Infinitesimal algebraic skeletons for a (2+1)-dimensional Toda type system

TL;DR

This paper develops a novel algebraic framework for a (2+1)-dimensional Toda system using series expansions and constructs infinitesimal algebraic skeletons, including a realization as a Kac-Moody algebra.

Contribution

It introduces a new method to construct algebraic skeletons for Toda systems, linking them to Kac-Moody algebras and expanding the algebraic understanding of such integrable models.

Findings

Constructed a series expansion-based tower for the Toda system.

Formulated an analog of the Baker-Campbell-Hausdorff formula for this context.

Realized the prolongation skeleton as a Kac-Moody algebra.

Abstract

A tower for a (2+1)-dimensional Toda type system is constructed in terms of a series expansion of operators which can be interpreted as generalized Bessel coefficients; the result is formulated as an analog of the Baker-Campbell-Hausdorff formula. We tackle the problem of the construction of infinitesimal algebraic skeletons for such a tower and discuss some open problems arising along our approach. In particular, we realize the prolongation skeleton as a Kac-Moody algebra.