Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-26T10:05:56.025Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantum groups and representations of monoidal categories

Published online by Cambridge University Press:  24 October 2008

David N. Yettera
David N. Yettera
Affiliation:
Department of Mathematics, Ohio State University, 1680 University Dr., Mansfield, OH 44907, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

This paper is intended to make explicit some aspects of the interactions which have recently come to light between the theory of classical knots and links, the theory of monoidal categories, Hopf-algebra theory, quantum integrable systems, the theory of exactly solvable models in statistical mechanics, and quantum field theories. The main results herein show an intimate relation between representations of certain monoidal categories arising from the study of new knot invariants or from physical considerations and quantum groups (that is, Hopf algebras). In particular categories of modules and comodules over Hopf algebras would seem to be much more fundamental examples of monoidal categories than might at first be apparent. This fundamental role of Hopf algebras in monoidal categories theory is also manifest in the Tannaka duality theory of Deligne and Mime [8a], although the relationship of that result and the present work is less clear than might be hoped.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 2 , September 1990 , pp. 261 - 290
Copyright
Copyright © Cambridge Philosophical Society 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Abe, E.. Hopf Algebras (Cambridge University Press, 1977).Google Scholar
[2]Akutsu, V. and Wadati, M.. Exactly solvable models and new link polynomials, I. J. Phys. Soc. Japan 56 (1987), 3039–3051.CrossRefGoogle Scholar
[3]Akutsu, V., Deguchi, T. and Wadati, M.. Exactly solvable models and new link polynomials, II–IV. J. Phys. Soc. Japan 56 (1987), 3464–3479CrossRefGoogle Scholar
[3]Akutsu, V., Deguchi, T. and Wadati, M.. Exactly solvable models and new link polynomials, II–IV. J. Phys. Soc. Japan 57 (1988), 757–776CrossRefGoogle Scholar
[3]Akutsu, V., Deguchi, T. and Wadati, M.. Exactly solvable models and new link polynomials, II–IV. J. Phys. Soc. Japan 57 (1988), 1173–1185.CrossRefGoogle Scholar
[4]Atiyah, M., Hitchin, N., Lawrence, R. and Segal, G.. Notes on the Oxford seminar on Jones–Witten theory (Michaelmas Term 1988).Google Scholar
[5]Baxter, R. I.. The partition function of the eight-vertex lattice model. Ann. Physics 70 (1972), 193–228.CrossRefGoogle Scholar
[6]Brustein, R., Ne'eman, V. and Sternberg, S.. Duality, crossing and Mac Lane's coherence. (Preprint.)Google Scholar
[7]Carboni, A.. Matrices, relations and group representations. (Preprint, 1988.)Google Scholar
[8]Deligne, P.. (Private communication.)Google Scholar
[8a]Deligne, P. and Milne, J.. Tannakian categories. In Hodge Cycles, Motives and Shimura Varieties, Lectures Notes in Math. vol. 900 (Springer-Verlag, 1982).CrossRefGoogle Scholar
[9]Drinfel'd, V. G.. Quantum groups. J. Soy. Math. 41 (2) (1988), 898–915.CrossRefGoogle Scholar
[10]Freyd, P. J. and Yetter, D. N.. Braided compact closed categories with applications to low-dimensional topology. Adv. in Math. 77 (1989), 156–182.CrossRefGoogle Scholar
[11]Freyd, P. J. and Yetter, D. N.. Coherence theorems via knot theory. J. Pure Appl. Algebra (To appear.)Google Scholar
[11]Joyal, A.. Lecture at McGill University (Autumn, 1987).Google Scholar
[12]Joyal, A. and Street, R.. Braided tensor categories. (Preprint.)Google Scholar
[13]Joyal, A. and Street, R.. Planar diagrams and tensor algebra. (Preprint.)Google Scholar
[14]Kauffman, L.. An invariant of regular isotopy. Trans. Amer. Math. Soc. (To appear.)Google Scholar
[15]Kauffman, L.. Statistical Mechanics and the Jones Polynomial. In Braids (eds. Birman, J. S. and Libgober, A.), Contemp. Math. vol. 78 (American Mathematical Society, 1988).CrossRefGoogle Scholar
[16]Kelly, G. M. and Laplaza, M. L.. Coherence for compact closed categories. J. Pure Appl. Algebra 19 (1980), 193–213.CrossRefGoogle Scholar
[17]Kirby, R.. A calculus for framed links in S 3. Invent. Math. 45 (1978), 35–36.CrossRefGoogle Scholar
[18]Kulish, P. P. and Sklyanin, E. K.. Solutions of the Yang–Baxter equation. Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov 95 (1980), 129–160. (in Russian).Google Scholar
[19]Lyubashenko, V. V.. Hopf algebras and vector symmetries. Russ. Math. Surveys 41 (1986), 153–154.CrossRefGoogle Scholar
[20]Mac Lane, S.. Categories for the Working Mathematician (Springer-Verlag, 1971).CrossRefGoogle Scholar
[21]Mac Lane, S.. Natural associativity and commutativity. Rice Univ. Stud. 49 (1963), 28–46.Google Scholar
[21a]Majid, S.. Doubles of quasitriangular Hopf algebras. (Preprint.)Google Scholar
[22]Manin, Yu. I.. Quantum groups and non-commutative geometry (Université de Montreal, 1988).Google Scholar
[23]Moore, G. and Seiberg, N.. Classical and quantum conformal field theory. (Preprint.)Google Scholar
[24]Penrose, R.. Applications of negative dimensional tensors. In Combinatorial Mathematics and its Applications (ed. Welsh, D. J. A.), (Academic Press, 1971), pp. 221–244.Google Scholar
[25]Reidemeister, K.. Knot Theory (B.S.C. Associates, 1983)Google Scholar
Reidemeister, K.. Knot Theory (translation of Knotentheorie (Springer-Verlag, 1932)).Google Scholar
[26]Segal, G.. The definition of conformal theory. (Preprint.)Google Scholar
[27]Shum, M.-C.. Tortile Tensor Categories. Ph.D. thesis, Macquarie University (1989).Google Scholar
[28]Street, R. S. (Private communication.)Google Scholar
[29]Sweedler, M. E.. Hopf Algebras (Benjamin, 1969).Google Scholar
[30]Thakhtakzhan, L. A. and Faddeev, L. B.. The quantum method of the inverse problem and the Heisenberg XYZ model. Russian Math. Surveys 34 (1979), 11–68.CrossRefGoogle Scholar
[31]Witten, E.. Quantum field theory and the Jones polynomial. (Preprint.)Google Scholar
[32]Yang, C. N.. Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett. 19 (1967), 1312–1315.CrossRefGoogle Scholar