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Fibers of characters in Gelfand-Tsetlin categories


Authors: Vyacheslav Futorny and Serge Ovsienko
Journal: Trans. Amer. Math. Soc. 366 (2014), 4173-4208
MSC (2010): Primary 16D60, 16D90, 16D70, 17B65
DOI: https://doi.org/10.1090/S0002-9947-2014-05938-2
Published electronically: April 7, 2014
MathSciNet review: 3206456
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Abstract: For a class of noncommutative rings, called Galois orders, we study the problem of an extension of characters from a commutative subalgebra. We show that for Galois orders this problem is always solvable in the sense that all characters can be extended, moreover, in finitely many ways, up to isomorphism. These results can be viewed as a noncommutative analogue of liftings of prime ideals in the case of integral extensions of commutative rings. The proposed approach can be applied to the representation theory of many infinite dimensional algebras including universal enveloping algebras of reductive Lie algebras (in particular $ \mathrm {gl}_n$), Yangians and finite $ W$-algebras. As an example we recover the theory of Gelfand-Tsetlin modules for $ \mathrm {gl}_n$.


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  • [AHS] Jiří Adámek, Horst Herrlich, and George E. Strecker, Abstract and concrete categories, Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1990. The joy of cats; A Wiley-Interscience Publication. MR 1051419 (91h:18001)
  • [AM] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802 (39 #4129)
  • [Ba] V. V. Bavula, Generalized Weyl algebras and their representations, Algebra i Analiz 4 (1992), no. 1, 75-97 (Russian); English transl., St. Petersburg Math. J. 4 (1993), no. 1, 71-92. MR 1171955 (93h:16043)
  • [BO] V. Bavula and F. van Oystaeyen, Simple modules of the Witten-Woronowicz algebra, J. Algebra 271 (2004), no. 2, 827-845. MR 2025552 (2004k:16067), https://doi.org/10.1016/j.jalgebra.2003.06.013
  • [D] Jacques Dixmier, Algèbres enveloppantes, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). Cahiers Scientifiques, Fasc. XXXVII. MR 0498737 (58 #16803a)
  • [Dr] Yu. A. Drozd, Representations of Lie algebras $ {\mathfrak{s}}{\mathfrak{l}}(2)$, Vīsnik Kiïv. Unīv. Ser. Mat. Mekh. 25 (1983), 70-77 (Ukrainian, with Russian summary). MR 746766 (86j:17010)
  • [DK] Yurij A. Drozd and Vladimir V. Kirichenko, Finite-dimensional algebras, Springer-Verlag, Berlin, 1994. Translated from the 1980 Russian original and with an appendix by Vlastimil Dlab. MR 1284468 (95i:16001)
  • [DFO1] Yu. A. Drozd, S. A. Ovsienko, and V. M. Futorny, On Gelfand-Zetlin modules, Proceedings of the Winter School on Geometry and Physics (Srní, 1990), 1991, pp. 143-147. MR 1151899 (93b:17021)
  • [DFO2] Yu. A. Drozd, V. M. Futorny, and S. A. Ovsienko, Harish-Chandra subalgebras and Gelfand-Zetlin modules, Finite-dimensional algebras and related topics (Ottawa, ON, 1992) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 79-93. MR 1308982 (95k:17016)
  • [Fe] S. L. Fernando, Lie algebra modules with finite-dimensional weight spaces. I, Trans. Amer. Math. Soc. 322 (1990), no. 2, 757-781. MR 1013330 (91c:17006), https://doi.org/10.2307/2001724
  • [FM] A. S. Miščenko and A. T. Fomenko, Euler equation on finite-dimensional Lie groups, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 2, 396-415, 471 (Russian). MR 0482832 (58 #2881)
  • [FMO] Vyacheslav Futorny, Alexander Molev, and Serge Ovsienko, Harish-Chandra modules for Yangians, Represent. Theory 9 (2005), 426-454. MR 2142818 (2006a:17009), https://doi.org/10.1090/S1088-4165-05-00195-0
  • [FMO1] Vyacheslav Futorny, Alexander Molev, and Serge Ovsienko, The Gelfand-Kirillov conjecture and Gelfand-Tsetlin modules for finite $ W$-algebras, Adv. Math. 223 (2010), no. 3, 773-796. MR 2565549 (2011d:17019), https://doi.org/10.1016/j.aim.2009.08.018
  • [FO1] Vyacheslav Futorny and Serge Ovsienko, Galois orders in skew monoid rings, J. Algebra 324 (2010), no. 4, 598-630. MR 2651560 (2011j:16051), https://doi.org/10.1016/j.jalgebra.2010.05.006
  • [FO2] Vyacheslav Futorny and Serge Ovsienko, Kostant's theorem for special filtered algebras, Bull. London Math. Soc. 37 (2005), no. 2, 187-199. MR 2119018 (2005m:16055), https://doi.org/10.1112/S0024609304003844
  • [FO3] Vyacheslav Futorny and Serge Ovsienko, Harish-Chandra categories, Proceedings of the XVIth Latin American Algebra Colloquium (Spanish), Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 2007, pp. 271-286. MR 2500363 (2011b:16016)
  • [Ga] Peter Gabriel, Finite representation type is open, Algebras (Carleton Univ., Ottawa, Ont., 1974) Carleton Univ., Ottawa, Ont., 1974, pp. 23 pp. Carleton Math. Lecture Notes, No. 9. MR 0376769 (51 #12944)
  • [GR] P. Gabriel and A. V. Roĭter, Representations of finite-dimensional algebras, Algebra, VIII, Encyclopaedia Math. Sci., vol. 73, Springer, Berlin, 1992, pp. 1-177. With a chapter by B. Keller. MR 1239447 (94h:16001b)
  • [GTs] I. M. Gelfand and M. L. Cetlin, Finite-dimensional representations of groups of orthogonal matrices, Doklady Akad. Nauk SSSR (N.S.) 71 (1950), 1017-1020 (Russian). MR 0034763 (11,639e)
  • [Gr1] M. I. Graev, Infinite-dimensional representations of the Lie algebra $ \mathfrak{gl}(n,{\mathbb{C}})$ related to complex analogs of the Gelfand-Tsetlin patterns and general hypergeometric functions on the Lie group $ {\rm GL}(n,{\mathbb{C}})$, Acta Appl. Math. 81 (2004), no. 1-3, 93-120. MR 2069333 (2005c:22024), https://doi.org/10.1023/B:ACAP.0000024196.16709.f6
  • [Gr2] M. I. Graev, A continuous analogue of Gelfand-Tsetlin schemes and a realization of the principal series of irreducible unitary representations of the group $ {\rm GL}(n,\mathbb{C})$ in the space of functions on the manifold of these schemes, Dokl. Akad. Nauk 412 (2007), no. 2, 154-158 (Russian); English transl., Dokl. Math. 75 (2007), no. 1, 31-35. MR 2451955, https://doi.org/10.1134/S1064562407010103
  • [Kh] O. Khomenko, Some applications of Gelfand-Zetlin modules, Representations of algebras and related topics, Fields Inst. Commun., vol. 45, Amer. Math. Soc., Providence, RI, 2005, pp. 205-213. MR 2146251 (2006k:17021)
  • [KW1] Bertram Kostant and Nolan Wallach, Gelfand-Zeitlin theory from the perspective of classical mechanics. I, Studies in Lie theory, Progr. Math., vol. 243, Birkhäuser Boston, Boston, MA, 2006, pp. 319-364. MR 2214253 (2007e:14072), https://doi.org/10.1007/0-8176-4478-4_12
  • [KW2] Bertram Kostant and Nolan Wallach, Gelfand-Zeitlin theory from the perspective of classical mechanics. II, The unity of mathematics, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 387-420. MR 2181811 (2007e:14073), https://doi.org/10.1007/0-8176-4467-9_10
  • [Ma] Olivier Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 537-592 (English, with English and French summaries). MR 1775361 (2001h:17017)
  • [Mat] H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, 1997.
  • [Maz1] Volodymyr Mazorchuk, Tableaux realization of generalized Verma modules, Canad. J. Math. 50 (1998), no. 4, 816-828. MR 1638623 (99g:17018), https://doi.org/10.4153/CJM-1998-043-x
  • [Maz2] Volodymyr Mazorchuk, Quantum deformation and tableaux realization of simple dense $ {\mathfrak{gl}}(n,\mathbb{C})$-modules, J. Algebra Appl. 2 (2003), no. 1, 1-20. MR 1964762 (2004b:17015), https://doi.org/10.1142/S0219498803000325
  • [Maz3] Volodymyr Mazorchuk, Orthogonal Gelfand-Zetlin algebras. I, Beiträge Algebra Geom. 40 (1999), no. 2, 399-415. MR 1720114 (2000j:17011)
  • [MazO] V. S. Mazorchuk and S. A. Ovsienko, Submodule structure of generalized Verma modules induced from generic Gelfand-Zetlin modules, Algebr. Represent. Theory 1 (1998), no. 1, 3-26. MR 1654598 (2000a:17006), https://doi.org/10.1023/A:1009929615175
  • [MazC] Volodymyr Mazorchuk and Catharina Stroppel, Cuspidal $ \mathfrak{sl}_n$-modules and deformations of certain Brauer tree algebras, Adv. Math. 228 (2011), no. 2, 1008-1042. MR 2822216 (2012f:17013), https://doi.org/10.1016/j.aim.2011.06.005
  • [M] A. I. Molev, Gelfand-Tsetlin bases for classical Lie algebras, Handbook of algebra. Vol. 4, Handb. Algebr., vol. 4, Elsevier/North-Holland, Amsterdam, 2006, pp. 109-170. MR 2523420 (2011k:17016), https://doi.org/10.1016/S1570-7954(06)80006-9
  • [OV] Andrei Okounkov and Anatoly Vershik, A new approach to representation theory of symmetric groups, Selecta Math. (N.S.) 2 (1996), no. 4, 581-605. MR 1443185 (99g:20024), https://doi.org/10.1007/PL00001384
  • [Ov1] Serge Ovsienko, Finiteness statements for Gelfand-Zetlin modules, Third International Algebraic Conference in the Ukraine (Ukrainian), Natsīonal. Akad. Nauk Ukraïni Īnst. Mat., Kiev, 2002, pp. 323-338. MR 2210503 (2006k:17023)
  • [Ov2] Serge Ovsienko, Strongly nilpotent matrices and Gelfand-Zetlin modules, Linear Algebra Appl. 365 (2003), 349-367. Special issue on linear algebra methods in representation theory. MR 1987348 (2004d:17018), https://doi.org/10.1016/S0024-3795(02)00675-4
  • [S] Jean-Pierre Serre, Local algebra, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. Translated from the French by CheeWhye Chin and revised by the author. MR 1771925 (2001b:13001)
  • [Vi] È. B. Vinberg, Some commutative subalgebras of a universal enveloping algebra, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 1, 3-25, 221 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 1, 1-22. MR 1044045 (91b:17015)
  • [Zh] Zhelobenko D.P. Compact Lie groups and their representations, Nauka, Moscow, 1970 (Translations of mathematical monographs, 40, AMS, Providence, Rhode Island, 1973). MR 0473097 (57:12776a)

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Additional Information

Vyacheslav Futorny
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, São Paulo, CEP 05315-970, Brasil
Email: futorny@ime.usp.br

Serge Ovsienko
Affiliation: Faculty of Mechanics and Mathematics, Kiev Taras Shevchenko University, Vladimirskaya 64, 00133, Kiev, Ukraine
Email: ovsiyenko.sergiy@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2014-05938-2
Received by editor(s): January 19, 2010
Received by editor(s) in revised form: June 18, 2010, August 14, 2012, and August 18, 2012
Published electronically: April 7, 2014
Additional Notes: The first author was supported in part by the CNPq grant (processo 301743/2007-0) and by the Fapesp grant (processo 2010/50347-9)
The second author is grateful to Fapesp for financial support (processos 2004/02850-2 and 2006/60763-4) and to the University of São Paulo for their hospitality during his visits
Article copyright: © Copyright 2014 American Mathematical Society
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