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Kirby Elements and Quantum Invariants
Author(s) -
Virelizier Alexis
Publication year - 2006
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611506015905
Subject(s) - ribbon , mathematics , hopf algebra , pure mathematics , class (philosophy) , element (criminal law) , algebra over a field , quantum , quantum group , algebraic structure , manifold (fluid mechanics) , combinatorics , geometry , physics , computer science , quantum mechanics , artificial intelligence , political science , law , mechanical engineering , engineering
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. 2000 Mathematics Subject Classification 57M27, 18D10, 81R50.
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