Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-06-01T20:01:04.233Z Has data issue: false hasContentIssue false
  • English
  • Français

The Representation Ring and the Centre of a Hopf Algebra

Published online by Cambridge University Press:  20 November 2018

Sarah J. Witherspoon
Sarah J. Witherspoon*
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 3G3 email: sjw@math.toronto.edu,
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.

When $H$ is a finite dimensional, semisimple, almost cocommutative Hopf algebra, we examine a table of characters which extends the notion of the character table for a finite group. We obtain a formula for the structure constants of the representation ring in terms of values in the character table, and give the example of the quantum double of a finite group. We give a basis of the centre of $H$ which generalizes the conjugacy class sums of a finite group, and express the class equation of $H$ in terms of this basis. We show that the representation ring and the centre of $H$ are dual character algebras (or signed hypergroups).

Keywords

16W3020N20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 4 , 01 August 1999 , pp. 881 - 896
Copyright
Copyright © Canadian Mathematical Society 1999

References

[1] Bannai, E. and Ito, T., Algebraic Combinatorics I. Association Schemes. Benjamin/Cummings, 1984.Google Scholar
[2] Bantay, P., Orbifolds, Hopf algebras, and the moonshine. Lett. Math. Phys. 22(1991), 187–194.Google Scholar
[3] Blau, H. I., Quotient structures in C-algebras. J. Algebra 175(1995), 24–64.Google Scholar
[4] Chari, V. and Pressley, A., A Guide to Quantum Groups. Cambridge University Press, 1994.Google Scholar
[5] Curtis, C. W. and Reiner, I., Methods of Representation Theory with Applications to Finite Groups and Orders, Volume I. Wiley, 1981.Google Scholar
[6] Dijkgraaf, R., Pasquier, V., and Roche, P., Quasi Hopf algebras, group cohomology and orbifold models. Nuclear Physics B (Proc. Suppl.) 18B(1990), 60–72.Google Scholar
[7] Kac, G. I., Certain arithmetic properties of ring groups. Funct. Anal. Appl. 6(1972), 158–160.Google Scholar
[8] Larson, R. G., Characters of Hopf algebras. J. Algebra 17(1971), 352–368.Google Scholar
[9] Larson, R. G. and Sweedler, M. E., An associative orthogonal bilinear form for Hopf algebras. Amer. J. Math. 91(1969), 75–94.Google Scholar
[10] Lorenz, M., Representations of finite-dimensional Hopf algebras. J. Algebra 188(1997), 476–505.Google Scholar
[11] Lorenz, M., On the class equation for Hopf algebras. Proc. Amer. Math. Soc. (10) 126(1998), 2841–2844.Google Scholar
[12] Lusztig, G., Leading coefficients of character values of Hecke algebras. Proc. Sympos. Pure Math. 47(1987), 235–262.Google Scholar
[13] Mason, G. E., The quantum double of a finite group and its role in conformal field theory. In: Groups ‘93, Galway/St. Andrews, London Math. Soc. Lecture Note Ser. 212, Cambridge University Press, 1995, 405–417.Google Scholar
[14] Masuoka, A., Selfdual Hopf algebras of dimension p3 obtained by extension. J. Algebra 178(1995), 791–806.Google Scholar
[15] Masuoka, A., The pn-theorem for semisimple Hopf algebras. Proc. Amer.Math. Soc. 124(1996), 735–737.Google Scholar
[16] Montgomery, S., Hopf Algebras and Their Actions on Rings. CBMS Regional Conference Series inMathematics 82, Amer. Math. Soc., 1993.Google Scholar
[17] Nichols, W. D. and Richmond, M. B., The Grothendieck group of a Hopf algebra. J. Pure Appl. Algebra 106(1996), 297–306.Google Scholar
[18] Nichols, W. D. and Richmond, M. B., The Grothendieck algebra of a Hopf algebra, I. Comm. Algebra (4) 26(1998), 1081–1095.Google Scholar
[19] Radford, D. E., Minimal quasitriangular Hopf algebras. J. Algebra 157(1993), 281–315.Google Scholar
[20] Radford, D. E., The trace function and Hopf algebras. J. Algebra 163(1994), 583–622.Google Scholar
[21] Schneider, H.-J., Lectures on Hopf Algebras. Lecture Notes, Universidad Nacional de Cordoba, 1995.Google Scholar
[22] Sommerhäuser, Y., On Kaplansky's fifth conjecture. J. Algebra (1) 204(1998), 202–224.Google Scholar
[23] Waerden, van der, Moderne Algebra II. Springer-Verlag, 1931.Google Scholar
[24] Wildberger, N. J., Finite commutative hypergroups and applications from group theory to conformal field theory. In: Applications of Hypergroups and Related Measure Algebras, Contemp. Math. 183, Amer.Math. Soc., 1995, 413–434.Google Scholar
[25] Witherspoon, S. J., The representation ring of the quantum double of a finite group. J. Algebra 179(1996), 305–329.Google Scholar
[26] Zhu, Y., Hopf algebras of prime dimension. Internat.Math. Res. Notices 1(1994), 53–59.Google Scholar