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Q-Deformation of sl(2,ℂ) × ZN and Link Invariants

  • H. C. Lee
Part of the NATO ASI Series book series (NSSB, volume 245)

Abstract

It is by now well known1–3 that for each simple Lie algebra or Kac-Moody algebra g there is a quantum group U p(g), and that for each representation of this group, a knot invariant of the Jones type can be defined. Thus the Jones polynomial4 corresponds to the 2 × 2 matrix representation of U p(sl(2, ℂ) and the Akutsu-Wadati polynomial5, the 3 × 3 matrix representation. Interestingly enough, the oldest knot polynomial, the Alexander-Conway polynomial6, is yet to be identified with a quantum group. On the other hand, it is already known that the Alexander-Conway polynomial is given by a two-state solution of the quantum Yang-Baxter equation. Recently, Kauffman7 derived the polynomial from a state model, and discovered a bialgebra associated with it. Lee has shown that the polynomial at least is associated with a pseudo Hopf algebra — the representation for the universal -matrix was given, but the bstract form of the antipode of the algebra was not.

Keywords

Hopf Algebra Quantum Group Open String Reidemeister Move Link Invariant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • H. C. Lee
    • 1
    • 2
  1. 1.Theoretical Physics Branch, Chalk River Nuclear LaboratoriesAtomic Energy of Canada Limited Research CompanyChalk RiverCanada
  2. 2.Department of Applied MathematicsUniversity of Western OntarioLondonCanada

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