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Lattice analogues of $$W$$-algebras and classical integrable equations. (English) Zbl 0905.17032
Summary: We propose a regular way to construct lattice versions of $$W$$-algebras, both for the quantum and the classical case. In the classical case we write the algebra explicitly and derive the lattice analogue of the Boussinesq equation from the Hamiltonian equations of motion. The connection between the lattice Faddeev-Takhtadjan-Volkov algebra and the $$q$$-deformed Virasoro algebra is also discussed.

##### MSC:
 17B68 Virasoro and related algebras 17B37 Quantum groups (quantized enveloping algebras) and related deformations 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 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
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