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Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity

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An example of a finite dimensional factorizable ribbon Hopf ℂ-algebra is given by a quotientH=u q (g) of the quantized universal enveloping algebraU q (g) at a root of unityq of odd degree. The mapping class groupM g,1 of a surface of genusg with one hole projectively acts by automorphisms in theH-moduleH *⊗g, ifH * is endowed with the coadjointH-module structure. There exists a projective representation of the mapping class groupM g,n of a surface of genusg withn holes labeled by finite dimensionalH-modulesX 1, ...,X n in the vector space Hom H (X 1 ⊗ ... ⊗X n ,H *⊗g). An invariant of closed oriented 3-manifolds is constructed. Modifications of these constructions for a class of ribbon Hopf algebras satisfying weaker conditions than factorizability (including most ofu q (g) at roots of unityq of even degree) are described.

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  1. Department of Mathematics, University of York, YO1 5DD, Heslington, York, England, U.K.

    Volodymyr V. Lyubashenko

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  1. Volodymyr V. Lyubashenko
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Communicated by G. Felder

This work was supported in part by the EPSRC research grant GR/G 42976.

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Lyubashenko, V.V. Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity. Commun.Math. Phys. 172, 467–516 (1995). https://doi.org/10.1007/BF02101805

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