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Compact quantum groups and their corepresentations

Published online by Cambridge University Press:  17 April 2009

Huu Hung Bui
Huu Hung Bui
Affiliation:
School of MathematicsThe University of New South WalesSydney NSW 2025Australia
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Abstract

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A compact quantum group is defined to be a unital Hopf C*–algebra generated by the matrix elements of a family of invertible corepresentations. We present a version of the Tannaka–Krein duality theorem for compact quantum groups in the context of abstract categories; this result encompasses the result of Woronowicz and the classical Tannaka-Krein duality theorem. We construct the orthogonality relations (similar to the case of compact groups). The Plancherel theorem is then established.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 57 , Issue 1 , February 1998 , pp. 73 - 91
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Bui, H.H., ‘Morita equivalence of twisted crossed products by coactions’, J. Funct. Anal. 123 (1994), 59–98.CrossRefGoogle Scholar
[2]Bui, H.H., Compact quantum groups associated with monoidal functors, Proc. Amer. Math. Soc. (to appear).Google Scholar
[3]Baaj, S. and Skandalis, G., ‘Unitaires multiplicatifs et dualite pour les produits crossés de C*–algebrès’, Ann. Sci. École. Norm. Sup. 26 (1993), 425–488.CrossRefGoogle Scholar
[4]Doplicher, S. and Roberts, J.E., ‘A new duality theory for compact groups’, Invent. Math. 98 (1989), 157–218.CrossRefGoogle Scholar
[5]Fell, J.M.G. and Doran, R.S., Representations of *–algebras, locally compact groups and Banach *–algebraic bundles, Vol. 2, Pure and Applied Math. 126 (Academic Press, New York, London, 1988).Google Scholar
[6]Joyal, A. and Street, R., ‘An introduction to the Tannaka duality and quantum groups’, in Lecture Notes in Mathematics 1488 (Springer-Verlag, Berlin, Heidelberg, New York, 1991), pp. 411–492.Google Scholar
[7]Joyal, A. and Street, R., ‘Braided tensor categories’, Adv. Math. 102 (1993), 20–78.CrossRefGoogle Scholar
[8]Kassel, C., Quantum groups, Graduate Texts in Mathematics 155 (Springer-Verlag, Berlin, Heidelberg, New York, 1995).CrossRefGoogle Scholar
[9]Kassel, C. and Turaev, V., ‘Double construction for monoidal categories’, Acta. Math. 175 (1995), 1–48.CrossRefGoogle Scholar
[10]Rudin, W., Functional analysis (McGraw-Hill Book Co., New York, 1991).Google Scholar
[11]Skandalis, G., ‘Operator algebras and duality’, in Proceedings of International Congress of Mathematicians 2, 1990, pp. 997–1009.Google Scholar
[12]Van Daele, A., ‘Dual pairs of Hopf *–algebras’, Bull. London Math. Soc. 25 (1993), 209–230.CrossRefGoogle Scholar
[13]Van Daele, A., ‘The Haar measure in a compact quantum group’, Proc. Amer. Math. Soc. 123 (1995), 3125–3128.CrossRefGoogle Scholar
[14]Woronowicz, S.L., ‘Compact matrix pseudogroups’, Comm. Math. Phys. 111 (1987), 613–665.CrossRefGoogle Scholar
[15]Woronowicz, S.L., ‘Tannaka–Krein duality for compact matrix pseudogroups. Twisted SU(N) groups’, Invent. Math. 93 (1988), 35–76.CrossRefGoogle Scholar
[16]Yetter, D.N., ‘Quantum groups and representation of monoidal categories’, Math. Proc. Cambridge Philos. Soc. 180 (1990), 261–290.CrossRefGoogle Scholar