Integrable systems on semidirect product Lie groups

TL;DR

This paper develops a framework for integrable systems on semidirect product Lie groups, using symplectic geometry and Dirac brackets, and introduces recursive structures for specific Hamiltonians.

Contribution

It constructs a class of integrable systems on semidirect product Lie groups via symplectic submanifolds and Dirac brackets, advancing the understanding of collective dynamics and recursive integrability.

Findings

Constructed symplectic submanifolds with Dirac brackets for integrable systems.

Addressed factorization, Poisson-Lie structures, and dressing actions.

Established recursive procedures for particular Hamilton functions.

Abstract

We study integrable systems on the semidirect product of a Lie group and its Lie algebra as the representation space of the adjoint action. Regarding the tangent bundle of a Lie group as phase space endowed with this semidirect product Lie group structure, we construct a class of symplectic submanifolds equipped with a Dirac bracket on which integrable systems (in the Adler-Kostant-Symes sense) are naturally built through collective dynamics. In doing so, we address other issues as factorization, Poisson-Lie structures and dressing actions. We show that the procedure becomes recursive for some particular Hamilton functions, giving rise to a tower of nested integrable systems.