Quantum groups and ribbon G-categories

Abstract

For a group G, the notion of a ribbon G-category was introduced in Turaev (Homotopy field theory in dimension 3 and crossed group-categories, preprint, math. GT/0005291) with a view towards constructing 3-dimensional homotopy quantum field theories (HQFTs) with target K( G,1). We discuss here how to derive ribbon G-categories from a simple complex Lie algebra g where G is the center of g . Our construction is based on a study of representations of the quantum group U q( g) at a root of unity ε. Under certain assumptions on ε, the resulting G-categories give rise to numerical invariants of pairs (a closed oriented 3-manifold M, an element of H 1( M; G)) and to 3-dimensional HQFTs.