Dual pairs for matrix Lie groups

TL;DR

This paper introduces two dual pairs for matrix Lie groups, serving as linear analogues to known fluid dynamics dual pairs, thereby advancing the mathematical framework connecting Lie groups and fluid mechanics.

Contribution

It presents novel dual pairs for matrix Lie groups that extend the understanding of duality concepts from fluid mechanics to linear algebraic structures.

Findings

Established two dual pairs for matrix Lie groups.

Connected linear dual pairs to fluid mechanics dual pairs.

Provided a new mathematical framework for Lie group analysis.

Abstract

In this paper we present two dual pairs that can be seen as the linear analogues of the following two dual pairs related to fluids: the EPDiff dual pair due to Holm and Marsden, and the ideal fluid dual pair due to Marsden and Weinstein.