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1997 | 40 | 1 | 51-58
Tytuł artykułu

Multiplier Hopf algebras and duality

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EN
Abstrakty
EN
We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf *-algebra with positive invariant functionals, then also the dual is a multiplier Hopf *-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C*-algebras.
Słowa kluczowe
Rocznik
Tom
40
Numer
1
Strony
51-58
Opis fizyczny
Daty
wydano
1997
Twórcy
autor
  • Department of Mathematics, University of Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium
Bibliografia
  • [1] E. Abe, Hopf Algebras, Cambridge University Press (1977).
  • [2] V. G. Drinfel'd, Quantum groups, Proceedings ICM Berkeley (1986) 798-820.
  • [3] E. Effros & Z.-J. Ruan, Discrete Quantum Groups I. The Haar Measure, Int. J. Math. 5 (1994) 681-723.
  • [4] Y. Nakagami, T. Masuda & S. L. Woronowicz, (in preparation).
  • [5] P. Podleś & S. L. Woronowicz, Quantum Deformation of Lorentz Group, Comm. Math. Phys. 130 (1990) 381-431.
  • [6] E. M. Sweedler, Hopf Algebras, Benjamin (1969).
  • [7] A. Van Daele, Dual Pairs of Hopf *-algebras, Bull. London Math. Soc. 25 (1993) 209-230.
  • [8] A. Van Daele, The Haar Measure on Finite Quantum Groups, to appear in Proc. Amer. Math. Soc.
  • [9] A. Van Daele, The Haar Measure on Compact Quantum Groups, Proc. Amer. Math. Soc. 123 (1995) 3125-3128.
  • [10] A. Van Daele, Multiplier Hopf Algebra, Trans. Amer. Math. Soc. 342 (1994) 917-932.
  • [11] A. Van Daele, Discrete Quantum Groups, J. of Algebra 180 (1996) 431-444.
  • [12] A. Van Daele, An Algebraic Framework for Group Duality, preprint K.U. Leuven (1996).
  • [13] B. Drabant & A. Van Daele, Pairing and The Quantum Double of Multiplier Hopf Algebras, preprint K.U. Leuven (1996).
  • [14] J. Kustermans & A. Van Daele, C*-algebraic Quantum Groups arising from Algebraic Quantum Groups, preprint K.U. Leuven (1996).
  • [15] S. L. Woronowicz, Compact Matrix Pseudo Groups, Comm. Math. Phys. 111 (1987) 613-665.
  • [16] S. L. Woronowicz, Compact Quantum Groups, preprint University of Warsaw (1992).
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Bibliografia
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bwmeta1.element.bwnjournal-article-bcpv40z1p51bwm
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