 274, 399–443 (1982)
Happel, D.: On the derived category of a finite-dimensional algebra. Comment. Math. Helv. 62, 339–389 (1987)
Happel, D., Unger, L.: Almost complete tilting modules. Proc. Am. Math. Soc. 107, 603–610 (1989)
Igusa, K.: Notes on the no loops conjecture. J. Pure Appl. Algebra 69, 161–176 (1990)
Iyama, O.: Finiteness of representation dimension. Proc. Am. Math. Soc. 131, 1011–1014 (2003) (electronic)
Iyama, O.: Higher dimensional Auslander–Reiten theory on maximal orthogonal subcategories, 24 pp., arXiv:math.RT/0407052
Iyama, O.: Higher dimensional Auslander correspondence, 30 pp., arXiv:math.RT/0411631
Iyama, O.: Private communication. ICRA XI (Mexico), August 2004
Keller, B.: On triangulated orbit categories. Doc. Math. 10, 551–581 (2005) (electronic)
Lenzing, H.: Nilpotente Elemente in Ringen von endlicher globaler Dimension (German). Math. Z. 108, 313–324 (1969)
Lusztig, G.: Quivers, perverse sheaves, and quantized enveloping algebras. J. Am. Math. Soc. 4, 365–421 (1991)
Lusztig, G.: Constructible functions on the Steinberg variety. Adv. Math. 130, 287–310 (1997)
Lusztig, G.: Semicanonical bases arising from enveloping algebras. Adv. Math. 151, 129–139 (2000)
Riedtmann, C.: Degenerations for representations of quivers with relations. Ann. Sci. Éc. Norm. Supér., IV. Sér,. 19, 275–301 (1986)
Riedtmann, C., Schofield, A.: On open orbits and their complements. J. Algebra 130, 388–411 (1990)
Riedtmann, C., Schofield, A.: On a simplicial complex associated with tilting modules. Comment. Math. Helv. 66, 70–78 (1991)
Ringel, C.M.: Tame algebras and integral quadratic forms. Lect. Notes Math.,vol. 1099, xiii+376 pp. Berlin: Springer 1984
Ringel, C.M.: Hall algebras and quantum groups. Invent. Math. 101, 583–591 (1990)
Ringel, C.M.: The preprojective algebra of a quiver. Algebras and modules, II (Geiranger 1996), pp. 467–480, CMS Conf. Proc., vol. 24. Providence, RI: Am. Math. Soc. 1998
Unger, L.: Schur modules over wild, finite-dimensional path algebras with three simple modules. J. Pure Appl. Algebra 64, 205–222 (1990)