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

@article{SemenovTianShansky1998DrinfeldSokolovRF, title={Drinfeld–Sokolov Reduction for Difference Operators and Deformations of W-Algebras¶ II. The General Semisimple Case }, author={Michael Arsen'evich Semenov-Tian-Shansky and A. V. Sevostyanov}, journal={Communications in Mathematical Physics}, year={1998}, volume={192}, pages={631-647} }

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 deformation of the Lie bialgebra structure on the associated loop algebra. The related classical r-matrix is explicitly described in terms of the Coxeter transformation. We also present a cross-section theorem for q-gauge transformations which generalizes a theorem due to R. Steinberg.

## 60 Citations

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

- Mathematics
- 1998

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…

Drinfeld―Sokolov Reduction for Difference Lax Operators with Periodic Boundary Conditions in the Case of gl(n,C((λ-1)))

- Mathematics
- 2001

Within the framework of the method of formal dressing transformations, we construct a family of zero-curvature equations with discrete space variable for the case of the algebra gl(n,ℂ((λ-1))). The…

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…

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…

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…

Algebraic Group Analogues of the Slodowy Slices and Deformations of Poisson W-algebras

- Mathematics
- 2010

We define algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra . The new slices are transversal to the conjugacy classes in an algebraic…

Deformations of W-algebras associated to simple Lie algebras

- Mathematics
- 1997

Deformed W-algebra Wq,t(g) associated to an arbitrary simple Lie alge- bra g is defined together with its free field realizations and the screening operators. Explicit formulas are given for…

The structure of the nilpotent cone, the Kazhdan-Lusztig map and algebraic group analogues of the Slodowy slices

- Mathematics
- 2008

We define algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g. The new slices are transversal to the conjugacy classes in an algebraic…

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…

Quadratic relations of the deformed W-superalgebra

- Mathematics
- 2020

This paper is a continuation of the study by Ding and Feigin, Contemp.Math. 248, 83 (1998). We find a bosonization of the deformed W -superalgebras Wq,t(sl(2|1)) that commute up to the total…

## References

SHOWING 1-10 OF 23 REFERENCES

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

- Mathematics
- 1998

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…

Lie algebras and equations of Korteweg-de Vries type

- Mathematics
- 1985

The survey contains a description of the connection between the infinite-dimensional Lie algebras of Kats-Moody and systems of differential equations generalizing the Korteweg-de Vries and…

Poisson Lie Groups, Quantum Duality Principle, and the Quantum Double

- Mathematics, Physics
- 1993

The Heisenberg double of a Hopf algebra may be regarded as a quantum analogue of the cotangent bundle of a Lie group. Quantum duality principle describes relations between a Hopf algebra, its dual,…

Deformations of Poisson brackets, Dirac brackets and applications

- Physics, Mathematics
- 1976

After a short review of results which we recently obtained on deformations of Lie algebras associated with symplectic manifolds, we discuss physical applications and treat some examples with deformed…

Central extensions of quantum current groups

- Mathematics
- 1990

We describe Hopf algebras which are central extensions of quantum current groups. For a special value of the central charge, we describe Casimir elements in these algebras. New types of generators…

Quantum-algebras and elliptic algebras

- Mathematics
- 1995

AbstractWe define a quantum-algebra associated to
$$\mathfrak{s}\mathfrak{l}_N $$
as an associative algebra depending on two parameters. For special values of the parameters, this algebra becomes…

Regular elements of semisimple algebraic groups

- Mathematics
- 1965

© Publications mathématiques de l’I.H.É.S., 1965, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://…

The local structure of Poisson manifolds

- Mathematics
- 1983

Varietes de Poisson et applications. Decomposition. Structures de Poisson lineaires. Approximation lineaire. Systemes hamiltoniens. Le probleme de linearisation. Groupes de fonction, realisations et…

Linear Algebraic Groups

- Mathematics
- 1991

Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality…