Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-27T22:40:57.388Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantum double finite group algebras and their representations

Published online by Cambridge University Press:  17 April 2009

M.D. Gould
M.D. Gould
Affiliation:
Department of Mathematics, The University of Queensland, Queensland 4072, Australia
Rights & Permissions [Opens in a new window]

Extract

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.

The quantum double construction is applied to the group algebra of a finite group. Such algebras are shown to be semi-simple and a complete theory of characters is developed. The irreducible matrix representations are classified and applied to the explicit construction of R-matrices: this affords solutions to the Yang-Baxter equation associated with certain induced representations of a finite group. These results are applied in the second paper of the series to construct unitary representations of the Braid group and corresponding link polynomials.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 48 , Issue 2 , October 1993 , pp. 275 - 301
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Akutsu, Y. and Wadati, M., ‘Knots, links, braids and exactly solvable models in statistical mechanics’, Comm. Math. Phys. 117 (1988), 243–259.CrossRefGoogle Scholar
[2]Alvarez-Gaumé, L., Gomes, C. and Sierra, G., ‘Duality and quantum groups’, Phys. Lett. B 220 (1989), 142–150.CrossRefGoogle Scholar
[3]Baxter, R.J., Exactly solved models in statistical mechanics (Academic press, 1982).Google Scholar
[4]Baxter, R.J., ‘The inversion relation method for some two-dimensional exactly solved models in lattice statistics’, J. Statist. Phys. 28 (1982), 1–14.CrossRefGoogle Scholar
[5]Curtis, C.W. and Reiner, I., Representation theory of finite groups and associative algebras (Wiley, New York, 1962).Google Scholar
[6]Drinfeld, V.G., ‘Quantum groups’, in Proc. ICM Berkeley 1, 1986, pp. 798–820.Google Scholar
[7]Faddeev, L.D., Reshetikhin, N. Yu. and Takhtajan, L.A., ‘Quantum groups’, Algebra and analysis 1 (1988), 129–139.Google Scholar
[8]Gould, M.D., Zhang, R.B. and Bracken, A.J., ‘Generalized Gelfand invariants and characteristic identities for Quantum groups’, J. Math. Phys. 32 (1991), 2298–2303.CrossRefGoogle Scholar
[9]Gould, M.D., Zhang, R.B. and Bracken, A.J., ‘Quantum double construction for graded Hopf algebras’, Bull. Austral. Math. Soc. 47 (1993), 353–375.CrossRefGoogle Scholar
[10]Hamermesh, M., Group theory (Addison-Wesley, Cambridge MA, 1962).Google Scholar
[11]Jimbo, M., ‘A q–difference analogue of U(g) and the Yang-Baxter equation’, Lett. Math. Phys. 10 (1985), 63–69.CrossRefGoogle Scholar
[12]Kulish, P.P. and Sklyanin, E.K., Integrable Quantum Field Theories, Lecture notes in Physics 151 (Springer-Verlag, New York, 1982).Google Scholar
[13]Larson, R.G., ‘Characters of Hopf algebras’, J. Algebra 17 (1971), 352–368.CrossRefGoogle Scholar
[14]Links, J.R. and Gould, M.D., ‘Casimir invariants for Hopf algebras’, Rep. Math. Phys. (to appear).Google Scholar
[15]Lusztig, G., ‘Leading coefficients of character values of Hecke algebras’, Proc. Sympos. Pure Math. 47 (1987), 235–262.CrossRefGoogle Scholar
[16]Sweedler, M.E., Hopf algebras (Benjamin, New York, 1969).Google Scholar
[17]Temperley, H.N.V. and Lieb, E.H., ‘Relations between the Perculation and colouring problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the perculation problem’, Proc. Roy. Soc. A332 (1971), 251–270.Google Scholar
[18]Turaev, V.G., ‘The Yang-Baxter equation and invariant of links’, Invent. Math. 92 (1988), 527–553.CrossRefGoogle Scholar
[19]Witten, E., ‘Quantum field theorey and the Jones polynomial’, Comm. Math. Phys. 121 (1989), 351–399.CrossRefGoogle Scholar
[20]Reshetikhin, N. Yu., ‘Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I, II’, L.O.M.I. Preprints (Leningrad) E-4–87, E-17–87.Google Scholar