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The duality theory of a finite dimensional discrete quantum group


Authors: Lining Jiang, Maozheng Guo and Min Qian
Journal: Proc. Amer. Math. Soc. 132 (2004), 3537-3547
MSC (2000): Primary 46L05; Secondary 16W30
DOI: https://doi.org/10.1090/S0002-9939-04-07397-6
Published electronically: July 14, 2004
MathSciNet review: 2084075
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Abstract: Suppose that $\mathcal{H}$ is a finite dimensional discrete quantum group and $K$ is a Hilbert space. This paper shows that if there exists an action $ \gamma $ of $\mathcal{H}$ on $L(K)$ so that $L(K)$ is a modular algebra and the inner product on $K$ is $\mathcal{H}$-invariant, then there is a unique C*-representation $\theta $ of $\mathcal{H}$ on $K$ supplemented by the $ \gamma .$ The commutant of $\theta \left( \mathcal{H}\right) $ in $L(K)$ is exactly the $\mathcal{H}$-invariant subalgebra of $L(K)$. As an application, a new proof of the classical Schur-Weyl duality theory of type A is given.


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Additional Information

Lining Jiang
Affiliation: Department of Mathematics, Beijing Institute of Technology, Beijing (100081), People’s Republic of China
Email: jiangjln@sina.com

Maozheng Guo
Affiliation: Department of Mathematics, Peking University, Beijing (100871), People’s Republic of China
Email: maguo@pku.edu.cn

Min Qian
Affiliation: Department of Mathematics, Peking University, Beijing (100871), People’s Republic of China

DOI: https://doi.org/10.1090/S0002-9939-04-07397-6
Keywords: Discrete quantum group, C*-homomorphism, duality
Received by editor(s): November 28, 2001
Received by editor(s) in revised form: December 25, 2002
Published electronically: July 14, 2004
Additional Notes: This project was supported by the National Natural Science Foundation of China (10301004)
Communicated by: David R. Larson
Article copyright: © Copyright 2004 American Mathematical Society

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