Infinite-dimensional Lie Groups Research Articles

Non-abelian extensions of infinite-dimensional Lie groups

Dans cet article nous etudions les extensions non abeliennes d'un groupe de Lie G modele sur un espace localement convexe par un groupe de Lie N. Les classes d'equivalence de telles extensions sont groupees en celles qui correspondent a la classe des actions dites des actions exterieures S de G sur N. Si S est donne, nous montrons que l'ensemble correspondant Ext(G, N) S des classes d'extensions est un espace homogene principal du groupe de cohomologie localement lisse H) 2 SS (G, Z(N)) s . Pour chaque S une obstruction localement lisse Χ(S) dans un groupe de cohomologie H) 3 SS (G, Z(N)) S est definie. Elle s'annule si et seulement si il existe une extension correspondante de G par N. Un point central est que nous ramenons plusieurs problemes concernant des extensions par des groupes non abeliens a des questions sur des extensions par des groupes abeliens, qui ont ete etudiees dans des travaux anterieurs. Un outil important est une notion de module croise lisse, relevant de la theorie de Lie, α: H → G, que nous voyons comme une extension centrale d'un sous-groupe normal de G.

Direct limit groups do not have small subgroups

We show that countable direct limits of finite-dimensional Lie groups do not have small subgroups. The same conclusion is obtained for suitable direct limits of infinite-dimensional Lie groups.

Connectivity properties of moment maps on based loop groups

For a compact, connected, simply-connected Lie group G, the loop group LG is the infinite-dimensional Hilbert Lie group consisting of H^1-Sobolev maps S^1-->G. The geometry of LG and its homogeneous spaces is related to representation theory and has been extensively studied. The space of based loops Omega(G) is an example of a homogeneous space of $LG$ and has a natural Hamiltonian T x S^1 action, where T is the maximal torus of G. We study the moment map mu for this action, and in particular prove that its regular level sets are connected. This result is as an infinite-dimensional analogue of a theorem of Atiyah that states that the preimage of a moment map for a Hamiltonian torus action on a compact symplectic manifold is connected. In the finite-dimensional case, this connectivity result is used to prove that the image of the moment map for a compact Hamiltonian T-space is convex. Thus our theorem can also be viewed as a companion result to a theorem of Atiyah and Pressley, which states that the image mu(Omega(G)) is convex. We also show that for the energy functional E, which is the moment map for the S^1 rotation action, each non-empty preimage is connected.

On the Birkhoff factorization problem for the Heisenberg magnet and nonlinear Schrödinger equations

A geometrical description of the Heisenberg magnet (HM) equation with classical spins is given in terms of flows on the quotient space G∕H+, where G is an infinite dimensional Lie group and H+ is a subgroup of G. It is shown that the HM flows are induced by an action of R2 on G∕H+, and that the HM equation can be integrated by solving a Birkhoff factorization problem for G. For the HM flows that are Laurent polynomials in the spectral variable, we derive an algebraic transformation between solutions of the nonlinear Schrödinger (NLS) and Heisenberg magnet equation. The Birkhoff factorization problem for G is treated in terms of the geometry of the Segal-Wilson Grassmannian Gr(H). The solution of the problem is given in terms of a pair of Baker functions for special subspaces in Gr(H). The Baker functions are constructed explicitly for subspaces that yield multisoliton solutions of NLS and HM equations.

Hölder continuous homomorphisms between infinite-dimensional Lie groups are smooth

Let f : G → H be a homomorphism between smooth Lie groups modelled on Mackey complete, locally convex real topological vector spaces. We show that if f is Hölder continuous at 1, then f is smooth.

Hilbert–Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvature is - ∞ .

Poincaré–Birkhoff–Witt theorems and generalized Casimir invariants for some infinite-dimensional Lie groups: II

Analogues of the Poincare-Birkhoff-Witt theorem for several pairs of Lie groups acting on complex manifolds are studied. These results lead to algebraic analogues of a theorem of I Segal on the two-sided regular representation of a unimodular locally compact group for dual representations of some infinite-dimensional Lie groups on generalized Bargmann-Segal-Fock spaces. Generalized Casimir invariants for these dual representations are also developed.

Exact anisotropic MHD equilibria

An infinite-dimensional Lie group of symmetries of the anisotropic Chew–Goldberger–Low (CGL) plasma equilibrium equations is introduced. The symmetries are used to construct families of new anisotropic plasma equilibria. An infinite-dimensional family of transformations between solutions to the isotropic magnetohydrodynamic (MHD) equilibrium equations and solutions to the anisotropic CGL plasma equilibrium equations is presented. The transformations depend on the topology of the original solutions and produce a wide class of anisotropic plasma equilibrium solutions, including 3D solutions with no geometrical symmetries.

An infinite-dimensional manifold structure for analytic Lie pseudogroups of infinite type

We construct an infinite-dimensional manifold structure adapted to analytic Lie pseudogroups of infinite type. More precisely, we prove that any isotropy subgroup of an analytic Lie pseudogroup of infinite type is a regular infinite-dimensional Lie group, modelled on a locally convex strict inductive limit of Banach spaces. This is an infinite-dimensional generalization to the case of Lie pseudogroups of the classical second fundamental theorem of Lie.

Self-Invariant Contact Symmetries

Every smooth second-order scalar ordinary differential equation (ODE) that is solved for the highest derivative has an infinite-dimensional Lie group of contact symmetries. However, symmetries other than point symmetries are generally difficult to find and use. This paper deals with a class of one-parameter Lie groups of contact symmetries that can be found and used. These symmetry groups have a characteristic function that is invariant under the group action; for this reason, they are called ‘self-invariant.’ Once such symmetries have been found, they may be used for reduction of order; a straightforward method to accomplish this is described. For some ODEs with a one-parameter group of point symmetries, it is necessary to use self-invariant contact symmetries before the point symmetries (in order to take advantage of the solvability of the Lie algebra). The techniques presented here are suitable for use in computer algebra packages. They are also applicable to higher-order ODEs

Lie Groups of Measurable Mappings

AbstractWe describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space (X; Σ, μ) and (possibly infinite-dimensional) Lie group G, we construct a Lie group L∞(X; G), which is a Fréchet-Lie group if G is so. We also show that the weak direct product of an arbitrary family (Gi)i∈I of Lie groups can be made a Lie group, modelled on the locally convex direct sum .

Holomorphic Line Bunbles on the loop space of the Riemann Sphere

The loop space $L\mathbb{P}_1$ of the Riemann sphere consisting of all $C^k$ or Sobolev $W^{k,p}$ maps from the circle $S^1$ to $\mathbb{P}_1$ is an infinite dimensional complex manifold. The loop group $LPGL(2,\mathbb{C})$ acts on $L\mathbb{P}_1$ . We prove that the group of $LPGL(2,\mathbb{C})$ invariant holomorphic line bundles on $L\mathbb{P}_1$ is isomorphic to an infinite dimensional Lie group. Further, we prove that the space of holomorphic sections of these bundles is finite dimensional, and compute the dimension for a generic bundle.

Geodesic flow on the diffeomorphism group of the circle

We show that certain right-invariant metrics endow the infinite-dimensional Lie group of all smooth orientation-preserving diffeomorphisms of the circle with a Riemannian structure. The study of the Riemannian exponential map allows us to prove infinite-dimensional counterparts of results from classical Riemannian geometry: the Riemannian exponential map is a smooth local diffeomorphism and the length-minimizing property of the geodesics holds.

Kazhdan's Property T for the Symplectic Group over a Ring

Kazhdan constants for $\mathop{\rm {Sp}}\left( n,\Omega \right) $ where $\Omega $ is a commutative topological ring with dense finitely generated subring with unity are determined. This implies Kazhdan's property T for these groups. As application explicit Kazhdan constants are determined for the loop groups corresponding to $\mathop{\rm {Sp}}\left( n,\mathbf{C}\right) $. These are further examples of groups with property T which are infinite dimensional Lie groups and not locally compact.

Flux Homomorphisms and Principal Bundles over Infinite Dimensional Manifolds

Flux homomorphisms for closed vector-valued differential forms on infinite dimensional manifolds are defined. We extend the relation between the kernel of the flux for a closed 2-form ω and Kostant’s exact sequence associated to a principal bundle with curvature ω to the context of infinite-dimensional fiber and base space. We then use these results to construct central extensions of infinite dimensional Lie groups.

Applications of the differential invariants of infinite dimensional groups in hydrodynamics

The bases of differential invariants for infinite-dimensional Lie groups, admitted by the Navier–Stokes and gas dynamics equations, are calculated. The possibilities of bases application to construct differentially invariant solutions are shown. In the terms of differential invariants the group interpretations of Karman’s solution, describing a liquid flow between rotating disks, and solution, giving a stationary cylindrical vortex in a viscous fluid, are given. The application of the bases of differential invariants to construct the group stratification of equations is shown by two examples. The examples are the group stratification for the stationary gas dynamics and for the transonic gas motion equations. The possibilities of group stratification use to form new group-invariant solutions are analyzed.

Complex homogeneous spaces of pseudo-restricted groups

a Abstract. One constructs invariant Kahler structures on homogeneous spaces of certain classes of infinite-dimensional Lie groups. The main new classes of examples which fall under these general constructions are homogeneous spaces of certain Banach-Lie groups described by means of admissible pairs of ideals of compact operators on Hilbert spaces.

Lie Group Structures on Quotient Groups and Universal Complexifications for Infinite-Dimensional Lie Groups

We characterize the existence of Lie group structures on quotient groups and the existence of universal complexifications for the class of Baker–Campbell–Hausdorff (BCH–) Lie groups, which subsumes all Banach–Lie groups and “linear” direct limit Lie groups, as well as the mapping groups C K r ( M, G)≔{γ∈ C r ( M, G) : γ∣ M\\ K =1}, for every BCH–Lie group G, second countable finite-dimensional smooth manifold M, compact subset K of M, and 0⩽ r⩽∞. Also the corresponding test function groups D r ( M, G)=⋃ K C K r ( M, G) are BCH–Lie groups.

Infinite-dimensional Lie groups without completeness restrictions

We describe a setting of infinite-dimensional smooth (resp., analytic) Lie groups modelled on arbitrary, not necessarily sequentially complete, locally convex spaces, generalizing the framework of Lie theory formulated in [R. Hamilton, <i>The inverse func

Central extensions of infinite-dimensional Lie groups

Le principal resultat de cet article est une suite exacte pour le groupe abelien des extensions centrales d'un groupe de Lie connexe G de dimension infinie par un groupe abelien de Lie Z pour lequel la composante connexe est un quotient d'un espace vectoriel par un sous-groupe discret. Un point essentiel de ce resultat est qu'il n'est pas restreint aux groupes lissement paracompacts. Par consequence, il s'applique a tous les groupes de Lie-Banach et de Lie-Frechet. La suite exacte codifie en particulier les obstructions precises pour l'integration d'un cocycle d'algebre de Lie a un cocycle localement lisse des groupes de Lie.