Gradient flows in the normal and K\"ahler metrics and triple bracket generated metriplectic systems

TL;DR

This paper explores various gradient and Hamiltonian flows on Lie groups and loop groups, comparing metrics like the normal and Kähler metrics, and introduces metriplectic systems generated by triple brackets with Lie algebraic insights.

Contribution

It provides a comprehensive comparison of gradient flows under different metrics, introduces a class of metriplectic systems from triple brackets, and offers explicit examples with Lie algebraic interpretations.

Findings

Comparison of normal and Kähler metrics on adjoint orbits

Introduction of metriplectic systems generated by triple brackets

Explicit examples illustrating different flow types

Abstract

The dynamics of gradient and Hamiltonian flows with particular application to flows on adjoint orbits of a Lie group and the extension of this setting to flows on a loop group are discussed. Different types of gradient flows that arise from different metrics including the so-called normal metric on adjoint orbits of a Lie group and the K\"ahler metric are compared. It is discussed how a K\"ahler metric can arise from a complex structure induced by the Hilbert transform. Hybrid and metriplectic flows that arise when one has both Hamiltonian and gradient components are examined. A class of metriplectic systems that is generated by completely antisymmetric triple brackets is described and for finite-dimensional systems given a Lie algebraic interpretation. A variety of explicit examples of the several types of flows are given.