Khoroshkin, Sergey; Nazarov, Maxim Mickelsson algebras and representations of Yangians. (English) Zbl 1244.17009 Trans. Am. Math. Soc. 364, No. 3, 1293-1367 (2012). This paper provides realizations (up to changing the action of the center) of irreducible finite dimensional modules over the Yangian \(Y(\mathfrak{gl}_n)\) and twisted Yangians \(Y(\mathfrak{sp}_n)\) and \(Y(\mathfrak{so}_n)\) as certain quotients of tensor products of symmetric and exterior powers of the vector space \(\mathbb{C}^n\) (in the last case of the twisted Yangians \(Y(\mathfrak{so}_n)\) the realization covers only modules on which the action of \(\mathfrak{so}_n\) integrates to the action of \(\mathrm{SO}_n\)). The construction and arguments are based on the theory of reductive dual pairs and on the representation theory of Mickelsson algebras. Reviewer: Volodymyr Mazorchuk (Uppsala) Cited in 1 ReviewCited in 16 Documents MSC: 17B35 Universal enveloping (super)algebras 17B37 Quantum groups (quantized enveloping algebras) and related deformations Keywords:Mickelsson algebra; Yangian; Howe duality; Cherednik functor; Drinfeld functor PDFBibTeX XMLCite \textit{S. Khoroshkin} and \textit{M. Nazarov}, Trans. Am. Math. Soc. 364, No. 3, 1293--1367 (2012; Zbl 1244.17009) Full Text: DOI arXiv References: [1] Tatsuya Akasaka and Masaki Kashiwara, Finite-dimensional representations of quantum affine algebras, Publ. Res. Inst. Math. Sci. 33 (1997), no. 5, 839 – 867. · Zbl 0915.17011 [2] Tomoyuki Arakawa and Takeshi Suzuki, Duality between \?\?_{\?}(\?) and the degenerate affine Hecke algebra, J. Algebra 209 (1998), no. 1, 288 – 304. · Zbl 0919.17005 [3] R.Asherova, Y.Smirnov and V.Tolstoy, A description of certain class of projection operators for complex semisimple Lie algebras, Math. Notes 26 (1980), 499-504. · Zbl 0427.17007 [4] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). · Zbl 0186.33001 [5] Jonathan Brundan and Alexander Kleshchev, Representations of shifted Yangians and finite \?-algebras, Mem. Amer. Math. Soc. 196 (2008), no. 918, viii+107. · Zbl 1169.17009 [6] Vyjayanthi Chari, Braid group actions and tensor products, Int. Math. Res. Not. 7 (2002), 357 – 382. · Zbl 0990.17009 [7] Vyjayanthi Chari and Andrew Pressley, Fundamental representations of Yangians and singularities of \?-matrices, J. Reine Angew. Math. 417 (1991), 87 – 128. · Zbl 0726.17014 [8] I. V. Cherednik, Factorizing particles on a half line, and root systems, Teoret. Mat. Fiz. 61 (1984), no. 1, 35 – 44 (Russian, with English summary). · Zbl 0575.22021 [9] I. V. Cherednik, A new interpretation of Gel\(^{\prime}\)fand-Tzetlin bases, Duke Math. J. 54 (1987), no. 2, 563 – 577. · Zbl 0645.17006 [10] V. G. Drinfel\(^{\prime}\)d, Degenerate affine Hecke algebras and Yangians, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 69 – 70 (Russian). [11] V. G. Drinfel\(^{\prime}\)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. [12] P. Etingof and A. Varchenko, Dynamical Weyl groups and applications, Adv. Math. 167 (2002), no. 1, 74 – 127. · Zbl 1033.17010 [13] Harish-Chandra, Representations of semisimple Lie groups. II, Trans. Amer. Math. Soc. 76 (1954), 26 – 65. · Zbl 0055.34002 [14] Roger Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), no. 2, 539 – 570. , https://doi.org/10.1090/S0002-9947-1989-0986027-X Roger Howe, Erratum to: ”Remarks on classical invariant theory”, Trans. Amer. Math. Soc. 318 (1990), no. 2, 823. · Zbl 0674.15021 [15] Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1 – 182. · Zbl 0194.53802 [16] S. M. Khoroshkin, An extremal projector and a dynamical twist, Teoret. Mat. Fiz. 139 (2004), no. 1, 158 – 176 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 139 (2004), no. 1, 582 – 597. · Zbl 1178.17014 [17] Sergey Khoroshkin and Maxim Nazarov, Yangians and Mickelsson algebras. I, Transform. Groups 11 (2006), no. 4, 625 – 658. · Zbl 1116.17009 [18] Sergey Khoroshkin and Maxim Nazarov, Yangians and Mickelsson algebras. II, Mosc. Math. J. 6 (2006), no. 3, 477 – 504, 587 (English, with English and Russian summaries). · Zbl 1141.17009 [19] Sergey Khoroshkin and Maxim Nazarov, Twisted Yangians and Mickelsson algebras. I, Selecta Math. (N.S.) 13 (2007), no. 1, 69 – 136. · Zbl 1172.17007 [20] M. Nazarov and S. Khoroshkin, Twisted Yangians and Mickelsson algebras. II, Algebra i Analiz 21 (2009), no. 1, 153 – 228 (Russian); English transl., St. Petersburg Math. J. 21 (2010), no. 1, 111 – 161. · Zbl 1211.17014 [21] Sergey Khoroshkin, Maxim Nazarov, and Ernest Vinberg, A generalized Harish-Chandra isomorphism, Adv. Math. 226 (2011), no. 2, 1168 – 1180. · Zbl 1231.17005 [22] S. Khoroshkin and O. Ogievetsky, Mickelsson algebras and Zhelobenko operators, J. Algebra 319 (2008), no. 5, 2113 – 2165. · Zbl 1236.16023 [23] P. P. Kulish, N. Yu. Reshetikhin, and E. K. Sklyanin, Yang-Baxter equations and representation theory. I, Lett. Math. Phys. 5 (1981), no. 5, 393 – 403. · Zbl 0502.35074 [24] J. Lepowsky and G. W. McCollum, On the determination of irreducible modules by restriction to a subalgebra, Trans. Amer. Math. Soc. 176 (1973), 45 – 57. · Zbl 0266.17014 [25] Jouko Mickelsson, Step algebras of semi-simple subalgebras of Lie algebras, Rep. Mathematical Phys. 4 (1973), 307 – 318. · Zbl 0285.17005 [26] A. I. Molev, Skew representations of twisted Yangians, Selecta Math. (N.S.) 12 (2006), no. 1, 1 – 38. · Zbl 1143.17005 [27] Alexander Molev, Yangians and classical Lie algebras, Mathematical Surveys and Monographs, vol. 143, American Mathematical Society, Providence, RI, 2007. · Zbl 1141.17001 [28] Alexander Molev and Grigori Olshanski, Centralizer construction for twisted Yangians, Selecta Math. (N.S.) 6 (2000), no. 3, 269 – 317. · Zbl 1026.17021 [29] A. I. Molev, V. N. Tolstoy, and R. B. Zhang, On irreducibility of tensor products of evaluation modules for the quantum affine algebra, J. Phys. A 37 (2004), no. 6, 2385 – 2399. · Zbl 1050.17012 [30] Maxim Nazarov, Representations of twisted Yangians associated with skew Young diagrams, Selecta Math. (N.S.) 10 (2004), no. 1, 71 – 129. · Zbl 1055.17004 [31] Maxim Nazarov and Vitaly Tarasov, On irreducibility of tensor products of Yangian modules, Internat. Math. Res. Notices 3 (1998), 125 – 150. · Zbl 0893.17011 [32] Maxim Nazarov and Vitaly Tarasov, On irreducibility of tensor products of Yangian modules associated with skew Young diagrams, Duke Math. J. 112 (2002), no. 2, 343 – 378. · Zbl 1027.17013 [33] G. I. Ol\(^{\prime}\)shanskiĭ, Extension of the algebra \?(\?) for infinite-dimensional classical Lie algebras \?, and the Yangians \?(\?\?(\?)), Dokl. Akad. Nauk SSSR 297 (1987), no. 5, 1050 – 1054 (Russian); English transl., Soviet Math. Dokl. 36 (1988), no. 3, 569 – 573. [34] G. I. Ol\(^{\prime}\)shanskiĭ, Twisted Yangians and infinite-dimensional classical Lie algebras, Quantum groups (Leningrad, 1990) Lecture Notes in Math., vol. 1510, Springer, Berlin, 1992, pp. 104 – 119. [35] E. K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A 21 (1988), no. 10, 2375 – 2389. · Zbl 0685.58058 [36] V. O. Tarasov, The structure of quantum \?-operators for the \?-matrix of the \?\?\?-model, Teoret. Mat. Fiz. 61 (1984), no. 2, 163 – 173 (Russian, with English summary). [37] V. O. Tarasov, Irreducible monodromy matrices for an \?-matrix of the \?\?\? model, and lattice local quantum Hamiltonians, Teoret. Mat. Fiz. 63 (1985), no. 2, 175 – 196 (Russian, with English summary). [38] V. Tarasov and A. Varchenko, Difference equations compatible with trigonometric KZ differential equations, Internat. Math. Res. Notices 15 (2000), 801 – 829. · Zbl 0971.39009 [39] V. Tarasov and A. Varchenko, Duality for Knizhnik-Zamolodchikov and dynamical equations, Acta Appl. Math. 73 (2002), no. 1-2, 141 – 154. The 2000 Twente Conference on Lie Groups (Enschede). · Zbl 1013.17006 [40] J. Tits, Normalisateurs de tores. I. Groupes de Coxeter étendus, J. Algebra 4 (1966), 96 – 116 (French). · Zbl 0145.24703 [41] Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. · Zbl 1024.20501 [42] D. P. Zhelobenko, Extremal cocycles on Weyl groups, Funktsional. Anal. i Prilozhen. 21 (1987), no. 3, 11 – 21, 95 (Russian). · Zbl 0633.17008 [43] D. P. Zhelobenko, Extremal projectors and generalized Mickelsson algebras on reductive Lie algebras, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 4, 758 – 773, 895 (Russian); English transl., Math. USSR-Izv. 33 (1989), no. 1, 85 – 100. · Zbl 0659.17010 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.