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

TL;DR

This paper constructs invariants of 3-manifolds and projective representations of mapping class groups using quantum groups at roots of unity, extending to weaker algebraic conditions and motivated by conformal field theory.

Contribution

It introduces new invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity, including cases with weaker algebraic conditions.

Findings

Constructed invariants of closed oriented 3-manifolds.

Established projective representations of mapping class groups.

Extended constructions to weaker Hopf algebra conditions.

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.