Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Mickelsson algebras and representations of Yangians


Authors: Sergey Khoroshkin and Maxim Nazarov
Journal: Trans. Amer. Math. Soc. 364 (2012), 1293-1367
MSC (2010): Primary 17B35, 81R50
DOI: https://doi.org/10.1090/S0002-9947-2011-05367-5
Published electronically: October 12, 2011
MathSciNet review: 2869178
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \operatorname {Y}(\mathfrak{gl}_n)$ be the Yangian of the general linear Lie algebra $ \mathfrak{gl}_n$. We denote by $ \operatorname {Y}(\mathfrak{sp}_n)$ and $ \operatorname {Y}(\mathfrak{so}_n)$ the twisted Yangians corresponding to the symplectic and orthogonal subalgebras in the Lie algebra $ \mathfrak{gl}_n$. These twisted Yangians are one-sided coideal subalgebras in the Hopf algebra $ \operatorname {Y}(\mathfrak{gl}_n)$. We provide realizations of irreducible modules of the algebras $ \operatorname {Y}(\mathfrak{sp}_n)$ and $ \operatorname {Y}(\mathfrak{so}_n)$ as certain quotients of tensor products of symmetic and exterior powers of the vector space $ \mathbb{C}^n$. For the Yangian $ \operatorname {Y} (\mathfrak{gl}_n)$ such realizations have been known, but we give new proofs of these results. For the twisted Yangian $ \operatorname {Y}(\mathfrak{sp}_n)$, we realize all irreducible finite-dimensional modules. For the twisted Yangian $ \operatorname {Y}(\mathfrak{so}_n)$, we realize all those irreducible finite-dimensional modules, where the action of the Lie algebra $ \mathfrak{so}_n$ integrates to an action of the special orthogonal Lie group $ \mathrm {SO}_n$. Our results are based on the theory of reductive dual pairs due to Howe, and on the representation theory of Mickelsson algebras.


References [Enhancements On Off] (What's this?)

  • [AK] T.Akasaka and M.Kashiwara, Finite-dimensional representations of quantum affine algebras, Publ. Res. Inst. Math. Sci. 33 (1997), 839-867. MR 1607008 (99d:17017)
  • [AS] T.Arakawa and T.Suzuki, Duality between $ \mathfrak{sl}_n(\mathbb{C})$ and the degenerate affine Hecke algebra, J. Algebra 209 (1998), 288-304. MR 1652134 (99h:17005)
  • [AST] 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.
  • [B] N.Bourbaki, Groupes et algèbres de Lie IV-VI, Hermann, Paris, 1968. MR 0240238 (39:1590)
  • [BK] J.Brundan and A.Kleshchev, Representations of shifted Yangians and finite W-algebras, Mem. Amer. Math. Soc. 196 (2008), 1-107. MR 2456464 (2009i:17020)
  • [C] V.Chari, Braid group actions and tensor products, Internat. Math. Res. Notices (2002), 357-382. MR 1883181 (2003a:17014)
  • [CP] V.Chari and A.Pressley, Fundamental representations of Yangians and singularities of $ R$-matrices, J. Reine Angew. Math. 417 (1991), 87-128. MR 1103907 (92h:17010)
  • [C1] I.Cherednik, Factorized particles on the half-line and root systems, Theor. Math. Phys. 61 (1984), 977-983. MR 774205 (86g:81148)
  • [C2] I.Cherednik, A new interpretation of Gelfand-Tzetlin bases, Duke Math. J. 54 (1987), 563-577. MR 899405 (88k:17005)
  • [D1] V.Drinfeld, Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl. 20 (1986), 56-58. MR 831053 (87m:22044)
  • [D2] V.Drinfeld, A new realization of Yangians and quantized affine algebras, Soviet Math. Dokl. 36 (1988), 212-216. MR 914215 (88j:17020)
  • [EV] P.Etingof and A.Varchenko, Dynamical Weyl groups and applications, Adv. Math. 167 (2002), 74-127. MR 1901247 (2003d:17004)
  • [H] Harish-Chandra, Representations of semisimple Lie groups II, Trans. Amer. Math. Soc. 76 (1954), 26-65. MR 0058604 (15:398a)
  • [H1] R.Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), 539-570. MR 986027 (90h:22015a)
  • [H2] R.Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, Israel Math. Conf. Proc. 8 (1995), 1-182. MR 1321638 (96e:13006)
  • [K] S.Khoroshkin, Extremal projector and dynamical twist, Theoret. Math. Phys. 139 (2004), 582-597. MR 2076916 (2005f:17016)
  • [KN1] S.Khoroshkin and M.Nazarov, Yangians and Mickelsson algebras I, Transformation Groups 11 (2006), 625-658. MR 2278142 (2008d:17016)
  • [KN2] S.Khoroshkin and M.Nazarov, Yangians and Mickelsson algebras II, Moscow Math. J. 6 (2006), 477-504. MR 2274862 (2008d:17017)
  • [KN3] S.Khoroshkin and M.Nazarov, Twisted Yangians and Mickelsson algebras I, Selecta Math. 13 (2007), 69-136 and 14 (2009), 321. MR 2330588 (2009d:17021)
  • [KN4] S.Khoroshkin and M.Nazarov, Twisted Yangians and Mickelsson algebras II, St. Petersburg Math. J. 21 (2010), 111-161. MR 2553055
  • [KNV] S.Khoroshkin, M.Nazarov and E.Vinberg, A generalized Harish-Chandra isomorphism, Adv. Math. 226 (2011), 1168-1180. MR 2737780
  • [KO] S.Khoroshkin and O.Ogievetsky, Mickelsson algebras and Zhelobenko operators, J. Algebra 319 (2008), 2113-2165. MR 2394693 (2009a:16040)
  • [KRS] P.Kulish, N.Reshetikhin and E.Sklyanin, Yang-Baxter equation and representation theory, Lett. Math. Phys. 5 (1981), 393-403. MR 649704 (83g:81099)
  • [LM] J.Lepowski and G.McCollum, On the determination of irreducible modules by restriction to a subalgebra, Trans. Amer. Math. Soc. 176 (1973), 45-57. MR 0323846 (48:2201)
  • [M] J.Mickelsson, Step algebras of semi-simple subalgebras of Lie algebras, Rep. Math. Phys. 4 (1973), 307-318. MR 0342057 (49:6803)
  • [M1] A.Molev, Skew representations of twisted Yangians, Selecta Math. 12 (2006), 1-38. MR 2244262 (2007h:17013)
  • [M2] A.Molev, Yangians and classical Lie algebras, Amer. Math. Soc., Providence, RI, 2007. MR 2355506 (2008m:17033)
  • [MO] A.Molev and G.Olshanski, Centralizer construction for twisted Yangians, Selecta Math. 6 (2000), 269-317. MR 1817615 (2002j:17013)
  • [MTZ] A.Molev, V.Tolstoy and R.Zhang, On irreducibility of tensor products of evaluation modules for the quantum affine algebra, J. Phys. A37 (2004), 2385-2399. MR 2045932 (2005g:17034)
  • [N] M.Nazarov, Representations of twisted Yangians associated with skew Young diagrams, Selecta Math. 10 (2004), 71-129. MR 2061224 (2005e:17026)
  • [NT1] M.Nazarov and V.Tarasov, On irreducibility of tensor products of Yangian modules, Internat. Math. Res. Notices (1998), 125-150. MR 1606387 (99b:17014)
  • [NT2] M.Nazarov and V.Tarasov, On irreducibility of tensor products of Yangian modules associated with skew Young diagrams, Duke Math. J. 112 (2002), 343-378. MR 1894364 (2003b:17021)
  • [O1] G.Olshanski, Extension of the algebra $ U(g)$ for infinite-dimensional classical Lie algebras $ g$, and the Yangians $ Y(gl(m))$, Soviet Math. Dokl. 36 (1988), 569-573. MR 936073 (89g:17017)
  • [O2] G.Olshanski, Twisted Yangians and infinite-dimensional classical Lie algebras, Lecture Notes Math. 1510 (1992), 103-120. MR 1183482 (93h:17039)
  • [S] E.Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A21 (1988), 2375-2389. MR 953215 (89h:81258)
  • [T1] V.Tarasov, Structure of quantum L-operators for the $ R$-matrix of the $ XXZ$-model, Theor. Math. Phys. 61 (1984), 1065-1071. MR 778541 (87c:82023)
  • [T2] V.Tarasov, Irreducible monodromy matrices for the $ R$-matrix of the $ XXZ$-model and lattice local quantum Hamiltonians, Theor. Math. Phys. 63 (1985), 440-454. MR 800062 (87d:82022)
  • [TV1] V.Tarasov and A.Varchenko, Difference equations compatible with trigonometric KZ differential equations, Internat. Math. Res. Notices (2000), 801-829. MR 1780748 (2001k:32025)
  • [TV2] V.Tarasov and A.Varchenko, Duality for Knizhnik-Zamolodchikov and dynamical equations, Acta Appl. Math. 73 (2002), 141-154. MR 1926498 (2003h:17024)
  • [T] J. Tits, Normalisateurs de tores I. Groupes de Coxeter étendu, J. Algebra 4 (1966), 96-116. MR 0206117 (34:5942)
  • [W] H.Weyl, Classical Groups, their Invariants and Representations, Princeton University Press, Princeton, 1946. MR 1488158 (98k:01049)
  • [Z1] D.Zhelobenko, Extremal cocycles on Weyl groups, Funct. Anal. Appl. 21 (1987), 183-192. MR 911771 (89g:17007)
  • [Z2] D.Zhelobenko, Extremal cocycles and generalized Mickelsson algebras over reductive Lie algebras, Math. USSR Izvestiya 33 (1989), 85-100. MR 966983 (89m:17021)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 17B35, 81R50

Retrieve articles in all journals with MSC (2010): 17B35, 81R50


Additional Information

Sergey Khoroshkin
Affiliation: Institute for Theoretical and Experimental Physics, Moscow 117259, Russia – and – Department of Mathematics, Higher School of Economics, Moscow 117312, Russia
Email: khor@itep.ru

Maxim Nazarov
Affiliation: Department of Mathematics, University of York, York YO10 5DD, England
Email: mln1@york.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-2011-05367-5
Keywords: Howe duality, Cherednik functor, Drinfeld functor
Received by editor(s): November 27, 2009
Received by editor(s) in revised form: April 7, 2010
Published electronically: October 12, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society