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 quotient H=u_q(g) of the quantized universal enveloping algebra U_q(g) at a root of unity q of odd degree. The mapping class group M_{g,1} of a surface of genus g with one hole projectively acts by automorphisms in the H-module H^{*\otimes g}, if H^* is endowed with the coadjoint H-module structure. There exists a projective representation of the mapping class group M_{g,n} of a surface of genus g with n holes labelled by finite dimensional H-modules X_1,...,X_n in the vector space Hom_H(X_1\otimes...\otimes X_n,H^{*\otimes 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 of u_q(g) at roots of unity q of even degree) are described. The results are motivated by CFT.

61 pages, 12 eps figures. Hennings' invariant is a particular case of the described invariant; references added

References in corpus (4)

• Mapping Class Group Actions on Quantum Doubles
• Fusion rules of chiral algebras
• Topological Representations of $U_q(s{l}_2)$ on the Torus and the Mapping Class Group
• Invariants of 3-Manifolds Derived From Finite Dimensional Hopf Algebras

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