Universal Drinfeld-Sokolov Reduction and Matrices of Complex Size

TL;DR

This paper develops a universal framework for Drinfeld--Sokolov reduction applicable to matrices of complex size, extending classical algebraic structures and connecting to fractional order pseudodifferential operators and Toda lattices.

Contribution

It introduces an affinization of complex size matrix algebras and extends Drinfeld--Sokolov reduction, unifying various algebraic and integrable systems.

Findings

Constructed affinization of $gl_{\lambda}$ containing $gl_n$ for integers.

Derived quadratic Gelfand--Dickey structure on pseudodifferential operators.

Extended the framework to orthogonal, symplectic, and Toda lattice systems.

Abstract

We construct affinization of the algebra $gl_{\lambda}$ 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 $\hat{gl_{\lambda}}$ 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.