Geodesic
A Note on the Antipode for Algebraic Quantum Groups
Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 260-270

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-2011-079-4
Mots-clés : 16W30, 46L65, multiplier Hopf algebras, algebraic quantum groups, the antipode 
Delvaux, L.; Daele, A. Van; Wang, Shuanhong. A Note on the Antipode for Algebraic Quantum Groups. Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 260-270. doi: 10.4153/CMB-2011-079-4
@article{10_4153_CMB_2011_079_4,
     author = {Delvaux, L. and Daele, A. Van and Wang, Shuanhong},
     title = {A {Note} on the {Antipode} for {Algebraic} {Quantum} {Groups}},
     journal = {Canadian mathematical bulletin},
     pages = {260--270},
     year = {2012},
     volume = {55},
     number = {2},
     doi = {10.4153/CMB-2011-079-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-079-4/}
}
TY  - JOUR
AU  - Delvaux, L.
AU  - Daele, A. Van
AU  - Wang, Shuanhong
TI  - A Note on the Antipode for Algebraic Quantum Groups
JO  - Canadian mathematical bulletin
PY  - 2012
SP  - 260
EP  - 270
VL  - 55
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-079-4/
DO  - 10.4153/CMB-2011-079-4
ID  - 10_4153_CMB_2011_079_4
ER  - 
%0 Journal Article
%A Delvaux, L.
%A Daele, A. Van
%A Wang, Shuanhong
%T A Note on the Antipode for Algebraic Quantum Groups
%J Canadian mathematical bulletin
%D 2012
%P 260-270
%V 55
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-079-4/
%R 10.4153/CMB-2011-079-4
%F 10_4153_CMB_2011_079_4

[1] [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. Google Scholar | DOI

[2] [2] Delvaux, L., The size of the intrinsic group of a multiplier Hopf algebra. Comm. Algebra 31(2003), no. 3, 1499–1514. Google Scholar | DOI

[3] [3] Delvaux, L. and Van Daele, A., Algebraic quantum hypergroups. Adv. Math. 226(2011), no. 2, 1134–1167. Google Scholar | DOI

[4] [4] Delvaux, L. and Van Daele, A., The Drinfel’d double of multiplier Hopf algebras. J. Algebra 272(2004), no. 1, 273–291. Google Scholar | DOI

[5] [5] Delvaux, L., Van Daele, A., and Wang, S., Bicrossed product of multiplier Hopf algebras. arXiv:0903.2974v1[math.RA]. Google Scholar

[6] [6] Drabant, B. and Van Daele, A.. Pairing and quantum double of multiplier Hopf algebras. Algebra. Represent. Theory 4(2001), no. 2, 109–132. Google Scholar | DOI

[7] [7] Drabant, B., Van Daele, A., and Zhang, Y., Actions of multiplier Hopf algebras. Comm. Algebra 27(1999), no. 9, 4117–4172. Google Scholar | DOI

[8] [8] Kustermans, J., The analytic structure of algebraic quantum groups. J. Algebra 259(2003), no. 2, 415–450. Google Scholar | DOI

[9] [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] [10] Kustermans, J. and Vaes, S., Locally compact quantum groups. Ann. Sci. école Norm. Sup. 33(2000), no. 6, 837–934. Google Scholar

[11] [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] [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. Google Scholar | DOI

[13] [13] Larson, R. G. Characters of Hopf algebras. J. Algebra 17(1971), 352–368. Google Scholar | DOI

[14] [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] [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. Google Scholar | DOI

[16] [16] Van Daele, A., Multiplier Hopf algebras. Trans. Amer. Math. Soc. 342(1994), no. 2, 917–932. Google Scholar | DOI

[17] [17] Van Daele, A., An algebraic framework for group duality. Adv. Math. 140(1998), no. 2, 323–366. Google Scholar | DOI

[18] [18] Van Daele, A., Locally compact quantum groups. A von Neumann algebra approach. arXiv:math/0602212v1[math.OA]. Google Scholar

[19] [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] [20] Voigt, C., Bornological quantum groups. arXiv:math/0511195v1[math.QA]. Google Scholar | DOI

[21] [21] Voigt, C., Equivariant cyclic homology for quantum groups. arXiv:math/0601725v1[math.KT]. Google Scholar

Cité par Sources :