Affine Mirković-Vilonen polytopes
Publications Mathématiques de l'IHÉS, Tome 120 (2014), pp. 113-205.

Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra 𝔤 has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given 𝔤, there is a limit crystal, usually denoted by B(−∞), which contains all the other crystals. When 𝔤 is finite dimensional, a convex polytope, called the Mirković-Vilonen polytope, can be associated to each element in B(−∞). This polytope sits in the dual space of a Cartan subalgebra of 𝔤, and its edges are parallel to the roots of 𝔤. In this paper, we generalize this construction to the case where 𝔤 is a symmetric affine Kac-Moody algebra. The datum of the polytope must however be complemented by partitions attached to the edges parallel to the imaginary root δ. We prove that these decorated polytopes are characterized by conditions on their normal fans and on their 2-faces. In addition, we discuss how our polytopes provide an analog of the notion of Lusztig datum for affine Kac-Moody algebras. Our main tool is an algebro-geometric model for B(−∞) constructed by Lusztig and by Kashiwara and Saito, based on representations of the completed preprojective algebra Λ of the same type as 𝔤. The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott’s tilting theory for the category Λ- mod . The partitions we need come from studying the category of semistable Λ-modules of dimension-vector a multiple of δ.

DOI : https://doi.org/10.1007/s10240-013-0057-y
MANUSCRIPT : 57
PUBLISHER-ID : s10240-013-0057-y
Mots clés : Irreducible Component, Simple Object, Torsion Pair, Convex Order, Jordan Type
@article{PMIHES_2014__120__113_0,
     author = {Baumann, Pierre and Kamnitzer, Joel and Tingley, Peter},
     title = {Affine {Mirkovi\'c-Vilonen} polytopes},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {113--205},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {120},
     year = {2014},
     doi = {10.1007/s10240-013-0057-y},
     zbl = {1332.17012},
     mrnumber = {3270589},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-013-0057-y/}
}
TY  - JOUR
AU  - Baumann, Pierre
AU  - Kamnitzer, Joel
AU  - Tingley, Peter
TI  - Affine Mirković-Vilonen polytopes
JO  - Publications Mathématiques de l'IHÉS
PY  - 2014
DA  - 2014///
SP  - 113
EP  - 205
VL  - 120
PB  - Springer Berlin Heidelberg
PP  - Berlin/Heidelberg
UR  - http://archive.numdam.org/articles/10.1007/s10240-013-0057-y/
UR  - https://zbmath.org/?q=an%3A1332.17012
UR  - https://www.ams.org/mathscinet-getitem?mr=3270589
UR  - https://doi.org/10.1007/s10240-013-0057-y
DO  - 10.1007/s10240-013-0057-y
LA  - en
ID  - PMIHES_2014__120__113_0
ER  - 
Baumann, Pierre; Kamnitzer, Joel; Tingley, Peter. Affine Mirković-Vilonen polytopes. Publications Mathématiques de l'IHÉS, Tome 120 (2014), pp. 113-205. doi : 10.1007/s10240-013-0057-y. http://archive.numdam.org/articles/10.1007/s10240-013-0057-y/

[1.] Amiot, C.; Iyama, O.; Reiten, I.; Todorov, G. Preprojective algebras and c-sortable words, Proc. Lond. Math. Soc. (3), Volume 104 (2012), pp. 513-539 | Article | MR 2900235 | Zbl 1246.13033

[2.] Anderson, J. E. A polytope calculus for semisimple groups, Duke Math. J., Volume 116 (2003), pp. 567-588 | Article | MR 1958098 | Zbl 1064.20047

[3.] Assem, I. et al. Tilting theory – an introduction, Topics in Algebra, Part 1 (1990), pp. 127-180 | MR 1171230 | Zbl 0726.16008

[4.] Baumann, P.; Kamnitzer, J. Preprojective algebras and MV polytopes, Represent. Theory, Volume 16 (2012), pp. 152-188 | Article | MR 2892443 | Zbl 1242.05273

[5.] P. Baumann, T. Dunlap, J. Kamnitzer and P. Tingley, Rank 2 affine MV polytopes, Represent. Theory, to appear. | MR 3084418 | Zbl 1302.17037

[6.] Beck, J. Convex bases of PBW type for quantum affine algebras, Commun. Math. Phys., Volume 165 (1994), pp. 193-199 | Article | MR 1298947 | Zbl 0828.17016

[7.] Beck, J.; Nakajima, H. Crystal bases and two-sided cells of quantum affine algebras, Duke Math. J., Volume 123 (2004), pp. 335-402 | MR 2066942 | Zbl 1062.17006

[8.] Beck, J.; Chari, V.; Pressley, A. An algebraic characterization of the affine canonical basis, Duke Math. J., Volume 99 (1999), pp. 455-487 | Article | MR 1712630 | Zbl 0964.17013

[9.] Berenstein, A.; Zelevinsky, A. Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math., Volume 143 (2001), pp. 77-128 | Article | MR 1802793 | Zbl 1061.17006

[10.] Bourbaki, N. Groupes et Algèbres de Lie (1968)

[11.] Braverman, A.; Gaitsgory, D. Crystals via the affine Grassmannian, Duke Math. J., Volume 107 (2001), pp. 561-575 | Article | MR 1828302 | Zbl 1015.20030

[12.] Braverman, A.; Finkelberg, M.; Gaitsgory, D. Uhlenbeck Spaces Via Affine Lie Algebra. The Unity of Mathematics (2006), pp. 17-135 | MR 2181803 | Zbl 1105.14013

[13.] Bridgeland, T. et al. Spaces of stability conditions, Algebraic Geometry (2009), pp. 1-21 | MR 2483930 | Zbl 1169.14303

[14.] Buan, A. B.; Iyama, O.; Reiten, I.; Scott, J. Cluster structures for 2-Calabi-Yau categories and unipotent groups, Compos. Math., Volume 145 (2009), pp. 1035-1079 | Article | MR 2521253 | Zbl 1181.18006

[15.] Cellini, P.; Papi, P. The structure of total reflection orders in affine root systems, J. Algebra, Volume 205 (1998), pp. 207-226 | Article | MR 1631338 | Zbl 0915.20017

[16.] W. Crawley-Boevey, Lectures on Representations of Quivers, Lecture Notes for a Course Given in Oxford in Spring 1992, available at http://www.amsta.leeds.ac.uk/~pmtwc/.

[17.] Crawley-Boevey, W. On the exceptional fibres of Kleinian singularities, Am. J. Math., Volume 122 (2000), pp. 1027-1037 | Article | MR 1781930 | Zbl 1001.14001

[18.] Crawley-Boevey, W.; Schröer, J. Irreducible components of varieties of modules, J. Reine Angew. Math., Volume 553 (2002), pp. 201-220 | MR 1944812 | Zbl 1062.16019

[19.] T. Dunlap, Combinatorial Representation Theory of Affine𝔰𝔩 2 via Polytope Calculus, PhD thesis, Northwestern University, 2010. | MR 2736801

[20.] Frenkel, I.; Malkin, A.; Vybornov, M. Affine Lie algebras and tame quivers, Sel. Math. New Ser., Volume 7 (2001), pp. 1-56 | Article | MR 1856552 | Zbl 1002.16012

[21.] Frenkel, I.; Savage, A. Bases of representations of type A affine Lie algebras via quiver varieties and statistical mechanics, Int. Math. Res. Not., Volume 2003 (2003), pp. 1521-1547 | Article | MR 1976600 | Zbl 1023.17010

[22.] Gabriel, P.; Roiter, A. V. Representations of finite-dimensional algebras, Algebra VIII (1992) | MR 1239447

[23.] Gaussent, S.; Littelmann, P. LS galleries, the path model, and MV cycles, Duke Math. J., Volume 127 (2005), pp. 35-88 | Article | MR 2126496 | Zbl 1078.22007

[24.] Gelfand, I. M.; Goresky, R. M.; MacPherson, R. D.; Serganova, V. V. Combinatorial geometries, convex polyhedra, and Schubert cells, Adv. Math., Volume 63 (1987), pp. 301-316 | Article | MR 877789 | Zbl 0622.57014

[25.] Geiß, C.; Leclerc, B.; Schröer, J. Rigid modules over preprojective algebras, Invent. Math., Volume 165 (2006), pp. 589-632 | Article | MR 2242628 | Zbl 1167.16009

[26.] Geiß, C.; Leclerc, B.; Schröer, J. Semicanonical basis and preprojective algebras. II: A multiplication formula, Compos. Math., Volume 143 (2007), pp. 1313-1334 | MR 2360317 | Zbl 1132.17004

[27.] Geiß, C.; Leclerc, B.; Schröer, J. Kac-Moody groups and cluster algebras, Adv. Math., Volume 228 (2011), pp. 329-433 | Article | MR 2822235 | Zbl 1232.17035

[28.] Ito, K. The classification of convex orders on affine root systems, Commun. Algebra, Volume 29 (2001), pp. 5605-5630 | Article | MR 1872815 | Zbl 0989.17016

[29.] Ito, K. Parametrizations of infinite biconvex sets in affine root systems, Hiroshima Math. J., Volume 35 (2005), pp. 425-451 | MR 2210718 | Zbl 1108.17005

[30.] Ito, K. A new description of convex bases of PBW type for untwisted quantum affine algebras, Hiroshima Math. J., Volume 40 (2010), pp. 133-183 | MR 2680654 | Zbl 1217.17008

[31.] Iyama, O.; Reiten, I. Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras, Am. J. Math., Volume 130 (2008), pp. 1087-1149 | Article | MR 2427009 | Zbl 1162.16007

[32.] Iyama, O.; Reiten, I. 2-Auslander algebras associated with reduced words in Coxeter groups, Int. Math. Res. Not., Volume 2011 (2011), pp. 1782-1803 | MR 2806521 | Zbl 1277.16012

[33.] Y. Jiang, Parametrizations of canonical bases and irreducible components of nilpotent varieties, Int. Math. Res. Not., to appear. | MR 3217661

[34.] Kac, V. G. Infinite Dimensional Lie Algebras (1990) | MR 1104219 | Zbl 0716.17022

[35.] Kamnitzer, J. The crystal structure on the set of Mirković-Vilonen polytopes, Adv. Math., Volume 215 (2007), pp. 66-93 | Article | MR 2354986 | Zbl 1134.14028

[36.] Kamnitzer, J. Mirković-Vilonen cycles and polytopes, Ann. Math., Volume 171 (2010), pp. 245-294 | Article | MR 2630039 | Zbl 1271.20058

[37.] Kashiwara, M. On crystal bases, Representations of Groups (1995), pp. 155-197 | MR 1357199 | Zbl 0851.17014

[38.] Kashiwara, M.; Saito, Y. Geometric construction of crystal bases, Duke Math. J., Volume 89 (1997), pp. 9-36 | Article | MR 1458969 | Zbl 0901.17006

[39.] Y. Kimura, Affine Quivers and Crystal Bases, Master thesis, University of Kyoto, Japan, 2007.

[40.] King, A. Moduli of representations of finite dimensional algebras, Q. J. Math. Oxf. (2), Volume 45 (1994), pp. 515-530 | Article | MR 1315461 | Zbl 0837.16005

[41.] Lusztig, G. Canonical bases arising from quantized enveloping algebras, J. Am. Math. Soc., Volume 3 (1990), pp. 447-498 | Article | MR 1035415 | Zbl 0703.17008

[42.] Lusztig, G. Canonical bases arising from quantized enveloping algebras II, Prog. Theor. Phys. Suppl., Volume 102 (1990), pp. 175-201 | Article | MR 1182165 | Zbl 0776.17012

[43.] Lusztig, G. Quivers, perverse sheaves, and quantized enveloping algebras, J. Am. Math. Soc., Volume 4 (1991), pp. 365-421 | Article | MR 1088333 | Zbl 0738.17011

[44.] Lusztig, G. Affine quivers and canonical bases, Publ. Math. Inst. Hautes Études Sci., Volume 76 (1992), pp. 111-163 | Article | MR 1215594 | Zbl 0776.17013

[45.] Lusztig, G. Braid group action and canonical bases, Adv. Math., Volume 122 (1996), pp. 237-261 | Article | MR 1409422 | Zbl 0861.17008

[46.] Mirković, I.; Vilonen, K. Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. Math., Volume 166 (2007), pp. 95-143 | Article | MR 2342692 | Zbl 1138.22013

[47.] Mumford, D. The Red Book of Varieties and Schemes (1999) | MR 1748380 | Zbl 0945.14001

[48.] Muthiah, D. Double MV cycles and the Naito-Sagaki-Saito crystal, Adv. Math. (2013) | MR 3046309 | Zbl 1292.20052

[49.] Muthiah, D.; Tingley, P. Affine PBW bases and MV polytopes in rank 2, Sel. Math. (2012) | MR 3147416 | Zbl 1315.17014

[50.] Naito, S.; Sagaki, D.; Saito, Y. et al. Toward Berenstein-Zelevinsky data in affine type A, parts I and II, Representation Theory of Algebraic Groups and Quantum Groups ’10 (2012), pp. 143-216 | MR 2932426

[51.] Naito, S.; Sagaki, D.; Saito, Y. et al. Toward Berenstein-Zelevinsky data in affine type A. Part III. Symmetries, integrable systems and representations, Springer Proceedings in Mathematics and Statistics (2013), pp. 361-402 | MR 3077692

[52.] Reineke, M. The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math., Volume 152 (2003), pp. 349-368 | Article | MR 1974891 | Zbl 1043.17010

[53.] Ringel, C. M. The preprojective algebra of a quiver, Algebras and Modules II, Eighth International Conference on Representations of Algebras (1998), pp. 467-480 | MR 1648647 | Zbl 0928.16012

[54.] Ringel, C. M. et al. The preprojective algebra of a tame quiver: the irreducible components of the module varieties, Trends in the Representation Theory of Finite Dimensional Algebras (1998), pp. 293-306 | MR 1676227 | Zbl 0929.16016

[55.] Rudakov, A. Stability for an Abelian category, J. Algebra, Volume 197 (1997), pp. 231-245 | Article | MR 1480783 | Zbl 0893.18007

[56.] Saito, Y. PBW basis of quantized universal enveloping algebras, Publ. Res. Inst. Math. Sci., Volume 30 (1994), pp. 209-232 | Article | MR 1265471 | Zbl 0812.17013

[57.] Sekiya, Y.; Yamaura, K. Tilting theoretical approach to moduli spaces over preprojective algebras, Algebr. Represent. Theory (2012) | MR 3127356 | Zbl 1297.14013

[58.] Shatz, S. The decomposition and specialization of algebraic families of vector bundles, Compos. Math., Volume 35 (1977), pp. 163-187 | MR 498573 | Zbl 0371.14010

Cité par Sources :