Classification and Casimir Invariants of Lie-Poisson Brackets

TL;DR

This paper classifies Lie-Poisson brackets derived from Lie algebra extensions, providing a method to find Casimir invariants and explicitly computing them for low-order cases, with applications to magnetohydrodynamics.

Contribution

It introduces a classification scheme for Lie-Poisson brackets from algebra extensions and a general method for deriving Casimir invariants, including explicit computations for low-order brackets.

Findings

Classified Lie-Poisson brackets for extensions of order less than five.

Derived a general method for finding Casimir invariants.

Explicitly computed Casimir invariants for low-order brackets.

Abstract

We classify Lie-Poisson brackets that are formed from Lie algebra extensions. The problem is relevant because many physical systems owe their Hamiltonian structure to such brackets. A classification involves reducing all brackets to a set of normal forms, and is achieved partially through the use of Lie algebra cohomology. For extensions of order less than five, the number of normal forms is small and they involve no free parameters. We derive a general method of finding Casimir invariants of Lie-Poisson bracket extensions. The Casimir invariants of all low-order brackets are explicitly computed. We treat in detail a four field model of compressible reduced magnetohydrodynamics.