## Nilpotent action on the KdV variables and 2-dimensional Drinfeld-Sokolov reduction.(English)Zbl 0833.35123

Theor. Math. Phys. 98, No. 3, 256-258 (1994) and Teor. Mat. Fiz. 98, No. 3, 375-378 (1994).
Summary: We note that a version “with spectral parameter” of the Drinfeld- Sokolov reduction gives a natural mapping from the KdV phase space to the group of loops with values in $$\widehat N_+/A$$, with $$\widehat N_+$$ affine nilpotent and $$A$$ the principal commutative (or anisotropic Cartan) subgroup; this mapping is connected to the conserved densities of the hierarchy. We compute the Feigin-Frenkel action of $$\widehat n_+$$ (defined in terms of screening operators) on the conserved densities in the $$\text{sl}_2$$ case.

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 17B37 Quantum groups (quantized enveloping algebras) and related deformations 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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### References:

 [1] V.G. Drinfeld and V.V. Sokolov, J. Sov. Math.,30 (1985), 1975-2036. · Zbl 0578.58040 [2] B. Feigin and E. Frenkel, Integrals of motions and quantum groups, HEP-TH 93100222. · Zbl 0885.58034 [3] L.A. Dickey, Soliton Equations and Hamiltonian Systems, Advanced Series in Mathematical Physics, vol. 12, World Sci., Singapore, 1992. · Zbl 0768.35072
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