Dual pairs for non-abelian fluids

TL;DR

This paper rigorously analyzes two dual pairs of momentum maps in non-abelian fluid dynamics, demonstrating their mutual orthogonality and establishing the dual pair property within the context of automorphism groups.

Contribution

It introduces and proves the dual pair property for momentum maps in non-abelian fluid equations, a novel theoretical result in geometric fluid mechanics.

Findings

Actions are mutually completely orthogonal

Dual pair property is established for the momentum maps

Provides a rigorous mathematical framework for non-abelian fluids

Abstract

This paper is a rigorous study of two dual pairs of momentum maps arising in the context of fluid equations whose configuration Lie group is the group of automorphism of a trivial principal bundle, generically called here non-abelian fluids. It is shown that the actions involved are mutually completely orthogonal, which directly implies the dual pair property.