Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T19:50:42.307Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on the Antipode for Algebraic Quantum Groups

Published online by Cambridge University Press:  20 November 2018

L. Delvaux ,
A. Van Daele  and
Shuanhong Wang
L. Delvaux
Affiliation:
Department of Mathematics, University of Hasselt, Agoralaan, B-3590 Diepenbeek, Belgiume-mail: [email protected]
A. Van Daele
Affiliation:
Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B, B-3001 Heverlee, Belgiume-mail: [email protected]
Shuanhong Wang
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, Chinae-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra.

In this note, we show that this formula can be proved for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a finite-dimensional Hopf algebra, but also that of any Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the theory of regular multiplier Hopf algebras with integrals.

We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.

Keywords

16W3046L65multiplier Hopf algebrasalgebraic quantum groupsthe antipode
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 2 , 01 June 2012 , pp. 260 - 270
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Beattie, M., Bulacu, D., and Torrecillas, B., Radford's S 4 formula for co-Frobenius Hopf algebras. J. Algebra 307(2007), no. 1, 330–342. http://dx.doi.org/10.1016/j.jalgebra.2006.06.004 Google Scholar
[2] Delvaux, L., The size of the intrinsic group of a multiplier Hopf algebra. Comm. Algebra 31(2003), no. 3, 1499–1514. http://dx.doi.org/10.1081/AGB-120017785 Google Scholar
[3] Delvaux, L. and Van Daele, A., Algebraic quantum hypergroups. Adv. Math. 226(2011), no. 2, 1134–1167. http://dx.doi.org/10.1016/j.aim.2010.07.015 Google Scholar
[4] Delvaux, L. and Van Daele, A., The Drinfel’d double of multiplier Hopf algebras. J. Algebra 272(2004), no. 1, 273–291. http://dx.doi.org/10.1016/j.jalgebra.2003.03.003 Google Scholar
[5] Delvaux, L., Van Daele, A., and Wang, S., Bicrossed product of multiplier Hopf algebras. arXiv:0903.2974v1[math.RA].Google Scholar
[6] Drabant, B. and Van Daele, A.. Pairing and quantum double of multiplier Hopf algebras. Algebra. Represent. Theory 4(2001), no. 2, 109–132. http://dx.doi.org/10.1023/A:1011470032416 Google Scholar
[7] Drabant, B., Van Daele, A., and Zhang, Y., Actions of multiplier Hopf algebras. Comm. Algebra 27(1999), no. 9, 4117–4172. http://dx.doi.org/10.1080/00927879908826688 Google Scholar
[8] Kustermans, J., The analytic structure of algebraic quantum groups. J. Algebra 259(2003), no. 2, 415–450. http://dx.doi.org/10.1016/S0021-8693(02)00570-7 Google Scholar
[9] Kustermans, J. and Vaes, S., A simple definition for locally compact quantum groups. C. R. Acad. Sci. Paris Sér I Math. 328(1999), no. 10, 871–876.Google Scholar
[10] Kustermans, J. and Vaes, S., Locally compact quantum groups. Ann. Sci. école Norm. Sup. 33(2000), no. 6, 837–934.Google Scholar
[11] Kustermans, J. and Vaes, S., Locally compact quantum groups in the von Neumann algebra setting. Math. Scand. 92(2003), no. 1, 68–92.Google Scholar
[12] Kustermans, J. and Van Daele, A., C*-algebraic quantum groups arising from algebraic quantum groups. Internat. J. Math. 8(1997), no. 8, 1067–1139. http://dx.doi.org/10.1142/S0129167X97000500 Google Scholar
[13] Larson, R. G. Characters of Hopf algebras. J. Algebra 17(1971), 352–368. http://dx.doi.org/10.1016/0021-8693(71)90018-4 Google Scholar
[14] Landstad, M. B. and Van Daele, A., Compact and discrete subgroups of algebraic quantum groups. I. arXiv:math/0702458v2[math.OA].Google Scholar
[15] Radford, D., The order of the antipode of any finite-dimensional Hopf algebra is finite. Amer. J. Math. 98(1976), no. 2, 333–355. http://dx.doi.org/10.2307/2373888 Google Scholar
[16] Van Daele, A., Multiplier Hopf algebras. Trans. Amer. Math. Soc. 342(1994), no. 2, 917–932. http://dx.doi.org/10.2307/2154659 Google Scholar
[17] Van Daele, A., An algebraic framework for group duality. Adv. Math. 140(1998), no. 2, 323–366. http://dx.doi.org/10.1006/aima.1998.1775 Google Scholar
[18] Van Daele, A., Locally compact quantum groups. A von Neumann algebra approach. arXiv:math/0602212v1[math.OA].Google Scholar
[19] Van Daele, A. and Zhang, Y., A survey on multiplier Hopf algebras. In: Hopf Algebras and Quantum Groups. Lecture Notes in Pure and Appl. Math. 206. Dekker, New York, 2000, pp. 269—309.Google Scholar
[20] Voigt, C., Bornological quantum groups. arXiv:math/0511195v1[math.QA]. http://dx.doi.org/10.2140/pjm.2008.235.93 Google Scholar
[21] Voigt, C., Equivariant cyclic homology for quantum groups. arXiv:math/0601725v1[math.KT].Google Scholar