Q-Deformation of sl(2,ℂ) × ZN and Link Invariants
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 ℛ-matrix was given, but the bstract form of the antipode of the algebra was not.
KeywordsHopf Algebra Quantum Group Open String Reidemeister Move Link Invariant
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