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Conformal field algebras with quantum symmetry from the theory of superselection sectors. (English) Zbl 0715.17028

From the paper: “According to the theory of superselection sectors of [S. Doplicher, R. Haag and J. E. Roberts, Local observables and particle statistics. I, II, Commun. Math. Phys. 23, 199- 230 (1971) and 35, 49-85 (1974)], field operators which make transitions between different superselection sectors, i.e. different irreducible representations of the observable algebra, are to be constructed by adjoining localized endomorphisms to the algebra of local observables. We find the relevant endomorphisms of the chiral algebra of observables in the minimal conformal model with central charge \(c=\) (Ising model). We show by explicit and elementary construction how they determine a representation of the braid group \(B_{\infty}\) which is associated with a Temperley-Lieb-Jones algebra. We recover fusion rules, and compute the quantum dimensions of the superselection sectors. We exhibit a field algebra which is quantum group covariant and acts in the Hilbert space of physical states. It obeys local braid relations in an appropriate weak sense.”
Reviewer: A.N.Pressley

MSC:

17B81 Applications of Lie (super)algebras to physics, etc.
81T05 Axiomatic quantum field theory; operator algebras
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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