An Infinitesimal Quantum Group Underlies Classical Fluid Mechanics

TL;DR

This paper reveals that classical fluid mechanics can be understood through an underlying infinitesimal quantum group structure, linking fluid dynamics with advanced algebraic concepts.

Contribution

It demonstrates that helicity induces a Lie algebra splitting forming a Manin triple, unveiling a quantum group structure underlying classical fluid mechanics.

Findings

Helicity causes a Lie algebra splitting into isotropic subspaces.

A Manin triple structure is identified in the Lie algebra of incompressible vector fields.

An underlying infinitesimal quantum group explains classical fluid dynamics phenomena.

Abstract

Arnold showed that the Euler equations of an ideal fluid describe geodesics in the Lie algebra of incompressible vector fields. We will show that helicity induces a splitting of the Lie algebra into two isotropic subspaces, forming a Manin triple. Viewed another way, this shows that there is an infinitesimal quantum group (a.k.a. Lie bi-algebra) underlying classical fluid mechanics.