Super-Affine Hierarchies and their Poisson Embeddings

TL;DR

This paper explores the connection between super-affine Lie algebras and integrable hierarchies, introducing Poisson embeddings to systematically construct and reduce superintegrable hierarchies using Lie-algebraic methods.

Contribution

It introduces Poisson embeddings as a systematic tool for constructing superintegrable hierarchies from super-affine Lie algebras, overcoming previous limitations in matrix Lax operator construction.

Findings

Poisson embeddings relate affine and conformal structures systematically.

A full class of hierarchies can be derived using Lie-algebraic notions.

Reduced hierarchies are obtained via coset conditions or Hamiltonian constraints.

Abstract

The link between (super)-affine Lie algebras as Poisson brackets structures and integrable hierarchies provides both a classification and a tool for obtaining superintegrable hierarchies. The lack of a fully systematic procedure for constructing matrix-type Lax operators, which makes the supersymmetric case essentially different from the bosonic counterpart, is overcome via the notion of Poisson embeddings (P.E.), i.e. Poisson mappings relating affine structures to conformal structures (in their simplest version P.E. coincide with the Sugawara construction). A full class of hierarchies can be recovered by using uniquely Lie-algebraic notions. The group-algebraic properties implicit in the super-affine picture allow a systematic derivation of reduced hierarchies by imposing either coset conditions or hamiltonian constraints (or possibly both).