Papers by Karl-hermann Neeb
In this note we show that for any proper action of a Banach-Lie group G on a Banach manifold M , ... more In this note we show that for any proper action of a Banach-Lie group G on a Banach manifold M , the corresponding tangent maps g → T x (M ) have closed range for each x ∈ M , i.e., the tangent spaces of the orbits are closed. As a consequence, for each free proper action on a Hilbert manifold, the quotient M/G carries a natural manifold structure. Mathematics Subject Index 2000: 22E65, 58B25, 57E20.
In this paper we give an almost complete classification of the Hspherical unitary highest weight ... more In this paper we give an almost complete classification of the Hspherical unitary highest weight representations of a hermitian Lie group G,
A real semisimple Lie algebra g admits a Cartan involution, θ , for which the corresponding eigen... more A real semisimple Lie algebra g admits a Cartan involution, θ , for which the corresponding eigenspace decomposition g=k+p has the property that all operators ad X , X∈p are diagonalizable over R . We call such elements hyperbolic, and the elements X∈k are elliptic in the sense that ad X is semisimple with purely imaginary eigenvalues. The pairs (g,θ) are examples of symmetric Lie algebras, i.e., Lie algebras endowed with an involutive automorphism, such that the -1 -eigenspace of θ contains only hyperbolic elements. Let (g,τ ) be a symmetric Lie algebra and g= h+q the corresponding eigenspace decomposition for τ . The existence of "enough" hyperbolic elements in q is important for the structural analysis of symmetric Lie algebras in terms of root decompositions with respect to abelian subspaces of q consisting of hyperbolic elements. We study the convexity properties of the action of Inn g (h) on the space q . The key role will be played by those invariant convex subsets of q whose interior points are hyperbolic.
Monatshefte f�r Mathematik, 2003
Flux homomorphisms for closed vector-valued differential forms on infinite dimensional manifolds ... more 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.
Monatshefte für Mathematik, 2010
In this paper we develop an abstract setup for hamiltonian group actions as follows: Starting wit... more 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) ⊇ 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 → ham(h, ω), i.e., abstract hamiltonian actions, corresponding central and abelian extensions of G and momentum maps J : g → V .
International Mathematics Research Notices, 2011
Let (G, θ) be a Banach-Lie group with involutive automorphism θ, g = h ⊕ q be the θ-eigenspaces i... more Let (G, θ) be a Banach-Lie group with involutive automorphism θ, g = h ⊕ q be the θ-eigenspaces in the Lie algebra g of G, and H = (G θ ) 0 be the identity component of its group of fixed points. An Olshanski semigroup is a semigroup S ⊆ G of the form
Annals of Global Analysis and Geometry, 2009
If K is a Lie group and q : P → M is a principal K-bundle over the compact manifold M , then any ... more If K is a Lie group and 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 1-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.
International Mathematics Research Notices, 2015
Let g be a locally finite split simple complex Lie algebra of type A J , B J , C J or D J and h ⊆... more Let g be a locally finite split simple complex Lie algebra of type A J , B J , C J or D J and h ⊆ g be a splitting Cartan subalgebra. Fix D ∈ der(g) with h ⊆ ker D (a diagonal derivation). Then every unitary highest weight representation (ρ λ , V λ ) of g extends to a representatioñ ρ λ of the semidirect product g ⋊ CD and we say thatρ λ is a positive energy representation if the spectrum of −iρ λ (D) is bounded from below. In the present note we characterise all pairs (λ, D) with λ bounded for which this is the case.
Canadian Journal of Mathematics, 2016
The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e.... more The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras k, also called affinisations of k. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the 7 families A (1) and BC (2) J for some infinite set J. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra k, which we call standard. In this paper, we give for each affinisation g of a simple Hilbert-Lie algebra k an explicit isomorphism from g to one of the standard affinisations of k. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitely as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of g. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of k.
Oberwolfach Reports, 2014
Representation Theory of the American Mathematical Society
Let G/H be a symmetric space admitting a G-invariant hyperbolic cone field. For each such cone fi... more Let G/H be a symmetric space admitting a G-invariant hyperbolic cone field. For each such cone field we construct a local tube domain Ξ containing G/H as a boundary component. The domain Ξ is an orbit of an Ol'shanskii type semi group Γ. We describe the structure of the group G and the domain Ξ. Furthermore we explore the correspondence between Γmodules of holomorphic sections of line bundles over Ξ and spherical highest weight modules.
Springer Proceedings in Mathematics & Statistics, 2014
ABSTRACT We introduce a notion of a weak Poisson structure on a manifold $M$ modeled on a locally... more ABSTRACT We introduce a notion of a weak Poisson structure on a manifold $M$ modeled on a locally convex space. This is done by specifying a Poisson bracket on a subalgebra $\cA \subeq C^\infty(M)$ which has to satisfy a non-degeneracy condition (the differentials of elements of $\cA$ separate tangent vectors) and we postulate the existence of smooth Hamiltonian vector fields. Motivated by applications to Hamiltonian actions, we focus on affine Poisson spaces which include in particular the linear and affine Poisson structures on duals of locally convex Lie algebras. As an interesting byproduct of our approach, we can associate to an invariant symmetric bilinear form $\kappa$ on a Lie algebra $\g$ and a $\kappa$-skew-symmetric derivation $D$ a weak affine Poisson structure on $\g$ itself. This leads naturally to a concept of a Hamiltonian $G$-action on a weak Poisson manifold with a $\g$-valued momentum map and hence to a generalization of quasi-hamiltonian group actions.
Journal of Pure and Applied Algebra, 2015
Motivated by the theory of unitary representations of finite dimensional Lie supergroups, we desc... more Motivated by the theory of unitary representations of finite dimensional Lie supergroups, we describe those Lie superalgebras which have a faithful finite dimensional unitary representation. We call these Lie superalgebras unitary. This is achieved by describing the classification of real finite dimensional compact simple Lie superalgebras, and analyzing, in a rather elementary and direct way, the decomposition of reductive Lie superalgebras (g is a semisimple g0-module) over fields of characteristic zero into ideals.
For every finite dimensional Lie supergroup $(G,\mathfrak g)$, we define a $C^*$-algebra $\mathca... more For every finite dimensional Lie supergroup $(G,\mathfrak g)$, we define a $C^*$-algebra $\mathcal A:=\mathcal A(G,\mathfrak g)$, and show that there exists a canonical bijective correspondence between unitary representations of $(G,\mathfrak g)$ and nondegenerate $*$-representations of $\mathcal A$. The proof of existence of such a correspondence relies on a subtle characterization of smoothing operators of unitary representations. For a broad class of Lie supergroups, which includes nilpotent as well as classical simple ones, we prove that the associated $C^*$-algebra is CCR. In particular, we obtain the uniqueness of direct integral decomposition for unitary representations of these Lie supergroups.
A host algebra of a (possibly infinite dimensional) Lie group $G$ is a $C^*$-algebra whose repres... more A host algebra of a (possibly infinite dimensional) Lie group $G$ is a $C^*$-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations $\pi \colon G \to \U(\cH)$. In this paper we present a new approach to host algebras for infinite dimensional Lie groups which is based on smoothing operators, i.e., operators whose range is contained in the space $\cH^\infty$ of smooth vectors. Our first major result is a characterization of smoothing operators $A$ that in particular implies smoothness of the maps $\pi^A \colon G \to B(\cH), g \mapsto \pi(g)A$. The concept of a smoothing operator is particularly powerful for representations $(\pi,\cH)$ which are semibounded, i.e., there exists an element $x_0 \in\g$ for which all operators $i\dd\pi(x)$, $x \in \g$, from the derived representation are uniformly bounded from above in some neighborhood of $x_0$. Our second main result asserts that this implies that $\cH^\infty$ coincides with the spa...
Lecture Notes in Mathematics, 2011
We start by introducing the notation and stating several facts which are used in this article. Th... more We start by introducing the notation and stating several facts which are used in this article. The reader is assumed to be familiar with basics of the theory of superalgebras, and therefore this section is rather terse. For more detailed accounts of the subject the reader is referred to
International Mathematics Research Notices, 2014
We give a complete classification of all positive energy unitary representations of the Virasoro ... more We give a complete classification of all positive energy unitary representations of the Virasoro group. More precisely, we prove that every such representation can be expressed in an essentially unique way as a direct integral of irreducible highest weight representations.
Representation Theory of the American Mathematical Society, 2015
In this article we show the integrability of two types of infinitesimally unitary representations... more In this article we show the integrability of two types of infinitesimally unitary representations of a Banach-Lie algebra of the form g c = h + iq which is dual to the symmetric Banach-Lie algebra g = h + q with the involution τ (x+y) = x−y for x ∈ h, y ∈ q. The first class are smooth positive definite kernels K on a locally convex manifold M satisfying a natural invariance condition with respect to an infinitesimal action β : g → V(M ) by locally integrable vector fields that is compatible with a smooth action of a Lie group H with Lie algebra h. The second class are positive definite kernels corresponding to positive definite distributions K ∈ C −∞ (M ×M ) on a finite dimensional smooth manifold M which satisfy a similar invariance condition with respect to a homomorphism β : g → V(M ). In both cases we assume that β| h integrates to a smooth action of a global Lie group H on M .
Complex Analysis and Operator Theory, 2014
The concept of reflection positivity has its origins in the work of Osterwalder-Schrader on const... more The concept of reflection positivity has its origins in the work of Osterwalder-Schrader on constructive quantum field theory. It is a fundamental tool to construct a relativistic quantum field theory as a unitary representation of the Poincaré group from a non-relativistic field theory as a representation of the euclidean motion group. This is the second article in a series on the mathematical foundations of reflection positivity. We develop the theory of reflection positive one-parameter groups and the dual theory of dilations of contractive hermitian semigroups. In particular, we connect reflection positivity with the outgoing realization of unitary oneparameter groups by Lax and Phillips. We further show that our results provide effective tools to construct reflection positive representations of general symmetric Lie groups, including the ax + b-group, the Heisenberg group, the euclidean motion group and the euclidean conformal group.
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Papers by Karl-hermann Neeb