Generalized Toda mechanics associated with classical Lie algebras and their reductions

TL;DR

This paper introduces a broad class of integrable generalized Toda systems linked to classical Lie algebras, providing explicit constructions, reductions, and quantum extensions, expanding the understanding of Toda mechanics.

Contribution

It constructs a universal framework for generalized Toda systems labeled by (m,n), including explicit examples, reductions, and quantum versions, advancing integrable systems theory.

Findings

Explicit Lax pairs and Hamiltonians for all (m,n) cases.

Reductions relate different (m,n)-Toda systems.

Quantum integrability persists for most cases.

Abstract

For any classical Lie algebra $g$, we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers $(m,n)$. The universal form of the Lax pair, equations of motion, Hamiltonian as well as Poisson brackets are provided, and explicit examples for $\mathfrak{g}=B_{r},C_{r},D_{r}$ with $m,n\leq3$ are also given. For all $m,n$, it is shown that the dynamics of the $(m,n-1)$- and the $(m-1,n)$-Toda chains are natural reductions of that of the $(m,n)$-chain, and for $m=n$, there is also a family of symmetrically reduced Toda systems, the $(m,m)_{\mathrm{Sym}}$-Toda systems, which are also integrable. In the quantum case, all $(m,n)$-Toda systems with $m>1$ or $n>1$ describe the dynamics of standard Toda variables coupled to noncommutative variables. Except for the symmetrically reduced cases, the integrability for all $(m,n)$-Toda systems survive after quantization.