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Topological quantum field theory and invariants of graphs for quantum groups. (English) Zbl 0828.57012

Summary: On the basis of generalized \(6j\)-symbols we give a formulation of topological quantum field theories for 3-manifolds including observables in the form of coloured graphs. It is shown that the \(6j\)-symbols associated with deformations of the classical groups at primitive even roots of unity provide examples of this construction. Calculational methods are developed which, in particlar, yield the dimensions of the state spaces as well as a rather simple proof of the relation, previously found by Turaev and Walker for the case of \(U_q (\text{sl} (2, \mathbb{C}))\), between these models and corresponding ones based on the ribbon graph construction of Reshetikhin and Turaev.

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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