Quantum groups and ribbon G-categories

✍ Scribed by Thang Le; Vladimir Turaev

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 198 KB
• Volume: 178
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(02)00191-3

✦ Synopsis

For a group G, the notion of a ribbon G-category was introduced in Turaev (Homotopy ÿeld theory in dimension 3 and crossed group-categories, preprint, math. GT/0005291) with a view towards constructing 3-dimensional homotopy quantum ÿeld 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 Uq(g) at a root of unity . Under certain assumptions on , 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.