Infinite-dimensional Lie Groups Research Articles

The action homomorphism, quasimorphisms and moment maps on the space of compatible almost complex structures

We extend the definition of Weinstein’s action homomorphism to Hamiltonian actions with equivariant moment maps of (possibly infinite-dimensional) Lie groups on symplectic manifolds, and show that under conditions including a uniform bound on the symplectic areas of geodesic triangles the resulting homomorphism extends to a quasimorphism on the universal cover of the group. We apply these principles to finite-dimensional Hermitian Lie groups like the linear symplectic group, reinterpreting the Guichardet–Wigner quasimorphisms, and to the infinite-dimensional groups of Hamiltonian diffeomorphisms of closed symplectic manifolds that act on the space of compatible almost complex structures with an equivariant moment map given by the theory of Donaldson and Fujiki. We show that the quasimorphism on the universal cover of the Hamiltonian group obtained in the second case is symplectically conjugation-invariant and compute its restrictions to the fundamental group via a homomorphism introduced by Lalonde–McDuff–Polterovich, answering a question of Polterovich; to the subgroup of Hamiltonian biholomorphisms via the Futaki invariant; and to subgroups of diffeomorphisms supported in an embedded ball via the Barge–Ghys average Maslov quasimorphism, the Calabi homomorphism and the average Hermitian scalar curvature. We show that when the first Chern class vanishes this quasimorphism is proportional to a quasimorphism of Entov and when the symplectic manifold is monotone, it is proportional to a quasimorphism due to Py. As an application we show that a Sobolev distance on the universal cover of the Hamiltonian group is unbounded, similarly to the results of Eliashberg–Ratiu.

Representations of Lie Groups and Supergroups

Representations of Lie Groups and Supergroups

Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups

We formulate Euler-Poincar\'e equations on the Lie group Aut(P) of automorphisms of a principal bundle P. The corresponding flows are referred to as EPAut flows. We mainly focus on geodesic flows associated to Lagrangians of Kaluza-Klein type. In the special case of a trivial bundle P, we identify geodesics on certain infinite-dimensional semidirect-product Lie groups that emerge naturally from the construction. This approach leads naturally to a dual pair structure containing \delta-like momentum map solutions that extend previous results on geodesic flows on the diffeomorphism group (EPDiff). In the second part, we consider incompressible flows on the Lie group of volume-preserving automorphisms of a principal bundle. In this context, the dual pair construction requires the definition of chromomorphism groups, i.e. suitable Lie group extensions generalizing the quantomorphism group.

Nonstandard hulls of locally uniform groups

We present a nonstandard hull construction for locally uniform groups in a spirit similar to Luxembourg’s construction of the nonstandard hull of a uniform space. Our nonstandard hull is a local group rather than a global group. We investigate how this construction varies as one changes the family of pseudometrics used to construct the hull. We use the nonstandard hull construction to give a nonstandard characterization of Enflo’s notion of groups that are uniformly free from small subgroups. We prove that our nonstandard hull is locally isomorphic to Pestov’s nonstandard hull for Banach-Lie groups. We also give some examples of infinite-dimensional Lie groups that are locally uniform.

The automorphism groups of foliations with transverse linear connection

AbstractThe category of foliations is considered. In this category morphisms are differentiable maps sending leaves of one foliation into leaves of the other foliation. We prove that the automorphism group of a foliation with transverse linear connection is an infinite-dimensional Lie group modeled on LF-spaces. This result extends the corresponding result of Macias-Virgós and Sanmartín Carbón for Riemannian foliations. In particular, our result is valid for Lorentzian and pseudo-Riemannian foliations.

Quasi-invariance for heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg groups

We study heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg-like Lie groups. In particular, we show that Cameron-Martin type quasi-invariance results hold in this subelliptic setting and give $L^p$-estimates for the Radon-Nikodym derivatives. The main ingredient in our proof is a generalized curvature-dimension estimate which holds on approximating finite-dimensional projection groups. Such estimates were first introduced by Baudoin and Garofalo in \cite{BaudoinGarofalo2011}.

A partially invariant solution of hydrodynamic equations for the atmosphere

UDC 517.944 + 532.5 The hydrodynamic equations for the atmosphere in f -plane approximation admit an infinite-dimensional Lie group of transformations. We analyze a partially invariant solution to these equations that possesses a certain functional arbitrariness. The problem of finding such a solution is reduced to the integration of the Hopf equations and finite formulas. Bibliography :8 titles. Illustrations :4 figures.

SDiff(2) and uniqueness of the Plebański equation

The group of area preserving diffeomorphisms showed importance in the problems of self-dual gravity and integrability theory. We discuss how representations of this infinite-dimensional Lie group can arise in mathematical physics from pure local considerations. Then using Lie algebra extensions and cohomology we derive the second Plebański equation and its geometry. We do not use Kähler or other additional structures but obtain the equation solely from the geometry of area preserving transformations group. We conclude that the Plebański equation is Lie remarkable.

Nonmeasurable automorphisms of lie groups relative to real- and non-archimedean-valued measures

In this paper, we study the problem on the existence of nonmeasurable automorphisms of finite-dimensional and infinite-dimensional Lie groups over the field of real numbers and also over non-Archimedean local fields. The nonmeasurability of automorphisms is considered relative to real-valued measures and also measures with values in non-Archimedean local fields. Their existence is proved and a procedure for their construction is given. Their application to the construction of nonmeasurable irreducible unitary representations is demonstrated.

Reconstructing quasimorphisms from associated partial orders and a question of Polterovich

We show that every continuous homogeneous quasimorphism on a finite-dimensional 1-connected simple Lie group arises as the relative growth of any continuous bi-invariant partial order on that group. More generally we show, that an arbitrary homogeneous quasimorphism can be reconstructed as the relative growth of a partial order subject to a certain sandwich condition. This provides a link between invariant orders and bounded cohomology and allows the concrete computation of relative growth for finite dimensional simple Lie groups as well as certain infinite-dimensional Lie groups arising from symplectic geometry.

A Smooth Model for the String Group

We construct a model for the string group as an infinite-dimensional Lie group. In a second step we extend this model by a contractible Lie group to a Lie 2-group model. To this end we need to establish some facts on the homotopy theory of Lie 2-groups. Moreover, we provide an explicit comparison of string structures for the two models and a uniqueness result for Lie 2-group models.

On a two-component [formula omitted]-Camassa–Holm system

A novel π-Camassa–Holm system is studied as a geodesic flow on a semidirect product obtained from the diffeomorphism group of the circle. We present the corresponding details of the geometric formalism for metric Euler equations on infinite-dimensional Lie groups and compare our results to what has already been obtained for the usual two-component Camassa–Holm equation. Our approach results in well-posedness theorems and explicit computations of the sectional curvature.

Weighted diffeomorphism groups of Banach spaces and weighted mapping groups

In this work, we construct and study certain classes of infinite dimensional Lie groups that are modelled on weighted function spaces. In particular, we construct a Lie group of weighted diffeomorphisms on a Banach space. Further, we also construct certain types of weighted mapping groups. Both the weighted diffeomorphism groups and the weighted mapping groups are shown to be regular Lie groups in Milnor's sense. We also discuss semidirect products of the former groups. Moreover, we study the integrability of Lie algebras of certain vector fields.

Surface Tension of Multi-phase Flow with Multiple Junctions Governed by the Variational Principle

We explore a computational model of an incompressible fluid with a multi-phase field in three-dimensional Euclidean space. By investigating an incompressible fluid with a two-phase field geometrically, we reformulate the expression of the surface tension for the two-phase field found by Lafaurie, Nardone, Scardovelli, Zaleski and Zanetti (J. Comp. Phys. \vol{113} \yr{1994} \pages{134-147}) as a variational problem related to an infinite dimensional Lie group, the volume-preserving diffeomorphism. The variational principle to the action integral with the surface energy reproduces their Euler equation of the two-phase field with the surface tension. Since the surface energy of multiple interfaces even with singularities is not difficult to be evaluated in general and the variational formulation works for every action integral, the new formulation enables us to extend their expression to that of a multi-phase ($N$-phase, $N\ge2$) flow and to obtain a novel Euler equation with the surface tension of the multi-phase field. The obtained Euler equation governs the equation of motion of the multi-phase field with different surface tension coefficients without any difficulties for the singularities at multiple junctions. In other words, we unify the theory of multi-phase fields which express low dimensional interface geometry and the theory of the incompressible fluid dynamics on the infinite dimensional geometry as a variational problem. We apply the equation to the contact angle problems at triple junctions. We computed the fluid dynamics for a two-phase field with a wall numerically and show the numerical computational results that for given surface tension coefficients, the contact angles are generated by the surface tension as results of balances of the kinematic energy and the surface energy.

Algebras of symbols associated with the Weyl calculus for Lie group representations

We develop our earlier approach to the Weyl calculus for representations of infinite-dimensional Lie groups by establishing continuity properties of the Moyal product for symbols belonging to various modulation spaces. For instance, we prove that the modulation space of symbols M ∞,1 is an associative Banach algebra and the corresponding operators are bounded. We then apply the abstract results to two classes of representations, namely the unitary irreducible representations of nilpotent Lie groups, and the natural representations of the semidirect product groups that govern the magnetic Weyl calculus. The classical Weyl–Hörmander calculus is obtained for the Schrödinger representations of the finite-dimensional Heisenberg groups, and in this case we recover the results obtained by J. Sjöstrand (Math Res Lett 1(2):185–192, 1994).

Infinite Dimensional Lie Theory

The workshop focussed on recent developments in infinite-dimensional Lie theory. The talks covered a broad range of topics, such as structure and classification theory of infinite-dimensional Lie algebras, geometry of infinite-dimensional Lie groups and homogeneous spaces and representations theory of infinite-dimensional Lie groups, Lie algebras and Lie-superalgebras.

On the genesis of the Cartan–Kähler theory

The theory of exterior differential systems plays a crucial role in Cartan’s whole mathematical production. As he once recognized, all the germs of his subsequent work were contained there. Indeed, it provided him with powerful technical tools that turned out to be very useful in many different fields such as the theory of partial differential equations, the theory of infinite dimensional Lie groups (Lie pseudogroups) and differential geometry. Nevertheless, scarce attention has been paid to this area of historical research thus far. Although authoritative scholars have investigated the foundation of exterior differential calculus in Cartan’s early papers, no specific analysis of Cartan’s subsequent works laying the foundations of what nowadays is known as the Cartan–Kahler theory has been yet provided. This article represents a first attempt to remedy this unsatisfactory state of affairs by focusing on Cartan’s work on Pfaffian systems at the very beginning of the past century. The analysis of Cartan’s relevant papers is preceded by a description of the historical context in which such contributions were conceived. In this respect, special emphasis will be put on some works by Engel and von Weber on Pfaffian systems, which laid the basis for the subsequent geometrical developments of Cartan’s theory of exterior differential systems.

Continuity of magnetic Weyl calculus

We investigate continuity properties of the operators obtained by the magnetic Weyl calculus on nilpotent Lie groups, using modulation spaces associated with unitary representations of certain infinite-dimensional Lie groups.

On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

Let G be a connected and simply connected Banach–Lie group or, more generally, a BCH–Lie group. On the complex enveloping algebra UC(g) of its Lie algebra g we define the concept of an analytic functional and show that every positive analytic functional λ is integrable in the sense that it is of the form λ(D) = 〈dπ(D)v, v〉 for an analytic vector v of a unitary representation of G. On the way to this result we derive criteria for the integrability of ∗-representations of infinite dimensional Lie algebras of unbounded operators to unitary group representations. For the matrix coefficient πv,v(g) = 〈π(g)v, v〉 of a vector v in a unitary representation of an analytic Frechet–Lie group G we show that v is an analytic vector if and only if πv,v is analytic in an identity neighborhood. Combining this insight with the results on positive analytic functionals, we derive that every local positive definite analytic function on a 1-connected Frechet–BCH–Lie group G extends to a global analytic function.

THE PICARD GROUP OF THE LOOP SPACE OF THE RIEMANN SPHERE

The loop space Lℙ1 of the Riemann sphere consisting of all Ck or Sobolev Wk, p maps S1 → ℙ1 is an infinite dimensional complex manifold. We compute the Picard group pic(Lℙ1) of holomorphic line bundles on Lℙ1 as an infinite dimensional complex Lie group with Lie algebra the Dolbeault group H0, 1(Lℙ1). The group G of Möbius transformations and its loop group LG act on Lℙ1. We prove that an element of pic(Lℙ1) is LG-fixed if it is G-fixed, thus completely answering the question of Millson and Zombro about the G-equivariant projective embedding of Lℙ1.