Kirby elements and quantum invariants

Abstract

We define the notion of a Kirby element of a ribbon category C (not necessarily semisimple). Kirby elements lead to 3-manifolds invariants. We characterize (in terms of the structure maps of some categorical Hopf algebra) a set of Kirby elements of C which is sufficiently large to recover the known quantum invariants computed from C. The cases of a semisimple ribbon category and of a category of representations are explored in details.