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HOPF IMAGES AND INNER FAITHFUL REPRESENTATIONS

Published online by Cambridge University Press:  25 August 2010

TEODOR BANICA  and
JULIEN BICHON
TEODOR BANICA
Affiliation:
Département de Mathématiques, Université de Cergy-Pontoise, 2 avenue Chauvin, 95302 Cergy-Pontoise, France e-mail: Teodor.Banica@u-cergy.fr
JULIEN BICHON
Affiliation:
Laboratoire de Mathématiques, Université Blaise Pascal, Clermont-Ferrand II, Campus des Cézeaux, 63177 Aubière Cedex, France e-mail: Julien.Bichon@math.univ-bpclermont.fr
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Abstract

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We develop a general theory of Hopf image of a Hopf algebra representation, with the associated concept of inner faithful representation, modelled on the notion of faithful representation of a discrete group. We study several examples, including group algebras, enveloping algebras of Lie algebras, pointed Hopf algebras, function algebras, twistings and cotwistings, and we present a Tannaka duality formulation of the notion of Hopf image.

Keywords

16W30
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 3 , September 2010 , pp. 677 - 703
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Amitsur, S. A., Groups with representations of bounded degree II, Illinois J. Math. 5 (1961), 198–205.CrossRefGoogle Scholar
2.Andruskiewitsch, N. and Devoto, J., Extensions of Hopf algebras, St. Petersburg Math. J. 7 (1996), 17–52.Google Scholar
3.Andruskiewitsch, N. and Schneider, H.-J., On the classification of finite-dimensional pointed Hopf algebras, Ann. Math. 171 (2010), 375–417.CrossRefGoogle Scholar
4.Artin, M., Schelter, W. and Tate, J., Quantum deformations of GLn, Commun. Pure Appl. Math. 44 (1991), 879–895.CrossRefGoogle Scholar
5.Auerbach, H., Sur les groupes linéaires bornés III, Studia Math. 5 (1935), 43–49.CrossRefGoogle Scholar
6.Banica, T., Hopf algebras and subfactors associated to vertex models, J. Funct. Anal. 159 (1998), 243–266.CrossRefGoogle Scholar
7.Banica, T., Representations of compact quantum groups and subfactors, J. Reine Angew. Math. 509 (1999), 167–198.CrossRefGoogle Scholar
8.Banica, T. and Bichon, J., Quantum groups acting on 4 points, J. Reine Angew. Math. 626 (2009), 75–114.CrossRefGoogle Scholar
9.Banica, T., Bichon, J. and Collins, B., The hyperoctahedral quantum group, J. Ramanujan Math. Soc. 22 (2007), 345–384.Google Scholar
10.Banica, T., Bichon, J. and Schlenker, J.-M., Representations of quantum permutations algebras, J. Funct. Anal. 257 (2009), 2864–2910.CrossRefGoogle Scholar
11.Banica, T. and Moroianu, S., On the structure of quantum permutation groups, Proc. Amer. Math. Soc. 135 (2007), 21–29.CrossRefGoogle Scholar
12.Banica, T. and Nicoara, R., Quantum groups and Hadamard matrices, Panamer. Math. J. 17 (2007), 1–24.Google Scholar
13.Bichon, J., Free wreath product by the quantum permutation group, Alg. Rep. Theory 7 (2004), 343–362.CrossRefGoogle Scholar
14.Chin, W. and Musson, I., The coradical filtration for quantized enveloping algebras. J. Lond. Math. Soc. 53 (1996), 50–62.CrossRefGoogle Scholar
15.Doi, Y., Braided bialgebras and quadratic algebras, Commun. Algebra 21 (1993), 1731–1749.CrossRefGoogle Scholar
16.Drinfeld, V. G., Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), 1419–1457.Google Scholar
17.Enock, M. and Vainerman, L., Deformation of a Kac algebra by an abelian subgroup, Commun. Math. Phys. 178 (1996), 571–596.CrossRefGoogle Scholar
18.de la Harpe, P., Topics in geometric group theory (University of Chicago Press, Chicago, 2000).Google Scholar
19.Klimyk, A. and Schmüdgen, K., Quantum groups and their representations, texts and monographs in physics (Springer-Verlag, Berlin, 1997).CrossRefGoogle Scholar
20.Kustermans, J. and Vaes, S., Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92 (2003), 68–92.CrossRefGoogle Scholar
21.Montgomery, S., Hopf algebras and their actions on rings, Amer. Math. Soc. (1993).CrossRefGoogle Scholar
22.Müller, E., The coradical filtration of Uq() at roots of unity, Commun. Algebra 28 (2000), 1029–1044.CrossRefGoogle Scholar
23.Nikshych, D., K 0-rings and twisting of finite-dimensional semisimple Hopf algebras, Commun. Algebra 26 (1998), 321–342.CrossRefGoogle Scholar
24.Schauenburg, P., Hopf bigalois extensions, Comm. Algebra 24 (1996), 3797–3825.CrossRefGoogle Scholar
25.Schauenburg, P., Faithful flatness over Hopf subalgebras: Counterexamples, Lecture Notes Pure Appl. Math. (Dekker) 210 (2000), 331–344.Google Scholar
26.Sweedler, M., Hopf algebras (Benjamin, New York, 1969).Google Scholar
27.Vaes, S., Strictly outer actions of groups and quantum groups, J. Reine Angew. Math. 578 (2005), 147–184.Google Scholar
28.Zakrzewski, S., Matrix pseudogroups associated with anti-commutative plane, Lett. Math. Phys. 21 (1991), 309–321.CrossRefGoogle Scholar