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Glueing operation for R-matrices, quantum groups and link-invariants of Hecke type

Published online by Cambridge University Press:  24 October 2008

Shahn Majid  and
Martin Markl
Shahn Majid
Affiliation:
Department of Applied Mathematics & Theoretical Physics, University of Cambridge, CB3 9EW. e-mail: majid@damtp.cambridge.ac.uk
Martin Markl
Affiliation:
Mathematical Institute of the Academy, Zitná 25, 115 67 Prague, Czech Republic. e-mail: markl@earn.cvut.cz
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Abstract

We introduce an associative glueing operation ⊕q on the space of solutions of the Quantum Yang–Baxter Equations of Hecke type. The corresponding glueing operations for the associated quantum groups and quantum vector spaces are also found. The former involves 2×2 quantum matrices whose entries are themselves square or rectangular quantum matrices. The corresponding glueing operation for link-invariants is introduced and involves a state-sum model with Boltzmann weights determined by the link invariants to be glued. The standard su(n) solution, its associated quantum matrix group, quantum space and link-invariant arise at once by repeated glueing of the one-dimensional case.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 1 , January 1996 , pp. 139 - 166
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[1]Deligne, P.. Catégories Tannakiènnes (Preprint, 1988).Google Scholar
[2]Drinfeld, V. G.. Quantum groups: in Proceedings of the ICM (ed. Gleason, A.). pp. 798–820 (AMS, 1987).Google Scholar
[3]Faddeev, L. D., Reshetikhin, N. Yu. and Takhtajan, L. A.. Quantization of Lie groups and Lie algebras. Leningrad Math. J. 1 (1990), 193–225.Google Scholar
[4]Freyd, P. and Yetter, D.. Braided compact closed categories with applications to low dimensional topology. Adv. Math. 77 (1989), 156–182.CrossRefGoogle Scholar
[5]Ge, M.-L., Jing, N. and Wu, Y.-S.. A new quantum group associated with ‘non-standard’ braid group representation. Lett. Math. Phys. 21 (1991), 193–203.Google Scholar
[6]Gurevich, D. I.. Hecke Symmetry and quantum determinants. Sov. Math. Dokl. 38 (1989), 555–559.Google Scholar
[7]Gurevich, D. I.. Algebraic aspects of the quantum Yang-Baxter equation. Leningrad Math. J. 2 (1991), 801–828.Google Scholar
[8]Hilton, P. J. and Stammbach, U. S.. A Course in homological algebra (Springer-Verlag, 1980).Google Scholar
[9]Joyal, A. and Street, R.. Braided monoidal categories. Mathematics Reports 86008 (Macquarie University, 1986).Google Scholar
[10]Joyal, A. and Street, R.. Tortile Yang-Baxter operators in tensor categories. J. Pure Applied Algebra 71 (1991), 43–51.Google Scholar
[11]Kauffman, L.. Knots and Physics, vol. 1 of Knots and everything (World Sci., 1991).Google Scholar
[12]Lyubashenko, V. V.. Hopf algebras and vector symmetries. Russ. Math. Surveys 41 (1986), 153–154.CrossRefGoogle Scholar
[13]Lyubashenko, V. V.. Superanalysis and solutions to the triangles equation (Ph.D. thesis, Kiev, 1986).Google Scholar
[14]Majid, S.. Algebras and Hopf algebras in braided categories. Advances in Hopf algebras Lect. Notes in Pure and Applied Math. 158 (1994), 55–105 (Marcel Dekker).Google Scholar
[15]Majid, S.. Anyonic quantum groups. In Spinors, twistors, Clifford algebras and quantum deformations (ed. Oziewicz, Z. et al. ), pp. 327–336 (Kluwer, 1992).Google Scholar
[16]Majid, S.. Braided momentum in the q–Poincaré group. J. Math. Phys. 34 (1993), 2045–2058.CrossRefGoogle Scholar
[17]Majid, S.. Examples of braided groups and braided matrices. J. Math. Phys. 32 (1991), 3246–3253.Google Scholar
[18]Majid, S.. Free braided differential calculus, braided binomial theorem and the braided exponential map. J. Math. Phys. 34 (1993), 4843–4856.CrossRefGoogle Scholar
[19]Majid, S.. More examples of bicross product and double cross product Hopf algebras. Isr. J. Math. 72 (1990), 133–148.Google Scholar
[20]Majid, S.. Quasitriangular Hopf algebras and Yang-Baxter equations. Int. J. Modern Physics A 5(1) (1990), 1–91.CrossRefGoogle Scholar
[21]Majid, S.. Quantum and braided linear algebra. J. Math. Phys. 34 (1993), 1176–1196.Google Scholar
[22]Reshetikhin, N. Yu. and Turaev, V. G.. Ribbon graphs and their invariants derived from quantum groups. Comm. Math. Phys. 127 (1990), 1–26.Google Scholar
[23]Rivano, N. Saavedra. Catégories Tannakiennes. Lect. Notes in Math. 265 (Springer, 1972).Google Scholar
[24]Yetter, D. N.. Framed tangles and a theorem of Deligne on braided deformations of Tannakian categories. Contemp. Mathematics 134 (1992), 325–349.Google Scholar