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Harish-Chandra modules for Yangians


Authors: Vyacheslav Futorny, Alexander Molev and Serge Ovsienko
Journal: Represent. Theory 9 (2005), 426-454
MSC (2000): Primary 17B35, 81R10, 17B67
DOI: https://doi.org/10.1090/S1088-4165-05-00195-0
Published electronically: June 2, 2005
MathSciNet review: 2142818
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Abstract: We study Harish-Chandra representations of the Yangian $\mathrm {Y}(\mathfrak {gl}_2)$ with respect to a natural maximal commutative subalgebra. We prove an analogue of the Kostant theorem showing that the restricted Yangian $\mathrm {Y}_p(\mathfrak {gl}_2)$ is a free module over the corresponding subalgebra $\Gamma$ and show that every character of $\Gamma$ defines a finite number of irreducible Harish-Chandra modules over $\mathrm {Y}_p(\mathfrak {gl}_2)$. We study the categories of generic Harish-Chandra modules, describe their simple modules and indecomposable modules in tame blocks.


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  • Maurice Auslander, Representation theory of Artin algebras. I, II, Comm. Algebra 1 (1974), 177–268; ibid. 1 (1974), 269–310. MR 349747, DOI https://doi.org/10.1080/00927877408548230
  • Vladimir Bavula and Viktor Bekkert, Indecomposable representations of generalized Weyl algebras, Comm. Algebra 28 (2000), no. 11, 5067–5100. MR 1785490, DOI https://doi.org/10.1080/00927870008827145
  • Vyjayanthi Chari and Andrew Pressley, Yangians and $R$-matrices, Enseign. Math. (2) 36 (1990), no. 3-4, 267–302. MR 1096420
  • I. V. Cherednik, A new interpretation of Gel′fand-Tzetlin bases, Duke Math. J. 54 (1987), no. 2, 563–577. MR 899405, DOI https://doi.org/10.1215/S0012-7094-87-05423-8
  • Ivan Cherednik, Quantum groups as hidden symmetries of classic representation theory, Differential geometric methods in theoretical physics (Chester, 1988) World Sci. Publ., Teaneck, NJ, 1989, pp. 47–54. MR 1124414
  • Jacques Dixmier, Algèbres enveloppantes, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). Cahiers Scientifiques, Fasc. XXXVII. MR 0498737
  • [D1]d:ha Drinfeld V.G., Hopf algebras and the quantum Yang–Baxter equation, Soviet Math. Dokl. 32 (1985), 254–258.
  • V. G. Drinfel′d, A new realization of Yangians and of quantum affine algebras, Dokl. Akad. Nauk SSSR 296 (1987), no. 1, 13–17 (Russian); English transl., Soviet Math. Dokl. 36 (1988), no. 2, 212–216. MR 914215
  • Ju. A. Drozd, Tame and wild matrix problems, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) Lecture Notes in Math., vol. 832, Springer, Berlin-New York, 1980, pp. 242–258. MR 607157
  • Yu. A. Drozd, S. A. Ovsienko, and V. M. Futorny, On Gel′fand-Zetlin modules, Proceedings of the Winter School on Geometry and Physics (Srní, 1990), 1991, pp. 143–147. MR 1151899
  • Yu. A. Drozd, V. M. Futorny, and S. A. Ovsienko, Harish-Chandra subalgebras and Gel′fand-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, DOI https://doi.org/10.1007/978-94-017-1556-0_5
  • Vyacheslav Futorny and Serge Ovsienko, Kostant’s theorem for special filtered algebras, Bull. London Math. Soc. 37 (2005), no. 2, 187–199. MR 2119018, DOI https://doi.org/10.1112/S0024609304003844
  • I. R. Shafarevich (ed.), Algebra. VIII, Encyclopaedia of Mathematical Sciences, vol. 73, Springer-Verlag, Berlin, 1992. Representations of finite-dimensional algebras; A translation of Algebra, VIII (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow; Translation edited by A. I. Kostrikin and I. R. Shafarevich. MR 1239446
  • [IK]ik:lm Izergin A.G., Korepin V.E., A lattice model related to the nonlinear Schrödinger equation, Sov. Phys. Dokl. 26 (1981) 653–654.
  • Bertram Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404. MR 158024, DOI https://doi.org/10.2307/2373130
  • P. P. Kulish and E. K. Sklyanin, Quantum spectral transform method. Recent developments, Integrable quantum field theories (Tvärminne, 1981) Lecture Notes in Phys., vol. 151, Springer, Berlin-New York, 1982, pp. 61–119. MR 671263
  • A. I. Molev, Gel′fand-Tsetlin basis for representations of Yangians, Lett. Math. Phys. 30 (1994), no. 1, 53–60. MR 1259196, DOI https://doi.org/10.1007/BF00761422
  • [M2]m:ce Molev A.I., Casimir elements for certain polynomial current Lie algebras, in “Group 21, Physical Applications and Mathematical Aspects of Geometry, Groups, and Algebras," Vol. 1, (H.-D. Doebner, W. Scherer, P. Nattermann, Eds). World Scientific, Singapore, 1997, 172–176.
  • A. I. Molev, Irreducibility criterion for tensor products of Yangian evaluation modules, Duke Math. J. 112 (2002), no. 2, 307–341. MR 1894363, DOI https://doi.org/10.1215/S0012-9074-02-11224-1
  • Maxim Nazarov and Vitaly Tarasov, Representations of Yangians with Gelfand-Zetlin bases, J. Reine Angew. Math. 496 (1998), 181–212. MR 1605817, DOI https://doi.org/10.1515/crll.1998.029
  • [Ov]o:fsOvsienko S., Finiteness statements for Gelfand–Tsetlin modules, In: Algebraic structures and their applications, Math. Inst., Kiev, 2002. [TF]tf:qi Takhtajan L.A., Faddeev L.D., Quantum inverse scattering method and the Heisenberg $XYZ$-model, Russian Math. Surv. 34 (1979), no. 5, 11–68. [T1]t:sq Tarasov V., Structure of quantum $L$-operators for the $R$-matrix of the $XXZ$-model, Theor. Math. Phys. 61 (1984), 1065–1071. [T2]t:im Tarasov V., Irreducible monodromy matrices for the $R$-matrix of the $XXZ$-model, and lattice local quantum Hamiltonians, Theor. Math. Phys. 63 (1985), 440–454.

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Additional Information

Vyacheslav Futorny
Affiliation: Institute of Mathematics and Statistics, University of São Paulo, Caixa Postal 66281-CEP 05315-970, São Paulo, Brazil
MR Author ID: 238132
Email: futorny@ime.usp.br

Alexander Molev
Affiliation: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
MR Author ID: 207046
Email: alexm@maths.usyd.edu.au

Serge Ovsienko
Affiliation: Faculty of Mechanics and Mathematics, Kiev Taras Shevchenko University, Vladimirskaya 64, 00133, Kiev, Ukraine
Email: ovsienko@sita.kiev.ua

Received by editor(s): May 25, 2003
Published electronically: June 2, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.