arXiv subjects

Karl-Hermann Neeb

Publications and source records attributed to Karl-Hermann Neeb.

At least 55 records · Page 3Linked to original sources

arXiv2019

A Survey on Invariant Cones in Infinite Dimensional Lie Algebras

For the Lie algebra $\g$ of a connected infinite-dimensional Lie group~$G$, there is a natural duality between so-called semi-equicontinuous weak-*-closed convex Ad^*(G)-invariant subsets of the dual space $\g'$ and Ad(G)-invariant lower semicontinuous positively homogeneous convex functions on open convex cones in $\g$. In this survey, we discuss various aspects of this duality and some of its applications to a more systematic understanding of open invariant cones and convexity properties of coadjoint orbits. In particular, we show that root decompositions with respect to elliptic Cartan subalgebras provide powerful tools for important classes of infinite Lie algebras, such as completions of locally finite Lie algebras, Kac--Moody algebras and twisted loop algebras with infinite-dimensional range spaces. We also formulate various open problems.

math.RT↗

arXiv2019

Finite dimensional semigroups of unitary endomorphisms of standard subspaces

Let $V$ be a standard subspace in the complex Hilbert space $H$ and $G$ be a finite dimensional Lie group of unitary and antiunitary operators on $H$ containing the modular group $(Δ_V^{it})_{t \in R}$ of $V$ and the corresponding modular conjugation~$J_V$. We study the semigroup \[ S_V = \{ g\in G \cap U(H) : gV \subseteq V\} \] and determine its Lie wedge $L(S_V) = \{ x \in L(G) : exp(R_+ x) \subseteq S_V\}$, i.e., the generators of its one-parameter subsemigroups in the Lie algebra $L(G)$ of~$G$. The semigroup $S_V$ is analyzed in terms of antiunitary representations and their analytic extension to semigroups of the form $G exp(iC)$, where $C \subseteq L(G)$ is an $Ad(G)$-invariant closed convex cone. Our main results assert that the Lie wedge $L(S_V)$ spans a $3$-graded Lie subalgebra in which it can be described explicitly in terms of the involution $τ$ of $L(G)$ induced by $J_V$, the generator $h \in L(G)^τ$ of the modular group, and the positive cone of the corresponding representation. We also derive some global information on the semigroup $S_V$ itself

math.OA↗

arXiv2018

Half-Lie groups

In this paper we study the Lie theoretic properties of a class of topological groups which carry a Banach manifold structure but whose multiplication is not smooth. If $G$ and $N$ are Banach-Lie groups and $π: G \to \mathrm{Aut}(N)$ is a homomorphism defining a continuous action of $G$ on $N$, then $H := N \rtimes_πG$ is a Banach manifold with a topological group structure for which the left multiplication maps are smooth, but the right multiplication maps need not to be. We show that these groups share surprisingly many properties with Banach-Lie groups: (a) for every regulated function $ξ: [0,1] \to T_1H$ the initial value problem $\dotγ(t) = γ(t)ξ(t)$, $γ(0)= 1_H$, has a solution and the corresponding evolution map from curves in $T_1H$ to curves in $H$ is continuous; (b) every $C^1$-curve $γ$ with $γ(0) = 1$ and $γ'(0) = x$ satisfies $\lim_{n \to \infty} γ(t/n)^n = \exp(tx)$; (c) the Trotter formula holds for $C^1$ one-parameter groups in $H$; (d) the subgroup $N^\infty$ of elements with smooth $G$-orbit maps in $N$ carries a natural Fréchet-Lie group structure for which the $G$-action is smooth; (e) the resulting Fréchet-Lie group $H^\infty := N^\infty \rtimes G$ is also regular in the sense of (a).

math.RT↗

arXiv2018

Reflection Positivity---A Representation Theoretic Perspective

Refection Positivity is a central theme at the crossroads of Lie group representations, euclidean and abstract harmonic analysis, constructive quantum field theory, and stochastic processes. This book provides the first presentation of the representation theoretic aspects of Refection Positivity and discusses its connections to those different fields on a level suitable for doctoral students and researchers in related fields.

math.RT↗

arXiv2018

Multiplicity freeness of unitary representations in sections of holomorphic Hilbert bundles

We prove several results asserting that the action of a Banach-Lie group on Hilbert spaces of holomorphic sections of a holomorphic Hilbert space bundle over a complex Banach manifold is multiplicity free. These results require the existence of compatible antiholomorphic bundle maps and certain multiplicity freeness assumptions for stabilizer groups. For the group action on the base, the notion of an $(S,σ)$-weakly visible action (generalizing T.Koboyashi's visible actions) provides an effective way to express the assumptions in an economical fashion. In particular, we derive a version for group actions on homogeneous bundles for larger groups. We illustrate these general results by several examples related to operator groups and von Neumann algebras.

math.RT↗

arXiv2017

On the First Order Cohomology of Infinite--Dimensional Unitary Groups

The irreducible unitary highest weight representations $(π_λ,\mathcal{H}_λ)$ of the group $U(\infty)$, which is the countable direct limit of the compact unitary groups $U(n)$, are classified by the orbits of the weights $λ\in \mathbb{Z}^{\mathbb{N}}$ under the Weyl group $S_{(\mathbb{N})}$ of finite permutations. Here, we determine those weights $λ$ for which the first cohomology space $H^1(U(\infty),π_λ,\mathcal{H}_λ)$ vanishes. For finitely supported $λ\neq 0$, we find that the first cohomology space $H^1(U(\infty),π_λ,\mathcal{H}_λ)$ never vanishes. For these $λ$, the highest weight representations extend to norm-continuous irreducible representations of the full unitary group $U(\mathcal{H})$ (for $\mathcal{H}:= \ell^2(\mathbb{N},\mathbb{C})$) endowed with the strong operator topology and to norm-continuous representations of the unitary groups $U_p(\mathcal{H})$ ($p\in [1,\infty]$) consisting of those unitary operators $g\in U(\mathcal{H})$ for which $g-\mathbb{1}$ is of $p$th Schatten class. However, not every 1-cocycle on $U(\infty)$ automatically extends to one on these unitary groups, so we may not conclude that the first cohomology spaces of the extended representations are non-vanishing. On the contrary, for the groups $U(\mathcal{H})$ and $U_\infty(\mathcal{H})$, all first cohomology spaces vanish. This is different for the groups $U_p(\mathcal{H})$ with $1\leq p <\infty$, where only the natural representation on $\mathcal{H}$ and on its dual have vanishing first cohomology spaces.

math.RT↗

arXiv2017

Full regularity for a C*-algebra of the Canonical Commutation Relations. (Erratum added)

The Weyl algebra,- the usual C*-algebra employed to model the canonical commutation relations (CCRs), has a well-known defect in that it has a large number of representations which are not regular and these cannot model physical fields. Here, we construct explicitly a C*-algebra which can reproduce the CCRs of a countably dimensional symplectic space (S,B) and such that its representation set is exactly the full set of regular representations of the CCRs. This construction uses Blackadar's version of infinite tensor products of nonunital C*-algebras, and it produces a "host algebra" (i.e. a generalised group algebra, explained below) for the σ-representation theory of the abelian group S where σ(.,.):=e^{iB(.,.)/2}. As an easy application, it then follows that for every regular representation of the Weyl algebra of (S,B) on a separable Hilbert space, there is a direct integral decomposition of it into irreducible regular representations (a known result). An Erratum for this paper is added at the end.

math.OA↗

arXiv2017

On the geometry of standard subspaces

A closed real subspace V of a complex Hilbert space H is called standard if V intersects iV trivially and and V + i V is dense in H. In this note we study several aspects of the geometry of the space Stand(H) of standard subspaces. In particular, we show that modular conjugations define the structure of a reflection space and that the modular automorphism groups extend this to the structure of a dilation space. Every antiunitary representation of a graded Lie group G leads to a morphism of dilation spaces Hom$_{gr}(R^x,G)$ to Stand(H). Here dilation invariant geodesics (with respect to the reflection space structure) correspond to antiunitary representations U of Aff(R) and they are decreasing if and only if U is a positive energy representation. We also show that the ordered symmetric spaces corresponding to euclidean Jordan algebras have natural order embeddings into Stand(H) obtained from any antiunitary positive energy representations of the conformal group.

math.OA↗

arXiv2017

Current superalgebras and unitary representations

In this paper we determine the projective unitary representations of finite dimensional Lie supergroups whose underlying Lie superalgebra is $\frak{g} = A \otimes \frak{k}$, where $\frak{k}$ is a compact simple Lie superalgebra and $A$ is a supercommutative associative (super)algebra; the crucial case is when $A = Λ_s(\mathbb{R})$ is a Graßmann algebra. Since we are interested in projective representations, the first step consists of determining the cocycles defining the corresponding central extensions. Our second main result asserts that, if $\frak{k}$ is a simple compact Lie superalgebra with $\frak{k}_1\neq \{0\}$, then each (projective) unitary representation of $Λ_s(\mathbb{R})\otimes \frak{k}$ factors through a (projective) unitary representation of $\frak{k}$ itself, and these are known by Jakobsen's classification. If $\frak{k}_1 = \{0\}$, then we likewise reduce the classification problem to semidirect products of compact Lie groups $K$ with a Clifford--Lie supergroup which has been studied by Carmeli, Cassinelli, Toigo and Varadarajan.

math.QA↗

arXiv2017

Smoothing operators and $C^*$-algebras for infinite dimensional Lie groups

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 $π\colon G \to \U(\cH)$. 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 $π^A \colon G \to B(\cH), g \mapsto π(g)A$. The concept of a smoothing operator is particularly powerful for representations $(π,\cH)$ which are semibounded, i.e., there exists an element $x_0 \in\g$ for which all operators $i\ddπ(x)$, $x \in \g$, 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 space of smooth vectors for the one-parameter group $π_{x_0}(t) = π(\exp tx_0)$. We then show that natural types of smoothing operators can be used to obtain host algebras and that, for every metrizable Lie group, the class of semibounded representations can be covered completely by host algebras. In particular, it permits direct integral decompositions.

math.RT↗

arXiv2017

Antiunitary representations and modular theory

Antiunitary representations of Lie groups take values in the group of unitary and antiunitary operators on a Hilbert space H. In quantum physics, antiunitary operators implement time inversion or a PCT symmetry, and in the modular theory of operator algebras they arise as modular conjugations from cyclic separating vectors of von Neumann algebras. We survey some of the key concepts at the borderline between the theory of local observables (Quantum Field Theory (QFT) in the sense of Araki--Haag--Kastler) and modular theory of operator algebras from the perspective of antiunitary group representations. Here a central point is to encode modular objects in standard subspaces V in H which in turn are in one-to-one correspondence with antiunitary representations of the multiplicative group R^x. Half-sided modular inclusions and modular intersections of standard subspaces correspond to antiunitary representations of Aff(R), and these provide the basic building blocks for a general theory started in the 90s with the ground breaking work of Borchers and Wiesbrock and developed in various directions in the QFT context. The emphasis of these notes lies on the translation between configurations of standard subspaces as they arise in the context of modular localization developed by Brunetti, Guido and Longo, and the more classical context of von Neumann algebras with cyclic separating vectors. Our main point is that configurations of standard subspaces can be studied from the perspective of antiunitary Lie group representations and the geometry of the corresponding spaces, which are often fiber bundles over ordered symmetric spaces. We expect this perspective to provide new and systematic insight into the much richer configurations of nets of local observables in QFT.

math.RT↗

arXiv2016

Polynomial representations of $C^*$-algebras and their applications

This is a sequel to our paper on nonlinear completely positive maps and dilation theory for real involutive algebras, where we have reduced all representation classification problems to the passage from a $C^*$-algebra ${\mathcal A}$ to its symmetric powers $S^n({\mathcal A})$, resp., to holomorphic representations of the multiplicative $*$-semigroup $({\mathcal A},\cdot)$. Here we study the correspondence between representations of ${\mathcal A}$ and of $S^n({\mathcal A})$ in detail. As $S^n({\mathcal A})$ is the fixed point algebra for the natural action of the symmetric group $S_n$ on ${\mathcal A}^{\otimes n}$, this is done by relating representations of $S^n({\mathcal A})$ to those of the crossed product ${\mathcal A}^{\otimes n} \rtimes S_n$ in which it is a hereditary subalgebra. For $C^*$-algebras of type I, we obtain a rather complete description of the equivalence classes of the irreducible representations of $S^n({\mathcal A})$ and we relate this to the Schur--Weyl theory for $C^*$-algebras. Finally we show that if ${\mathcal A}\subseteq B({\mathcal H})$ is a factor of type II or III, then its corresponding multiplicative representation on ${\mathcal H}^{\otimes n}$ is a factor representation of the same type, unlike the classical case ${\mathcal A}=B({\mathcal H})$.

math.OA↗

arXiv2016

Reflection Positive Stochastic Processes Indexed by Lie Groups

Reflection positivity originates from one of the Osterwalder-Schrader axioms for constructive quantum field theory. It serves as a bridge between euclidean and relativistic quantum field theory. In mathematics, more specifically, in representation theory, it is related to the Cartan duality of symmetric Lie groups (Lie groups with an involution) and results in a transformation of a unitary representation of a symmetric Lie group to a unitary representation of its Cartan dual. In this article we continue our investigation of representation theoretic aspects of reflection positivity by discussing reflection positive Markov processes indexed by Lie groups, measures on path spaces, and invariant gaussian measures in spaces of distribution vectors. This provides new constructions of reflection positive unitary representations.

math-ph↗

arXiv2016

Diffeomorphism groups of compact convex sets

For a compact convex subset K with non-empty interior in a finite-dimensional vector space, let G be the group of all smooth diffeomorphisms of K which fix the boundary of K pointwise. We show that G is a C^0-regular infinite-dimensional Lie group. As a byproduct, we obtain results concerning solutions to ordinary differential equations on compact convex sets.

math.GR↗

arXiv2015

On the characterization of trace class representations and Schwartz operators

In this note we collect several characterizations of unitary representations $(π, \mathcal{H})$ of a finite dimensional Lie group $G$ which are trace class, i.e., for each compactly supported smooth function $f$ on $G$, the operator $π(f)$ is trace class. In particular we derive the new result that, for some $m \in \mathbb{N}$, all operators $π(f)$, $f \in C^m_c(G)$, are trace class. As a consequence the corresponding distribution character $θ_π$ is of finite order. We further show $π$ is trace class if and only if every operator $A$, which is smoothing in the sense that $A\mathcal{H}\subseteq \mathcal{H}^\infty$, is trace class and that this in turn is equivalent to the Fréchet space $\mathcal{H}^\infty$ being nuclear, which in turn is equivalent to the realizability of the Gaussian measure of $\mathcal{H}$ on the space $\mathcal{H}^{-\infty}$ of distribution vectors. Finally we show that, even for infinite dimensional Fréchet-Lie groups, $A$ and $A^*$ are smoothing if and only if $A$ is a Schwartz operator, i.e., all products of $A$ with operators from the derived representation are bounded.

math.RT↗

arXiv2015

Positive energy representations of double extensions of Hilbert loop algebras

A real Lie algebra with a compatible Hilbert space structure (in the sense that the scalar product is invariant) is called a Hilbert-Lie algebra. Such Lie algebras are natural infinite-dimensional analogues of the compact Lie algebras; in particular, any infinite-dimensional simple Hilbert-Lie algebra $\mathfrak{k}$ is of one of the four classical types $A_J$, $B_J$, $C_J$ or $D_J$ for some infinite set $J$. Imitating the construction of affine Kac-Moody algebras, one can then consider affinisations of $\mathfrak{k}$, that is, double extensions of (twisted) loop algebras over $\mathfrak{k}$. Such an affinisation $\mathfrak{g}$ of $\mathfrak{k}$ possesses a root space decomposition with respect to some Cartan subalgebra $\mathfrak{h}$, whose corresponding root system yields one of the seven locally affine root systems (LARS) of type $A_J^{(1)}$, $B^{(1)}_J$, $C^{(1)}_J$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ or $BC_J^{(2)}$. Let $D\in\mathrm{der}(\mathfrak{g})$ with $\mathfrak{h}\subseteq\mathrm{ker}D$ (a diagonal derivation of $\mathfrak{g}$). Then every highest weight representation $(ρ_λ,L(λ))$ of $\mathfrak{g}$ with highest weight $λ$ can be extended to a representation $\widetildeρ_λ$ of the semi-direct product $\mathfrak{g}\rtimes \mathbb{R} D$. In this paper, we characterise all pairs $(λ,D)$ for which the representation $\widetildeρ_λ$ is of positive energy, namely, for which the spectrum of the operator $-i\widetildeρ_λ(D)$ is bounded from below.

math.RT↗

arXiv2015

Isomorphisms of twisted Hilbert loop algebras

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 $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. 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 $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$.

math.RA↗

arXiv2015

Bounded and Semi-bounded Representations of Infinite Dimensional Lie Groups

In this note we describe the recent progress in the classification of bounded and semibounded representations of infinite dimensional Lie groups. We start with a discussion of the semiboundedness condition and how the new concept of a smoothing operator can be used to construct $C^*$-algebras (so called host algebras) whose representations are in one-to-one correspondence with certain semibounded representations of an infinite dimensional Lie group $G$. This makes the full power of $C^*$-theory available in this context. Then we discuss the classification of bounded representations of several types of unitary groups on Hilbert spaces and of gauge groups. After explaining the method of holomorphic induction as a means to pass from bounded representations to semibounded ones, we describe the classification of semibounded representations for hermitian Lie groups of operators, loop groups (with infinite dimensional targets), the Virasoro group and certain infinite dimensional oscillator groups.

math.RT↗