# Drinfeld–Sokolov Reduction for Difference Operators and Deformations of -Algebras¶I. The Case of Virasoro Algebra

@article{Frenkel1998DrinfeldSokolovRF, title={Drinfeld–Sokolov Reduction for Difference Operators and Deformations of -Algebras¶I. The Case of Virasoro Algebra }, author={Edward Vladimir Frenkel and Nikolai Reshetikhin and Michael Arsen'evich Semenov-Tian-Shansky}, journal={Communications in Mathematical Physics}, year={1998}, volume={192}, pages={605-629} }

Abstract:We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-deformations of the classical -algebras by reduction from Poisson-Lie loop groups. We consider in detail the case of SL2. The nontrivial consistency conditions fix the choice of the classical r-matrix defining the Poisson-Lie structure on the loop group LSL2, and this leads to a new elliptic classical r-matrix. The reduced Poisson algebra coincides with the deformation of the classical Virasoro…

## 7 Citations

Drinfeld–Sokolov Reduction for Difference
Operators and Deformations of W-Algebras¶ II. The General Semisimple Case

- Mathematics
- 1998

Abstract:The paper is the sequel to [9]. We extend the Drinfeld--Sokolov reduction procedure to q-difference operators associated with arbitrary semisimple Lie algebras. This leads to a new elliptic…

Drinfeld–Sokolov reduction for quantum groups and deformations of W-algebras

- Mathematics
- 2001

Abstract. We define deformations of W-algebras associated to complex semisimple Lie algebras by means of quantum Drinfeld-Sokolov reduction procedure for affine quantum groups. We also introduce…

Drinfeld-Sokolov reduction for quantum groups

- Mathematics
- 1998

We quantize the Lie-Poisson version of the Drinfeld-Sokolov reduction by applying the homological reduction technique developed in QA.math/9805134 for arbitrary first-class constraints.
An important…

Discrete Quantum Drinfeld–Sokolov Correspondence

- Mathematics
- 2002

Abstract: We construct a discrete quantum version of the Drinfeld–Sokolov correspondence for the sine-Gordon system. The classical version of this correspondence is a birational Poisson morphism…

A pr 2 00 0 Towards Drinfeld – Sokolov reduction for quantum groups

- Mathematics
- 2008

In this paper we study the Poisson–Lie version of the Drinfeld– Sokolov reduction defined in [13], [23]. Using the bialgebra structure related to the new Drinfeld realization of affine quantum groups…

The Deformed Virasoro Algebra at Roots of Unity

- Mathematics
- 1998

Abstract:We discuss some aspects of the representation theory of the deformed Virasoro algebra $\virpq$. In particular, we give a proof of the formula for the Kac determinant and then determine the…

Drinfeld currents and vertex operators

- Mathematics
- 1998

We investigate the structure of the elliptic algebra Uq,p(ŝl2) introduced earlier by one of the authors. Our construction is based on a new set of generating series in the quantum affine algebra…

Q-pseudodifference Drinfeld-Sokolov reduction for algebra of complex size matrices

- Mathematics
- 1999

The q-deformed version of the Drinfeld-Sokolov reduction is extended to the case of the algebra of 'complex size matrices'; this construction generalizes earlier results of B.Khesin and F.Malikov on…

Deformation approach to quantisation of field models

- Mathematics
- 2015

Associativity-preserving deformation quantisation × 7→ ⋆~ via Kontsevich’s summation over weighted graphs is lifted from the algebras of functions on finite-dimensional Poisson manifolds to the…