Kirby Elements and Quantum Invariants

We define the notion of a Kirby element of a ribbon category C (not necessarily semisimple). Kirby elements lead to 3-manifold invariants. We characterize a class of Kirby elements, the algebraic Kirby elements , in terms of the structure maps of a Hopf algebra in C. This class is sufficiently large to recover the quantum invariants of 3-manifolds of Reshetikhin and Turaev, of Hennings, Kauffman and Radford, and of Lyubashenko when these are well defined. The cases of a semisimple ribbon category and of a category of representations are explored in detail.