Central extensions of current groups

 In this paper we study central extensions of the identity component G of the Lie group C ^∞ (M,K) of smooth maps from a compact manifold M into a Lie group K which might be infinite-dimensional. We restrict our attention to Lie algebra cocycles of the form ω(ξ,η)=[κ(ξ,dη)], where κ:𝔨×𝔨→Y is a symmetric invariant bilinear map on the Lie algebra 𝔨 of K and the values of ω lie in Ω¹(M,Y)/dC ^∞ (M,Y). For such cocycles we show that a corresponding central Lie group extension exists if and only if this is the case for M=𝕊¹. If K is finite-dimensional semisimple, this implies the existence of a universal central Lie group extension of G. The groups Diff(M) and C ^∞ (M,K) act naturally on G by automorphisms. We also show that these smooth actions can be lifted to smooth actions on the central extension if it also is a central extension of the universal covering group G˜ of G. Received: 11 April 2002 / Revised version: 28 August 2002 / Published online: 28 March 2003     

Towards a Lie theory of locally convex groups

In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We further describe how regularity or local exponentiality of a Lie group can be used to obtain quite satisfactory answers to some of the fundamental problems. These results are illustrated by specialization to some specific classes of Lie groups, such as direct limit groups, linear Lie groups, groups of smooth maps and groups of diffeomorphisms.     

Lie group structures on symmetry groups of principal bundles

In this paper we describe how one can obtain Lie group structures on the group of (vertical) bundle automorphisms for a locally convex principal bundle P over the compact manifold M. This is done by first considering Lie group structures on the group of vertical bundle automorphisms Gau(P). Then the full automorphism group Aut(P) is considered as an extension of the open subgroup DiffP(M) of diffeomorphisms of M preserving the equivalence class of P under pull-backs, by the gauge group Gau(P). We derive explicit conditions for the extensions of these Lie group structures, show the smoothness of some natural actions and relate our results to affine Kac-Moody algebras and groups.     

Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds

We study Lie group structures on groups of the form C ^∞(M, K), where M is a non-compact smooth manifold and K is a, possibly infinite-dimensional, Lie group. First we prove that there is at most one Lie group structure with Lie algebra for which the evaluation map is smooth. We then prove the existence of such a structure if the universal cover of K is diffeomorphic to a locally convex space and if the image of the left logarithmic derivative in is a smooth submanifold, the latter being the case in particular if M is one-dimensional. We also obtain analogs of these results for the group of holomorphic maps on a complex manifold with values in a complex Lie group K. We further show that there exists a natural Lie group structure on if K is Banach and M is a non-compact complex curve with finitely generated fundamental group.      

Integration of Geodesic Flows on Homogeneous Spaces: The Case of a Wild Lie Group

We obtain necessary and sufficient conditions for the integrability in quadratures of geodesic flows on homogeneous spaces M with invariant and central metrics. The proposed integration algorithm consists in using a special canonical transformation in the space T ^* M based on constructing the canonical coordinates on the orbits of the coadjoint representation and on the simplectic sheets of the Poisson algebra of invariant functions. This algorithm is applicable to integrating geodesic flows on homogeneous spaces of a wild Lie group.     

Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds

We study compact complex 3-manifolds M admitting a (locally homogeneous) holomorphic Riemannian metric g. We prove the following: (i) If the Killing Lie algebra of g has a non trivial semi-simple part, then it preserves some holomorphic Riemannian metric on M with constant sectional curvature; (ii) If the Killing Lie algebra of g is solvable, then, up to a finite unramified cover, M is a quotient Γ\G, where Γ is a lattice in G and G is either the complex Heisenberg group, or the complex SOL group. S. Dumitrescu was partially supported by the ANR Grant BLAN 06-3-137237.     

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. To K.H. Hofmann on the occasion of his 75th birthday     

Locally Vacant Double Lie Groupoids and the Integration of Matched Pairs of Lie Algebroids

We prove a general integrability result for matched pairs of Lie algebroids. (Matched pairs of Lie algebras are also known as double Lie algebras or twilled extensions of Lie algebras.) The method used is an extension of a method introduced by Lu and Weinstein in the case of Poisson Lie groups, and yields double groupoids which satisfy an étale form of the vacancy condition.     

Weyl Groups are Finite — and Other Finiteness Properties of Cartan Subalgebras

For each pair (??,??) consisting of a real Lie algebra ?? and a subalgebra a of some Cartan subalgebra ?? of ?? such that [??, ??]∪ [??, ??] we define a Weyl group W(??, ??) and show that it is finite. In particular, W(??, ??,) is finite for any Cartan subalgebra h. The proof involves the embedding of 0 into the Lie algebra of a complex algebraic linear Lie group to which the structure theory of Lie algebras and algebraic groups is applied. If G is a real connected Lie group with Lie algebra ??, the normalizer N(??, G) acts on the finite set Λ of roots of the complexification ??c with respect to hc, giving a representation π : N(??, G)→ S(Λ) into the symmetric group on the set Λ. We call the kernel of this map the Cartan subgroup C(??) of G with respect to h; the image is isomorphic to W(??, ??), and C(??)= {g G : Ad(g)(h)— h ε [h,h] for all h ε h }. All concepts introduced and discussed reduce in special situations to the familiar ones. The information on the finiteness of the Weyl groups is applied to show that under very general circumstance, for b ∪ ?? the set ??? ?(b) remains finite as ? ranges through the full group of inner automorphisms of ??.     

The Cohomology of the Complex of G-Invariant Forms onG -Manifolds. II

The cohomology of G-manifolds of the type M=P× _K (G/H), where G is a reductive Lie group, H and N are its closed subgroups, H is a normal subgroup of N, K=N/H, and P is a smooth principal K-bundle, are considered. In the case when the Lie algebras of H and N are reductive, the differential graded algebra C(M) introduced in the previous paper with the same title and having the same minimal model as one of the algebra of G-invariant forms on M is investigated. Moreover, the main theorem on the cohomology algebra of C(M) is proved under weaker conditions than those of the previous paper.     

On the McCool group M3 and its associated Lie algebra

We prove that the Lie algebra of the McCool group M₃ is torsion free. As a result, we are able to give a presentation for the Lie algebra of M₃. Furthermore, M₃ is a Magnus group.     

Universal Central Extensions of Lie Groups

We call a central Z-extension of a group G weakly universal for an Abelian group A if the correspondence assigning to a homomorphism ZA the corresponding A-extension yields a bijection of extension classes. The main problem discussed in this paper is the existence of central Lie group extensions of a connected Lie group G which is weakly universal for all Abelian Lie groups whose identity components are quotients of vector spaces by discrete subgroups. We call these Abelian groups regular. In the first part of the paper we deal with the corresponding question in the context of topological, Fréchet, and Banach–Lie algebras, and in the second part we turn to the groups. Here we start with a discussion of the weak universality for discrete Abelian groups and then turn to regular Lie groups A. The main results are a Recognition and a Characterization Theorem for weakly universal central extensions.     

Metric Lie algebras and quadratic extensions

The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a nondegenerate invariant symmetric bilinear form. We show that any metric Lie algebra g without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary Lie algebra l by an orthogonal l-module a in a canonical way. Identifying equivalence classes of quadratic extensions of l by a with a certain cohomology set H²_Q(l,a), we obtain a classification scheme for general metric Lie algebras and a complete classification of metric Lie algebras of index 3.     

Symmetric nonself-adjoint operators in an enveloping algebra

Nelson and Stinespring proved that in any unitary representation of a Lie group with compact Lie algebra the representation of Hermitian elements in the enveloping algebra are essentially self-adjoint. If the Lie algebra is noncompact, we construct in its enveloping algebra a Hermitian element u such that in any locally faithful unitary representation the representative of u has no self-adjoint extension.     

Integrability of invariant geodesic flows on <Emphasis Type="Italic">n</Emphasis>-symmetric spaces

In this article, by modifying the argument shift method, we prove Liouville integrability of geodesic flows of normal metrics (invariant Einstein metrics) on the Ledger–Obata n-symmetric spaces K ⁿ /diag(K), where K is a semisimple (respectively, simple) compact Lie group.     

Two-Orbit Kähler Manifolds and Morse Theory

We deal with compact Kähler manifolds M acted on by a compact Lie group K of isometries, whose complexification K has exactly one open and one closed orbit in M. If the K-action is Hamiltonian, we investigate topological and cohomological properties of M.     

Two-Orbit K?hler Manifolds and Morse Theory

We deal with compact Kähler manifolds M acted on by a compact Lie group K of isometries, whose complexification K has exactly one open and one closed orbit in M. If the K-action is Hamiltonian, we investigate topological and cohomological properties of M.     

Special symplectic Lie groups and hypersymplectic Lie groups

A special symplectic Lie group is a triple ${(G,\omega,\nabla)}$ such that G is a finite-dimensional real Lie group and ω is a left invariant symplectic form on G which is parallel with respect to a left invariant affine structure ${\nabla}$ . In this paper starting from a special symplectic Lie group we show how to “deform” the standard Lie group structure on the (co)tangent bundle through the left invariant affine structure ${\nabla}$ such that the resulting Lie group admits families of left invariant hypersymplectic structures and thus becomes a hypersymplectic Lie group. We consider the affine cotangent extension problem and then introduce notions of post-affine structure and post-left-symmetric algebra which is the underlying algebraic structure of a special symplectic Lie algebra. Furthermore, we give a kind of double extensions of special symplectic Lie groups in terms of post-left-symmetric algebras.     

The universal central extension of the holomorphic current algebra

We identify the universal differential module ¹(A) for the Fréchet algebra A of holomorphic functions on a complex Stein manifold X, and more generally on a Riemannian domain R over X and for the algebra of germs of holomorphic functions on a compact subset K ⁿ . It turns out to be isomorphic to the Fréchet space of holomorphic 1-forms on X, resp. R, resp. to the space ¹(K) of germs of holomorphic 1-forms in K. This determines the center of the universal central extension of the Lie algebra (R, of holomorphic maps from R to a finite-dimensional simple complex Lie algebra .