Abstracts Mathematics

Using the Reshetikhin-Turaev Link Invariant Approach with Non-Semisimple Categories

by Adam Robertson

Invariants of knots and links are useful because they give rise to invariants of 3-manifolds. In particular, combinatorial link invariants give rise to combinatorial invariants of 3-manifolds, which are hard to come by using traditional methods from classical topology. The Reshetikhin–Turaev approach, which is based in quantum topology, develops link invariants using semisimple ribbon categories. However, a large class of algebraically interesting ribbon categories are non-semisimple and so give trivial link invariants via the Reshetikhin–Turaev method. We modify the Reshetikhin–Turaev method to make it suitable for non-semisimple ribbon categories. We discuss explicitly the following three examples: semisimple modules for the abelian quantum group, non-semisimple modules for Uq(gl(1|1)), and non-semisimple modules for the unrolled quantum group of sl2(C).