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

Enriquez, B.

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

Keywords:

Drinfeld-Sokolov reduction; group of loops; Feigin-Frenkel action

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