Abstract:

Helicity is the integral of the dot product of velocity and vorticity. It is famous as a conserved quantity of three dimensional hydrodynamics, shown by Moffat to be the average linking number of fluid paths.  I will give an introduction to Arnoldâs formulation of ideal fluid mechanics in terms of the Lie algebra of incompressible vector fields.

I will show that helicity is also an invariant inner product (not

positive) in this Lie algebra.  Then I will show that there are two sub-Lie-algebras dual to each other under this inner product, which form a Manin triple.  This shows that the the set of incompressible vector fields is a Lie bi-algebra or infinitesimal quantum group in the language of Drinfeld. When the fluid satisfies periodic boundary conditions, this Lie algebra also admits a central extension, making it an appealing generalization of the Virasoro algebra to three dimensions. arxiv.org/abs/2005.12125v1