Drinfeld-Sokolov reduction for difference operators and deformations of W-algebras I. The case of Virasoro algebra

E. Frenkel; N. Reshetikhin; M. A. Semenov-Tian-Shansky

doi:10.1007/s002200050311

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Drinfeld-Sokolov reduction for difference operators and deformations of W-algebras I. The case of Virasoro algebra

E. Frenkel·N. Reshetikhin·M. A. Semenov-Tian-Shansky

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Abstract

We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-deformations of the classical W-algebras by reduction from Poisson-Lie loop groups. We consider in detail the case of SL(2). The nontrivial consistency conditions fix the choice of the classical r-matrix defining the Poisson-Lie structure on the loop group LSL(2), and this leads to a new elliptic classical r-matrix. The reduced Poisson algebra coincides with the deformation of the classical Virasoro algebra previously defined in q-alg/9505025. We also consider a discrete analogue of this Poisson algebra. In the second part (q-alg/9702016) the construction is generalized to the case of an arbitrary semisimple Lie algebra.

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E. Frenkel, N. Reshetikhin, M. A. Semenov-Tian-Shansky. 1997-04-11. Drinfeld-Sokolov reduction for difference operators and deformations of W-algebras I. The case of Virasoro algebra. https://doi.org/10.1007/s002200050311

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