Representations as elements in affine composition algebras

Zhang, Pu

doi:10.1090/S0002-9947-00-02613-1

Representations as elements in affine composition algebras
HTML articles powered by AMS MathViewer

by Pu Zhang PDF

Trans. Amer. Math. Soc. 353 (2001), 1221-1249 Request permission

Abstract:

Let $A$ be the path algebra of a Euclidean quiver over a finite field $k$. The aim of this paper is to classify the modules $M$ with the property $[M]\in \mathcal {C}(A)$, where $\mathcal {C}(A)$ is Ringel’s composition algebra. Namely, the main result says that if $|k| \ne 2, 3$, then $[M]\in \mathcal {C}(A)$ if and only if the regular direct summand of $M$ is a direct sum of modules from non-homogeneous tubes with quasi-dimension vectors non-sincere. The main methods are representation theory of affine quivers, the structure of triangular decompositions of tame composition algebras, and the invariant subspaces of skew derivations. As an application, we see that $\mathcal {C}(A) = \mathcal {H}(A)$ if and only if the quiver of $A$ is of Dynkin type.

References

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
W. W. Crawley-Boevey, Lectures on representations of quivers; More lectures on representations of quivers, preprints, Oxford (1992).
Hidegorô Nakano, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357–375 (German). MR 94
X. Chen and J. Xiao: Exceptional sequences in Hall algebras and quantum groups, Compositio Math. 117(2)(1999), 165-191.
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
Peter Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71–103; correction, ibid. 6 (1972), 309 (German, with English summary). MR 332887, DOI 10.1007/BF01298413
James A. Green, Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120 (1995), no. 2, 361–377. MR 1329046, DOI 10.1007/BF01241133
James A. Green, Quantum groups, Hall algebras and quantized shuffles, Finite reductive groups (Luminy, 1994) Progr. Math., vol. 141, Birkhäuser Boston, Boston, MA, 1997, pp. 273–290. MR 1429876, DOI 10.1007/s10107-012-0519-x
Werner Geigle and Helmut Lenzing, Perpendicular categories with applications to representations and sheaves, J. Algebra 144 (1991), no. 2, 273–343. MR 1140607, DOI 10.1016/0021-8693(91)90107-J
Jin Yun Guo and Liangang Peng, Universal PBW-basis of Hall-Ringel algebras and Hall polynomials, J. Algebra 198 (1997), no. 2, 339–351. MR 1489901, DOI 10.1006/jabr.1997.7065
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
Dieter Happel, Silke Hartlieb, Otto Kerner, and Luise Unger, On perpendicular categories of stones over quiver algebras, Comment. Math. Helv. 71 (1996), no. 3, 463–474. MR 1418949, DOI 10.1007/BF02566431
Nathan Jacobson, Basic algebra. I, 2nd ed., W. H. Freeman and Company, New York, 1985. MR 780184
Victor G. Kac, Infinite-dimensional Lie algebras, Progress in Mathematics, vol. 44, Birkhäuser Boston, Inc., Boston, MA, 1983. An introduction. MR 739850, DOI 10.1007/978-1-4757-1382-4
V. G. Kac, Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), no. 1, 57–92. MR 557581, DOI 10.1007/BF01403155
V. G. Kac, Infinite root systems, representations of graphs and invariant theory. II, J. Algebra 78 (1982), no. 1, 141–162. MR 677715, DOI 10.1016/0021-8693(82)90105-3
M. M. Kapranov, Eisenstein series and quantum affine algebras, J. Math. Sci. (New York) 84 (1997), no. 5, 1311–1360. Algebraic geometry, 7. MR 1465518, DOI 10.1007/BF02399194
Otto Kerner, Representations of wild quivers, Representation theory of algebras and related topics (Mexico City, 1994) CMS Conf. Proc., vol. 19, Amer. Math. Soc., Providence, RI, 1996, pp. 65–107. MR 1388560
G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), no. 2, 365–421. MR 1088333, DOI 10.1090/S0894-0347-1991-1088333-2
George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
Robert V. Moody, Euclidean Lie algebras, Canadian J. Math. 21 (1969), 1432–1454. MR 255627, DOI 10.4153/CJM-1969-158-2
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
Claus Michael Ringel, Hall algebras, Topics in algebra, Part 1 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 433–447. MR 1171248
Claus Michael Ringel, Hall polynomials for the representation-finite hereditary algebras, Adv. Math. 84 (1990), no. 2, 137–178. MR 1080975, DOI 10.1016/0001-8708(90)90043-M
Claus Michael Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583–591. MR 1062796, DOI 10.1007/BF01231516
Claus Michael Ringel, From representations of quivers via Hall and Loewy algebras to quantum groups, Proceedings of the International Conference on Algebra, Part 2 (Novosibirsk, 1989) Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 381–401. MR 1175845
Claus Michael Ringel, Lie algebras arising in representation theory, Representations of algebras and related topics (Kyoto, 1990) London Math. Soc. Lecture Note Ser., vol. 168, Cambridge Univ. Press, Cambridge, 1992, pp. 284–291. MR 1211484
Claus Michael Ringel, The composition algebra of a cyclic quiver. Towards an explicit description of the quantum group of type $\~A_n$, Proc. London Math. Soc. (3) 66 (1993), no. 3, 507–537. MR 1207546, DOI 10.1112/plms/s3-66.3.507
Claus Michael Ringel, The Hall algebra approach to quantum groups, XI Latin American School of Mathematics (Spanish) (Mexico City, 1993) Aportaciones Mat. Comun., vol. 15, Soc. Mat. Mexicana, México, 1995, pp. 85–114. MR 1360930
Claus Michael Ringel, Hall algebras revisited, Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992) Israel Math. Conf. Proc., vol. 7, Bar-Ilan Univ., Ramat Gan, 1993, pp. 171–176. MR 1261907
Claus Michael Ringel, PBW-bases of quantum groups, J. Reine Angew. Math. 470 (1996), 51–88. MR 1370206, DOI 10.1515/crll.1996.470.51
Claus Michael Ringel, Green’s theorem on Hall algebras, Representation theory of algebras and related topics (Mexico City, 1994) CMS Conf. Proc., vol. 19, Amer. Math. Soc., Providence, RI, 1996, pp. 185–245. MR 1388564
Claus Michael Ringel, The braid group action on the set of exceptional sequences of a hereditary Artin algebra, Abelian group theory and related topics (Oberwolfach, 1993) Contemp. Math., vol. 171, Amer. Math. Soc., Providence, RI, 1994, pp. 339–352. MR 1293154, DOI 10.1090/conm/171/01786
Christine Riedtmann, Lie algebras generated by indecomposables, J. Algebra 170 (1994), no. 2, 526–546. MR 1302854, DOI 10.1006/jabr.1994.1351
Aidan Schofield, Semi-invariants of quivers, J. London Math. Soc. (2) 43 (1991), no. 3, 385–395. MR 1113382, DOI 10.1112/jlms/s2-43.3.385
Pu Zhang, Triangular decomposition of the composition algebra of the Kronecker algebra, J. Algebra 184 (1996), no. 1, 159–174. MR 1402573, DOI 10.1006/jabr.1996.0252
P. Zhang: Indecomposables in the composition algebra of the Kronecker algebra, Comm. in Algebra 27 (1999), 4733–4639.
P. Zhang: Composition algebras of affine types, J. Algebra 206(1998), 505-540.
P. Zhang and S. H. Zhang: Indecomposables as elements in affine composition algebras, J. Algebra 210(1998), 614-629.