Infinite-dimensional Lie Groups Research Articles

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.

On differentiable vectors for representations of infinite dimensional Lie groups

In this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations π : G → GL ( V ) of an infinite dimensional Lie group G on a locally convex space V. The first class of results concerns the space V ∞ of smooth vectors. If G is a Banach–Lie group, we define a topology on the space V ∞ of smooth vectors for which the action of G on this space is smooth. If V is a Banach space, then V ∞ is a Fréchet space. This applies in particular to C ∗ -dynamical systems ( A , G , α ) , where G is a Banach–Lie group. For unitary representations we show that a vector v is smooth if the corresponding positive definite function 〈 π ( g ) v , v 〉 is smooth. The second class of results concerns criteria for C k -vectors in terms of operators of the derived representation for a Banach–Lie group G acting on a Banach space V. In particular, we provide for each k ∈ N examples of continuous unitary representations for which the space of C k + 1 -vectors is trivial and the space of C k -vectors is dense.

Homotopy Theory of Function Spaces and Related Topics

This workshop brought together researchers studying a variety of problems related to the homotopy theory of function spaces. Topics covered included: evaluation maps and Gottlieb groups, the classification of gauge groups and of other function space components, algebraic models for function spaces both in the rational and in the p-local settings, operads, configuration spaces, free and based loop spaces and infinite-dimensional Lie groups.

Investigation of the geodesic flow on an infinite-dimensional Lie group by means of the coadjoint action operator

We consider a representation of the Euler equations as the geodesic flow on an infinite-dimensional Lie group. In these terms, we establish properties of solutions, which are provided by local existence and uniqueness theorems, at a limit point.

Uncertainty principles for magnetic structures on certain coadjoint orbits

By building on our earlier work, we establish uncertainty principles in terms of Heisenberg inequalities and of the ambiguity functions associated with magnetic structures on certain coadjoint orbits of infinite-dimensional Lie groups. These infinite-dimensional Lie groups are semidirect products of nilpotent Lie groups and invariant function spaces thereon. The recently developed magnetic Weyl calculus is recovered in the special case of function spaces on abelian Lie groups.

Heat kernel analysis on semi-infinite Lie groups

This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron–Martin type quasi-invariance theorem for the heat kernel measure and give estimates on the L p norms of the Radon–Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds in this setting.

Iwasawa decompositions of some infinite-dimensional Lie groups

We set up an abstract framework that allows the investigation of Iwasawa decompositions for involutive infinite-dimensional Lie groups modeled on Banach spaces. This provides a method to construct Iwasawa decompositions for classical real or complex Banach-Lie groups associated with the Schatten ideals $\mathfrak {S}_p(\mathcal {H})$ on a complex separable Hilbert space $\mathcal {H}$ if $1<p<\infty$.

Analytic mappings between LB-spaces and applications in infinite-dimensional Lie theory

We give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces. This tool is then used to construct new classes of (real and complex) infinite dimensional Lie groups: The group DiffGerm(K, X) of germs of analytic diffeomorphisms around a compact set K in a Banach space X and the group S n∈N Gn where the Gn are Banach Lie groups.

Central extensions of groups of sections

If q : P -> M is a principal K-bundle over the compact manifold M, then any invariant symmetric V-valued bilinear form on the Lie algebra k of K defines a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo exact forms. In the present paper we analyze the integrability of this extension to a Lie group extension for non-connected, possibly infinite-dimensional Lie groups K. If K has finitely many connected components we give a complete characterization of the integrable extensions. Our results on gauge groups are obtained by specialization of more general results on extensions of Lie groups of smooth sections of Lie group bundles. In this more general context we provide sufficient conditions for integrability in terms of data related only to the group K.

Magnetic pseudo-differential Weyl calculus on nilpotent Lie groups

We develop a pseudo-differential Weyl calculus on nilpotent Lie groups, which allows one to deal with magnetic perturbations of right invariant vector fields. For this purpose, we investigate an infinite-dimensional Lie group constructed as the semidirect product of a nilpotent Lie group and an appropriate function space thereon. We single out an appropriate coadjoint orbit in the semidirect product and construct our pseudo-differential calculus as a Weyl quantization of that orbit.

Pro-Lie groups which are infinite-dimensional Lie groups

AbstractA pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if G is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighbourhood in the Lie algebra diffeomorphically onto an open identity neighbourhood of the group.

Heat kernel analysis on infinite-dimensional Heisenberg groups

We introduce a class of non-commutative Heisenberg-like infinite-dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, { ν t } t > 0 , are also studied. We show that these heat kernel measures admit: (1) Gaussian like upper bounds, (2) Cameron–Martin type quasi-invariance results, (3) good L p -bounds on the corresponding Radon–Nikodym derivatives, (4) integration by parts formulas, and (5) logarithmic Sobolev inequalities. The last three results heavily rely on the boundedness of the Ricci tensor.

Symmetry transformation of solutions for the Navier–Stokes equations

It is shown in this study that the Navier–Stokes equations allows an infinite-dimensional Lie group of symmetries, i.e., a group transforming solutions amongst each other. The Lie algebra of this symmetry group here depends on four arbitrary functions of time. Some new deformed solutions of Navier–Stokes equations in two and three-dimensions are obtained by applying some of the element of the symmetry group of these equations to their basic solutions. In order to explore the properties of deformed solutions, the analytic solutions are analyzed. It is noted that the corresponding deformed solutions behave as the basic solutions in the limiting sense for large time (i.e. t → ∞ ).

Harmonische Analysis und Darstellungstheorie Topologischer Gruppen

This international conference was organized by Toshiyuki Kobayashi (University of Tokyo), Bernhard Krötz (MPIM Bonn), Erez Lapid (Hebrew University) and Charles Torossian (Paris VI). The general theme was representation theory of real and p-adic Lie groups, especially in connection to automorphic forms. In addition there were very interesting presentations on infinite dimensional Lie groups and quantization. It was very pleasing that new results of the highest caliber were presented by the youngest of the participants. There were many discussions at this relaxed and influential meeting. We would like to thank the MFO for having given us this wonderful opportunity. The organizers would like to thank Dmitry Gourevitch for his careful work in compiling and editing this report.

Manifold of nondegenerate affinor fields

We consider the quotient set of the set of nondegenerate affinor fields with respect to the action of the group of nowhere vanishing functions. This set is endowed with a structure of infinite-dimensional Lie group. On this Lie group, we construct an object of linear connection with respect to which all left-invariant vector fields are covariantly constant (the Cartan connection).

A complex semigroup approach to group algebras of infinite dimensional Lie groups

A host algebra of a topological group G is a C*-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations of G. In this paper we present an approach to host algebras for infinite dimensional Lie groups which is based on complex involutive semigroups. Any locally bounded absolute value α on such a semigroup S leads in a natural way to a C*-algebra C*(S,α), and we describe a setting which permits us to conclude that this C*-algebra is a host algebra for a Lie group G. We further explain how to attach to any such host algebra an invariant weak-*-closed convex set in the dual of the Lie algebra of G enjoying certain nice convex geometric properties. If G is the additive group of a locally convex space, we describe all host algebras arising this way. The general non-commutative case is left for the future.

Solutions to open problems in Neeb’s recent survey on infinite-dimensional Lie groups

We solve three open problems concerning infinite-dimensional Lie groups posed in a recent survey article by K.-H. Neeb (Towards a Lie theory of locally convex groups, Jpn. J. Math. 1 (2006), 291–468). Moreover, we prove a result by the author announced in the survey, which answers a question posed in an earlier, unpublished version thereof.

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.

Exceptional and semi simple Lie groups hierarchies and the maximum number of elementary particles beyond the standard model of high energy physics

In this work and following some major developments in our understanding of the connection between the exceptional Lie symmetry groups hierarchy and physics, we present finite and infinite dimensional Lie groups and their connections to the orthogonal and unitary groups. The dimension of the Lie group E 7 1 2 is included in the derivation of the number of elementary particle-like super states 576 and 672. Connections to the dimensions of Lie groups and the 72 and 84 elementary particles recently considered by the Author are discussed.

Infinite-Dimensional Lie Theory

Nowadays infinite-dimensional Lie theory is a core area of modern mathematics, covering a broad range of branches, such as the structure and classification theory of infinite-dimensional Lie algebras, geometry of infinite-dimensional Lie groups and their homogeneous spaces, and representation theory of infinite-dimensional Lie groups, Lie algebras and Lie-superalgebras. The focus of this workshop was on recent developments in all of these areas with particular emphasis on connections with other branches of mathematics, such as algebraic groups and Galois cohomology. The meeting was attended by 52 participants from many European countries, Canada, the USA, Brazil, Japan and Australia. The meeting was organized around a series of 23 lectures each of 50 minutes duration representing the major recent advances in the area. We feel that the meeting was exciting and highly successful. The quality of the lectures, several of which surveyed recent developments, was outstanding. The exceptional atmosphere of the Oberwolfach Institute provided an optimal environment for bringing people working in algebraically, geometrically or analytically oriented areas of infinite-dimensional Lie theory together, and to create an atmosphere of scientific interaction and cross-fertilization. Without going too much into detail, let us mention some important new developments that were showcased during the workshop. In the structure theory of infinite-dimensional Lie algebras, the classification of extended affine Lie algebras, based on the notion of a Lie torus has now reached a mature state (Neher). In the classification theory of infinite-dimensional Lie algebras, several deep results were obtained recently with Galois cohomology methods exhibiting exciting connections between forms of multiloop algebras and the Galois theory of forms of algebras over rings (Allison, Gille, Chernousov). This branch of structure theory is complemented by the connection between the classification of generalized Kac–Moody algebras and automorphic forms (Scheithauer). In the representation theory of infinite-dimensional Lie algebras, the most exciting new developments concern various kinds of categories of representations of current algebras and Kac–Moody–Lie (super-)algebras (Benkart, Chari, Futorny, Gorelik, Kumar, Littelmann, Serganova). Another interesting, recently very active direction of Kac–Moody theory are Kac–Moody groups over finite fields, which leads to a new class of infinite simple groups (Caprace). On geometric and analytic Lie theory, we had exiting talks on new geometric directions in the representation theory of Banach–Lie groups, related to Banach–Lie–Poisson spaces (Ratiu), and applications of heat kernel measures in the representation theory of loop groups (Pickrell). On the opposite side of the spectrum of Lie group theory, namely direct limit theory, crucial progress has been made on direct limits of unitary representations, as well as in the context of direct limits of infinite-dimensional groups (Wolf, Glöckner). We further had several contributions dealing with geometric aspects such as Chern forms, gerbes and generalized projective geometries (Paycha, Schweigert, Bertram). Finally, we had several exciting talks about several more particular results, dealing with vertex operator algebras, polyzeta values and quantization (Billig, Mathieu, Omori). More specific information is contained in the abstracts which follow in this volume.