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On \(q\)-deformed infinite-dimensional \(n\)-algebra. (English) Zbl 1332.17016

Summary: The \(q\)-deformation of the infinite-dimensional \(n\)-algebras is investigated. Based on the structure of the \(q\)-deformed Virasoro-Witt algebra, we derive a nontrivial \(q\)-deformed Virasoro-Witt \(n\)-algebra which is nothing but a sh-\(n\)-Lie algebra. Furthermore in terms of the pseud-differential operators, we construct the (co)sine \(n\)-algebra and the \(q\)-deformed \(S \mathrm{Diff}(T^2) n\)-algebra. We find that they are the sh-\(n\)-Lie algebras for the \(n\) even case. In terms of the magnetic translation operators, an explicit physical realization of the (co)sine \(n\)-algebra is given.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B65 Infinite-dimensional Lie (super)algebras
17B68 Virasoro and related algebras
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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