Communications in Mathematical Physics

, Volume 168, Issue 2, pp 353–388 | Cite as

Mappin class group actions on quantum doubles

  • Thomas Kerler


We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non-semisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT's is circumvented. We find compact formulae for theS±1-matrices using the canonical, non-degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford's relations between the canonical forms and the moduli of integrals. We analyze the projectiveSL(2, Z)-action on the center ofUq(sl2) forq anl=2m+1st root of unity. It appears that the 3m+1-dimensional representation decomposes into anm+1-dimensional finite representation and a2m-dimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation ofSL(2, Z) and the finite,m-dimensional representation, obtained from the truncated TQFT of the semisimplified representation category ofUq(sl2).


Irreducible Representation Canonical Form Hopf Algebra Class Group Dimensional Representation 
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Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Thomas Kerler
    • 1
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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