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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
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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.


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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.