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Quantum affine algebras and deformations of the Virasoro and \({\mathcal W}\)-algebras. (English) Zbl 0869.17014
There is a description of the center \(Z_h(\widehat{sl}_2)\) of a completion \(\widetilde{U}_h (\widehat{sl}_2)_{cr}\) of the quantum affine algebra \(U_q (\widehat{sl}_2)\) at the critical level, its Poisson structure and spectrum. A homomorphism of \(\widetilde{U}_h (\widehat{sl}_2)_{cr}\) to the tensor product of a certain Heisenberg algebra and a Heisenberg-Poisson algebra \({\mathfrak H}(sl_2)\) are constructed such that the restriction to the center is a homomorphism of Poisson algebras, which is a \(q\)-deformation of the Miura transformation. By a computation of Poisson brackets, it turns out that the Poisson structure of \(Z_h (\widehat{sl}_2)\) is closely related to a \(q\)-deformation of the classical Virasoro algebra.
In the last part of the paper the results are generalized to the case of \(\widetilde{U}_h (\widehat{sl}_N)_{cr}\), and there is a conjecture that analogous results may be obtained in the general case of an arbitrary simple Lie algebra \({\mathfrak g}\), a central subalgebra \(Z_h(\widehat {\mathfrak g})\) of \(\widetilde{U}_h (\widehat{\mathfrak g})_{cr}\) and the Heisenberg-Poisson algebra \({\mathfrak H}_h ({\mathfrak g})\).

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B68 Virasoro and related algebras
17B81 Applications of Lie (super)algebras to physics, etc.
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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