1993/01/29 by Sanjaye Ramgoolam, Ramgoolam, Sanjaye
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Quantum Algebra (math.QA) #hep-th #math.QA
paper · pdf · doi:10.48550/arxiv.hep-th/9301121
30 pages (minor typos corrected, refs added)
openalex publication_date 1993/01/29 · arxiv created 1993/02/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
We show that the Hopf link invariants for an appropriate set of finite dimensional representations of Uq SL(2) are identical, up to overall normalisation, to the modular S matrix of Kac and Wakimoto for rational k \widehat sl(2) representations. We use this observation to construct new modular Hopf algebras, for any root of unity q=e-iπm/r, obtained by taking appropriate quotients of Uq SL(2), that give rise to 3-manifold invariants according to the approach of Reshetikin and Turaev. The phase factor correcting for the `framing anomaly' in these invariants is equal to e^- i π \over 4 ( 3k \over k+2), an analytic continuation of the anomaly at integer k. As expected, the Verlinde formula gives fusion rule multiplicities in agreement with the modular Hopf algebras. This leads to a proposal, for (k+2)=r/m rational with an odd denominator, for a set of \widehat sl(2) representations obtained by dropping some of the highest weight representations in the Kac-Wakimoto set and replacing them with lowest weight representations. For this set of representations the Verlinde formula gives non-negative integer fusion rule multiplicities. We discuss the consistency of the truncation to highest and lowest weight representations in conformal field theory.