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Infinite symmetry of the spin systems with inverse square interactions. (English) Zbl 0972.82506
Summary: A 1-dimensional quantum particle system with \(\text{SU}(\nu)\) spins interacting through inverse square interactions is studied. We reveal the algebraic structure of the system: a hidden symmetry is the \(\text{U}(\nu)\cong \text{SU}(\nu)\otimes \text{U}(1)\) current algebra. This is consistent with the fact that the ground state wave function is a solution of the Knizhnik-Zamolodchikov equation. Furthermore we show that the system has a higher symmetry, which is the \(w_{1+\infty}\) algebra. With this \(W\)-algebra we can clarify simultaneously the structures of the Calogero type \((1/x^2\)-interactions) and Sutherland type \((1/\sin^2 x\)-interactions). The Yangian symmetry is briefly discussed.

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81R12 Groups and algebras in quantum theory and relations with integrable systems
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