Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-25T12:59:27.028Z Has data issue: false hasContentIssue false

Quantum double construction for graded Hopf algebras

Published online by Cambridge University Press:  17 April 2009

M.D. Gould
Affiliation:
Department of Mathematics, The University of Queensland, Queensland 4072, Australia
R.B. Zhang
Affiliation:
Department of Mathematics, The University of Queensland, Queensland 4072, Australia
A.J. Bracken
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.

A detailed proof of the quantum double construction is given for Z2 -graded Hopf algebras, and an explicit formula for the graded universal R−matrix is obtained in a general fashion.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Abe, E., Hopf algebras (Cambridge University Press, 1980).Google Scholar
[2]Bracken, A.J., Gould, M.D. and Zhang, R.B., ‘Quantum supergroups and solutions of the Yang-Baxter equation’, Modern Phys. Lett. A 5 (1990), 831840.CrossRefGoogle Scholar
[3]Chaichian, M. and Kulish, P.P., ‘Quantum Lie superalgebras and q−oscillators’, Phys. Lett. B 234 (1990), 7280.CrossRefGoogle Scholar
[4]Deguchi, T., Fujii, A., and Ito, K., ‘Quantum superalgebra U q osp (2,2)’, Phys. Lett. B 238 (1990), 242246.CrossRefGoogle Scholar
[5]Drinfeld, V.G., ‘Quantum groups’, Proc. I.C.M. Berkeley 1 (1986), 798820.Google Scholar
[6]Jimbo, M., ‘A q−difference analogue of U(g) and the Yang-Baxter equation’, Lett. Math. Phys 10 (1985), 6369.CrossRefGoogle Scholar
[7]Kirillov, A.M. and Reshetikhim, N.Yu., ‘q−Weyl group and a multiplicative formula for universal R−matrices’, (preprint).Google Scholar
[8]Kulish, P.P. and Reshetikhin, N.Yu., ‘Universal R−matrix of the quantum superalgebra osp (211)’, Lett.Math. Phys. 18 (1989), 143149.CrossRefGoogle Scholar
[9]Majid, S., ‘Physics for algebraists’, J. Algebra 130 (1990), 1764.CrossRefGoogle Scholar
[10]Milner, J.W. and More, J.C., ‘On the structure of Hopf algebras’, Ann. Math. 81 (1965), 211264.CrossRefGoogle Scholar
[11]Reshetikhin, N. Yu., ‘Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links’ (L.O.M.I., Leningrad).Google Scholar
[12]Rosso, M., ‘An analogue of PBW theorem and the universal R−matrix for U hsl(n + 1)’, Comm. Math. Phys. 124 (1990), 307318.CrossRefGoogle Scholar
[13]Saleur, H., ‘Quantum osp (1/2) and solutions of the graded Yang-Baxter equation’, Nuclear Phys. B 336 (1990), 363372.CrossRefGoogle Scholar
[14]Sweedler, M.E., Hopf algebras (Benjamin, New York, 1969).Google Scholar
[15]Zhang, R.B., Gould, M.D. and Bracken, A.J., ‘Solutions of the graded classical Yang-Baxter equation and integrable models’, J. Phys. A 24 (1991), 11851197.CrossRefGoogle Scholar