Cornelia Vizman

Geodesic Equations on Diffeomorphism Groups

We bring together those systems of hydrodynamical type that can be written as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups with right invariant L 2 or H 1 metrics. We present their formal derivation starting from Eulers equation, the first order equation satisfied by the right logarithmic derivative of a geodesic in Lie groups with right invariant metrics.

Totally geodesic subgroups of diffeomorphisms

Abstract We determine the Riemannian manifolds for which the group of exact volume preserving diffeomorphisms is a totally geodesic subgroup of the group of volume preserving diffeomorphisms, considering right invariant L2-metrics. The same is done for the subgroup of Hamiltonian diffeomorphisms as a subgroup of the group of symplectic diffeomorphisms in the Kahler case. These are special cases of totally geodesic subgroups of diffeomorphisms with Lie algebras big enough to detect the vanishing of a symmetric 2-tensor field.

The path group construction of Lie group extensions

Abstract We present an explicit realization of abelian extensions of infinite dimensional Lie groups using abelian extensions of path groups, by generalizing Mickelsson’s approach to loop groups and the approach of Losev–Moore–Nekrasov–Shatashvili to current groups. We apply our method to coupled cocycles on current Lie algebras and to Lichnerowicz cocycles on the Lie algebra of divergence free vector fields.

Abelian extensions via prequantization

We generalize the prequantization central extension of a group of diffeomorphisms preserving a closed 2-form ω, to an abelian extension of a group of diffeomorphisms preserving a closed vector valued 2-form ω up to a linear isomorphism (ω-equivariant diffeomorphisms). Every abelian extension of a simply connected Lie group can be obtained as the pull-back of such a prequantization abelian extension.

Lagrangian Reduction on Homogeneous Spaces with Advected Parameters

We study the Euler{Lagrange equations for a parameter dependent G-invariant Lagrangian on a homogeneous G-space. We consider the pullback of the parameter depen- dent Lagrangian to the Lie group G, emphasizing the special invariance properties of the associated Euler{Poincar e equations with advected parameters.

Cocycles and stream functions in quasigeostrophic motion

Abstract We present a geometric version of the Lie algebra 2-cocycle connected to quasigeostrophic motion in the β-plane approximation. We write down an Euler equation for the fluid velocity, corresponding to the evolution equation for the stream function in quasigeostrophic motion.

Dual Pairs for Non-Abelian Fluids

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 automorphisms 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.

Invariant Forms on Lie Algebra Extensions

For a perfect ideal 𝔥 of the Lie algebra 𝔤 with a 𝔤-invariant symmetric bilinear form ⟨·, ·⟩, we consider the continuous cohomology class in defined by the 2-cocycles of the form ⟨[X, ·], ·⟩ on 𝔥, X ∈ 𝔤. We determine the obstruction for extending this class to 𝔤. Invariant symmetric bilinear forms on corresponding Abelian extensions of 𝔤 by (𝔤/𝔥)* are constructed. The result is applied to central extensions of the Lie algebra of symplectic vector fields and of the Lie algebra of divergence free vector fields.