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Representation Theory

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BMW algebra, quantized coordinate algebra and type $C$ Schur–Weyl duality

Author: Jun Hu
Journal: Represent. Theory 15 (2011), 1-62
MSC (2000): Primary 17B37, 20C20; Secondary 20C08
Published electronically: January 10, 2011
MathSciNet review: 2754334
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Abstract: We prove an integral version of the Schur–Weyl duality between the specialized Birman–Murakami–Wenzl algebra $\mathfrak {B}_n(-q^{2m+1},q)$ and the quantum algebra associated to the symplectic Lie algebra $\mathfrak {sp}_{2m}$. In particular, we deduce that this Schur–Weyl duality holds over arbitrary (commutative) ground rings, which answers a question of Lehrer and Zhang in the symplectic case. As a byproduct, we show that, as a $\mathbb {Z}[q,q^{-1}]$-algebra, the quantized coordinate algebra defined by Kashiwara (which he denoted by $A_q^{\mathbb {Z}}(g)$) is isomorphic to the quantized coordinate algebra arising from a generalized Faddeev–Reshetikhin–Takhtajan construction.

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Additional Information

Jun Hu
Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia

Keywords: Birman–Murakami–Wenzl algebra, modified quantized enveloping algebra, canonical bases
Received by editor(s): March 8, 2009
Received by editor(s) in revised form: October 14, 2009
Published electronically: January 10, 2011
Additional Notes: This research was supported by National Natural Science Foundation of China (Project 10771014), the Program NCET and partly by an Australian Research Council discovery grant. The author also acknowledges the support of the Chern Institute of Mathematics during his visit in March of 2007.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.