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Twisted tensor coproduct of multiplier Hopf algebras. (English) Zbl 1071.16032
Let \(A\) and \(B\) be multiplier Hopf algebras, and let \(T\) be a cotwisting map, i.e. \(T\colon A\otimes B\to B\otimes A\) is an algebra isomorphism satisfying certain conditions. The author defines a comultiplication on the tensor product of algebras \(A\otimes B\) making it a multiplier Hopf algebra, called a twisted tensor coproduct. Skew copairings are introduced for multiplier Hopf algebras, and a cotwisting map is associated to any skew copairing. If \(A\) is an algebraic quantum group, i.e. a regular multiplier Hopf algebra with non-zero integrals, let \(\widehat A\) be the dual algebraic quantum group and \(D=\widehat A\times A^{cop}\) be the associated Drinfeld double, which is also an algebraic quantum group. The dual algebraic quantum group \(\widehat D\) is characterized as a twisted tensor coproduct.

MSC:
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16W35 Ring-theoretic aspects of quantum groups (MSC2000)
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[1] Beattie, M.; Dǎscǎlescu, S.; Grünenfelder, L.; Nǎstǎsescu, C., Finiteness conditions, co-Frobenius Hopf algebras and quantum groups, J. algebra, 200, 312-333, (1998) · Zbl 0902.16028
[2] Baaj, S.; Skandalis, G., Unitaires multiplicatifs et dualité pour LES produits croisés de \(C\^{}\{∗\}\)-algèbres, Ann. sci. école norm. sup. (4), 26, 425-488, (1993) · Zbl 0804.46078
[3] Delvaux, L., Pairing and Drinfel’d double of ore-extensions, Comm. algebra, 29, 7, 3167-3177, (2001) · Zbl 0985.16020
[4] Delvaux, L., Semi-direct products of multiplier Hopf algebras: smash coproducts, Comm. algebra, 30, 12, 5979-5997, (2002) · Zbl 1038.16028
[5] L. Delvaux, Twisted tensor product of multiplier Hopf (^∗)-algebras, Preprint LUC, 2002, J. Algebra, submitted for publication · Zbl 1036.16030
[6] L. Delvaux, A. Van Daele, The Drinfel’d double of multiplier Hopf algebras, Preprint LUC, 2002, J. Algebra, submitted for publication · Zbl 1044.16028
[7] L. Delvaux, A. Van Daele, The Drinfel’d double versus the Heisenberg double for an algebraic quantum group, Preprint LUC, 2003, submitted for publication · Zbl 1056.16027
[8] Drabant, B.; Van Daele, A., Pairing and quantum double of multiplier Hopf algebras, Algebr. represent. theory, 4, 2, 109-132, (2001) · Zbl 0993.16024
[9] Drabant, B.; Van Daele, A.; Zhang, Y., Actions of multiplier Hopf algebras, Comm. algebra, 27, 9, 4117-4127, (1999) · Zbl 0951.16013
[10] Kustermans, J., The analytic structure of algebraic quantum groups, J. algebra, 259, 415-450, (2003) · Zbl 1034.46064
[11] Kurose, H.; Van Daele, A.; Zhang, Y., Corepresentation theory of multiplier Hopf algebras II, Internat. J. math., 11, 2, 233-278, (2000) · Zbl 1108.16302
[12] Lu, J.-H., On the Drinfel’d double and the Heisenberg double of a Hopf algebra, Duke math. J., 74, 763-776, (1994) · Zbl 0815.16020
[13] Majid, S., Foundations of quantum group theorey, (1995), Cambridge Univ. Press Cambridge
[14] Sweedler, M.E., Hopf algebras, (1965), Benjamin · Zbl 0194.32901
[15] Van Daele, A., Multiplier Hopf algebras, Trans. amer. math. soc., 342, 2, 917-932, (1994) · Zbl 0809.16047
[16] Van Daele, A., An algebraic framework for group duality, Adv. math., 140, 323-366, (1998) · Zbl 0933.16043
[17] Van Daele, A.; Zhang, Y., A survey on multiplier Hopf algebras, (), 256-309 · Zbl 1020.16032
[18] Van Daele, A.; Zhang, Y., Galois theory for multiplier Hopf algebras with integrals, Algebr. represent. theory, 2, 83-106, (1999) · Zbl 0929.16038
[19] Zhang, Y., The quantum double of a co-Frobenius Hopf algebra, Comm. algebra, 27, 3, 1413-1427, (1999) · Zbl 0921.16027
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