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KIRBY ELEMENTS AND QUANTUM INVARIANTS

Published online by Cambridge University Press:  07 August 2006

ALEXIS VIRELIZIER
Affiliation:
Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720-3840, USAvirelizi@math.berkeley.edu
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Abstract

We define the notion of a Kirby element of a ribbon category $\mathcal{C}$ (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 $\mathcal{C}$. 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.

Keywords

Type
Research Article
Copyright
2006 London Mathematical Society

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