Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity |
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Authors: | Volodymyr V Lyubashenko |
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Institution: | (1) Department of Mathematics, University of York, YO1 5DD Heslington, York, England, U.K. |
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Abstract: | An example of a finite dimensional factorizable ribbon Hopf -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.This work was supported in part by the EPSRC research grant GR/G 42976. |
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Keywords: | |
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