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Universal Drinfeld-Sokolov Reduction and Matrices of Complex Size

Boris Khesin, Feodor Malikov

TL;DR

The paper develops an affinization $\hat{\hbox{gl}}_{\lambda}$ of complex-size matrix algebras $gl_{\lambda}$ and proves that Drinfeld--Sokolov reduction on this space yields the quadratic Gelfand--Dickey Poisson structure on the Poisson--Lie group of pseudodifferential symbols of fractional order. It extends the reduction to a deformation framework that includes self--adjoint (orthogonal/symplectic) cases, producing corresponding GD brackets on self-adjoint pseudodifferential symbols, and demonstrates a continuous deformation of Toda lattices within this setting. When $\lambda$ is integral, the construction recovers the classical $\hat{gl}_{n}$ DS reduction and embeds ordinary differential operators $DO_n$ into the DS family, while for generic $\lambda$ the Miura transform is unavailable, highlighting a broader, parameter-dependent DS theory. The work thus unifies complex-size matrix algebras, affine and DS reductions, and infinite-dimensional integrable systems, offering a continuum of GD structures and Toda-type hierarchies. The results have implications for the study of $W$-algebras, called also Adler--Gelfand--Dickey structures, in a fractional-order and deformation-theoretic context.

Abstract

We construct affinization of the algebra $gl_λ$ of ``complex size'' matrices, that contains the algebras $\hat{gl_n}$ for integral values of the parameter. The Drinfeld--Sokolov Hamiltonian reduction of the algebra $\hat{gl_λ}$ results in the quadratic Gelfand--Dickey structure on the Poisson--Lie group of all pseudodifferential operators of fractional order. This construction is extended to the simultaneous deformation of orthogonal and simplectic algebras that produces self-adjoint operators, and it has a counterpart for the Toda lattices with fractional number of particles.

Universal Drinfeld-Sokolov Reduction and Matrices of Complex Size

TL;DR

The paper develops an affinization of complex-size matrix algebras and proves that Drinfeld--Sokolov reduction on this space yields the quadratic Gelfand--Dickey Poisson structure on the Poisson--Lie group of pseudodifferential symbols of fractional order. It extends the reduction to a deformation framework that includes self--adjoint (orthogonal/symplectic) cases, producing corresponding GD brackets on self-adjoint pseudodifferential symbols, and demonstrates a continuous deformation of Toda lattices within this setting. When is integral, the construction recovers the classical DS reduction and embeds ordinary differential operators into the DS family, while for generic the Miura transform is unavailable, highlighting a broader, parameter-dependent DS theory. The work thus unifies complex-size matrix algebras, affine and DS reductions, and infinite-dimensional integrable systems, offering a continuum of GD structures and Toda-type hierarchies. The results have implications for the study of -algebras, called also Adler--Gelfand--Dickey structures, in a fractional-order and deformation-theoretic context.

Abstract

We construct affinization of the algebra of ``complex size'' matrices, that contains the algebras for integral values of the parameter. The Drinfeld--Sokolov Hamiltonian reduction of the algebra results in the quadratic Gelfand--Dickey structure on the Poisson--Lie group of all pseudodifferential operators of fractional order. This construction is extended to the simultaneous deformation of orthogonal and simplectic algebras that produces self-adjoint operators, and it has a counterpart for the Toda lattices with fractional number of particles.

Paper Structure

This paper contains 20 sections, 12 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Poisson manifolds and Drinfeld-Sokolov reduction
  3. Preliminaries on Poisson geometry and Hamiltonian reduction
  4. Drinfeld-Sokolov construction
  5. Differential operators of complex order and matrices of complex size.
  6. Poisson-Lie group of pseudodifferential symbols
  7. Definition of $\hbox{gl}_{\lambda},\;\lambda\in\hbox{${\bf C}$}$, and its extensions
  8. What is a matrix of complex size?
  9. $\hbox{gl}_{\infty}$ and extensions of $\hbox{gl}_{\lambda}$.
  10. Affinization and Coadjoint Representation of $\bar{\hbox{gl}}_{\lambda}$
  11. Drinfeld-Sokolov reduction on $\hat{\hbox{gl}}_{\lambda}$
  12. Proofs
  13. Affinization and Coadjoint Representation of $\hbox{gl}$
  14. Affinization
  15. Coadjoint Representation
  16. ...and 5 more sections

Key Result

Theorem 1.1

The classical Drinfeld--Sokolov reduction defined on $\hat{\hbox{gl}_n}$ admits a one-parameter deformation to the Hamiltonian reduction on $\hat{\hbox{gl}_\lambda}$. As a Poisson manifold the result of the reduction coincides with the entire Poisson--Lie group of pseudodifferential operators equipp

Theorems & Definitions (23)

  • Theorem 1.1
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Proposition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 3.1
  • Proposition 3.2
  • Lemma 3.4
  • ...and 13 more