Toda systems for Takiff algebras

TL;DR

This paper extends Toda systems to Takiff algebras, providing explicit solutions, applications to a 3-body problem, soliton extensions, and quantization of the associated Poisson algebra.

Contribution

It introduces new integrable systems based on Takiff algebras, generalizing Toda systems and solving related classical and quantum problems.

Findings

Explicit solutions for equations of motion using jet transformations

Application to a 3-body problem with $ ext{sl}(2)$

Extension of soliton solutions to Takiff algebras

Abstract

We study completely integrable systems attached to Takiff algebras $\mathfrak{g}_N$, extending open Toda systems of split simple Lie algebras $\mathfrak{g}$. With respect to Darboux coordinates on coadjoint orbits $\mathcal{O}$, the potentials of the hamiltonians are products of polynomial and exponential functions. General solutions for equations of motion for $\mathfrak{g}_N$ are obtained using differential operators called jet transformations. These results are applied to a $3$-body problem based on $\mathfrak{sl}(2)$, and to an extension of soliton solutions for $A_\infty$ to associated Takiff algebras. The new classical integrable systems are then lifted to families of commuting operators in an enveloping algebra, solving a Vinberg problem and quantizing the Poisson algebra of functions on $\mathcal{O}$.