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Cornelia Vizman

Publications and source records attributed to Cornelia Vizman.

43 records · Page 3Linked to original sources
arXiv2008

An abstract setting for hamiltonian actions

In this paper we develop an abstract setup for hamiltonian group actions as follows: Starting with a continuous 2-cochain $ω$ on a Lie algebra $h$ with values in an $h$-module $V$, we associate subalgebras $sp(h,ω) \supeq ham(h,ω)$ of symplectic, resp., hamiltonian elements. Then $ham(h,ω)$ has a natural central extension which in turn is contained in a larger abelian extension of $sp(h,ω)$. In this setting, we study linear actions of a Lie group $G$ on $V$ which are compatible with a homomorphism $g \to ham(h,ω)$, i.e. abstract hamiltonian actions, corresponding central and abelian extensions of $G$ and momentum maps $J : g \to V$.

math.SG↗
arXiv2008

The path group construction of Lie group extensions

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.

math.DG↗
arXiv2003

Non-linear Grassmannians as coadjoint orbits

For a given manifold $M$ we consider the non-linear Grassmann manifold $Gr_n(M)$ of $n$-dimensional submanifolds in $M$. A closed $(n+2)$-form on $M$ gives rise to a closed 2-form on $Gr_n(M)$. If the original form was integral, the 2-form will be the curvature of a principal $S^1$-bundle over $Gr_n(M)$. Using this $S^1$-bundle one obtains central extensions for certain groups of diffeomorphisms of $M$. We can realize $Gr_{m-2}(M)$ as coadjoint orbits of the extended group of exact volume preserving diffeomorphisms and the symplectic Grassmannians $SGr_{2k}(M)$ as coadjoint orbits in the group of Hamiltonian diffeomorphisms. We also generalize the vortex filament equation as a Hamiltonian equation on $Gr_{m-2}(M)$.

math.DG↗
arXiv2001

Totally geodesic subgroups of diffeomorphisms

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 $L^2$-metrics. The same is done for the subgroup of Hamiltonian diffeomorphisms as a subgroup of the group of symplectic diffeomorphisms in the Kähler 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.

math.DG↗
arXiv2001

Geodesics and curvature of semidirect product groups

Geodesics and curvature of semidirect product groups with right invariant metrics are determined. In the special case of an isometric semidirect product, the curvature is shown to be the sum of the curvature of the two groups. A series of examples, like the magnetic extension of a group, are then considered.

math.DG↗
arXiv2001

Geodesics on extensions of Lie groups and stability; the superconductivity equation

The equations of motion of a charged ideal fluid, respectively the superconductivity equation (both in a given magnetic field) are showed to be geodesic equations on a general, respectively central extension of the group of volume preserving diffeomorphisms with right invariant metric. For this, quantization of the magnetic flux is required. We do curvature computations in both cases in order to get some informations about the stability.

math.DG↗