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

Abstract

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.