Searcharxiv⌕ Search

arXiv subjects

Karl-Hermann Neeb

Publications and source records attributed to Karl-Hermann Neeb.

At least 73 records · Page 4Linked to original sources
arXiv2015

Momentum Maps for Smooth Projective Unitary Representations

For a smooth projective unitary representation of a locally convex Lie group G, the projective space of smooth vectors is a locally convex Kaehler manifold. We show that the action of G on this space is weakly Hamiltonian, and lifts to a Hamiltonian action of the central U(1)-extension of G obtained from the projective representation. We identify the non-equivariance cocycles obtained from the weakly Hamiltonian action with those obtained from the projective representation, and give some integrality conditions on the image of the momentum map.

math.RT↗
arXiv2015

Covariant central extensions of gauge Lie algebras

Motivated by positive energy representations, we classify those continuous central extensions of the compactly supported gauge Lie algebra that are covariant under a 1-parameter group of transformations of the base manifold.

math.RT↗
arXiv2015

Kaehler Geometry, Momentum Maps and Convex Sets

These notes grew out of an expose on M. Gromov's paper "Convex sets and Kähler manifolds'' ("Advances in Differential Geometry and Topology,'' World Scientific, 1990) at the DMV-Seminar on "Combinatorical Convex Geometry and Toric Varieties'' in Blaubeuren in April `93. Gromov's paper deals with a proof of Alexandrov--Fenchel type inequalities and the Brunn--Minkowski inequality for finite dimensional compact convex sets and their variants for compact Kähler manifolds. The emphasis of these notes lies on basic details and the techniques from various mathematical areas involved in Gromov's arguments.

math.SG↗
arXiv2015

Positive energy representations for locally finite split Lie algebras

Let $\mathfrak g$ be a locally finite split simple complex Lie algebra of type $A_J$, $B_J$, $C_J$ or $D_J$ and $\mathfrak h \subseteq \mathfrak g$ be a splitting Cartan subalgebra. Fix $D \in \mathrm{der}(\mathfrak g)$ with $\mathfrak h \subseteq \ker D$ (a diagonal derivation). Then every unitary highest weight representation $(ρ_λ, V^λ)$ of $\mathfrak g$ extends to a representation $\tildeρ_λ$ of the semidirect product $\mathfrak g \rtimes \mathbb C D$ and we say that $\tildeρ_λ$ is a positive energy representation if the spectrum of $-i\tildeρ_λ(D)$ is bounded from below. In the present note we characterise all pairs $(λ,D)$ with $λ$ bounded for which this is the case. If $U_1(\mathcal H)$ is the unitary group of Schatten class $1$ on an infinite dimensional real, complex or quaternionic Hilbert space and $λ$ is bounded, then we accordingly obtain a characterisation of those highest weight representations $π_λ$ satisfying the positive energy condition with respect to the continuous $\mathbb R$-action induced by $D$. In this context the representation $π_λ$ is norm continuous and our results imply the remarkable result that, for positive energy representations, adding a suitable inner derivation to $D$, we can achieve that the minimal eigenvalue of $\tildeρ_λ(D)$ is $0$ (minimal energy condition). The corresponding pairs $(λ,D)$ satisfying the minimal energy condition are rather easy to describe explicitly.

math.RT↗
arXiv2015

$C^*$-algebras and direct integral decomposition for Lie supergroups

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.

math.RT↗
arXiv2015

Finite dimensional compact and unitary Lie superalgebras

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 $\g_{\bar 0}$-module) over fields of characteristic zero into ideals.

math.QA↗
arXiv2015

Towards a Lie theory for 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 satisfying 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.

math.RT↗
arXiv2015

Projective unitary representations of infinite dimensional Lie groups

For an infinite dimensional Lie group $G$ modelled on a locally convex Lie algebra $\mathfrak{g}$, we prove that every smooth projective unitary representation of $G$ corresponds to a smooth linear unitary representation of a Lie group extension $G^{\sharp}$ of $G$. (The main point is the smooth structure on $G^{\sharp}$.) For infinite dimensional Lie groups $G$ which are 1-connected, regular, and modelled on a barrelled Lie algebra $\mathfrak{g}$, we characterize the unitary $\mathfrak{g}$-representations which integrate to $G$. Combining these results, we give a precise formulation of the correspondence between smooth projective unitary representations of $G$, smooth linear unitary representations of $G^{\sharp}$, and the appropriate unitary representations of its Lie algebra $\mathfrak{g}^{\sharp}$.

math.RT↗
arXiv2014

Nonlinear completely positive maps and dilation theory for real involutive algebras

A real seminormed involutive algebra is a real associative algebra ${\mathcal A}$ endowed with an involutive antiautomorphism $*$ and a submultiplicative seminorm $p$ with $p(a^*) =p(a)$ for $a\in {\mathcal A}$. Then ${\mathop{\tt ball}\nolimits}({\mathcal A},p) := \{a \in {\mathcal A} \colon p(a) < 1\}$ is an involutive subsemigroup. For the case where ${\mathcal A}$ is unital, our main result asserts that a function $ϕ\colon{\mathop{\tt ball}\nolimits}({\mathcal A},p) \to B(V)$, $V$ a Hilbert space, is completely positive (defined suitably) if and only if it is positive definite and analytic for any locally convex topology for which ${\mathop{\tt ball}\nolimits}({\mathcal A},p)$ is open. If $η_{\mathcal A} \colon {\mathcal A} \to C^*({\mathcal A},p)$ is the enveloping $C^*$-algebra of $({\mathcal A},p)$ and $e^{C^*({\mathcal A},p)}$ is the $c_0$-direct sum of the symmetric tensor powers $S^n(C^*({\mathcal A},p))$, then the above two properties are equivalent to the existence of a factorization $ϕ= Φ\circ Γ$, where $Φ\colon e^{C^*({\mathcal A},p)} \to B(V)$ is linear completely positive and $Γ(a) = \sum_{n = 0}^\infty η_{\mathcal A}(a)^{\otimes n}$. We also obtain a suitable generalization to non-unital algebras. An important consequence of this result is a description of the unitary representations of ${\rm U}({\mathcal A})$ with bounded analytic extensions to ${\mathop{\tt ball}\nolimits}({\mathcal A},p)$ in terms of representations of the $C^*$-algebra $e^{C^*({\mathcal A},p)}$.

math.OA↗
arXiv2014

Reflection positivity for the circle group

In this note we characterize those unitary one-parameter groups U^c which admit euclidean realizations in the sense that they are obtained by the analytic continuation process corresponding to reflection positivity from a unitary representation $U$ of the circle group. These are precisely the ones for which there exists an anti-unitary involution $J$ commuting with $U^c$. This provides an interesting link with the modular data arising in Tomita--Takesaki theory. Introducing the concept of a positive definite function with values in the space of sesquilinear forms, we further establish a link between KMS states and reflection positivity on the circle.

math.RT↗
arXiv2014

Reflection positive one-parameter groups and dilations

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 Poincare 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 one-parameter 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.

math.RT↗
arXiv2014

Integrability of unitary representations on reproducing kernel spaces

Let g be a Banach Lie algebra and τ: g ---> g an involution. Write g=h+q for the eigenspace decomposition of g with respect to τand g^c := h+iq for the dual Lie algebra. In this article we show the integrability of two types of infinitesimally unitary representations of g^c. The first class of representation is determined by a smooth positive definite kernel K on a locally convex manifold M. The kernel is assumed to satisfying a natural invariance condition with respect to an infinitesimal action β: g \to V(M) by locally integrable vector fields that is compatible with a smooth action of a connected Lie group $H$ with Lie algebra h. The second class is constructed from a positive definite kernel corresponding to a positive definite distribution K \in C^{-\infty}(M \times M) on a finite dimensional smooth manifold M which satisfies a similar invariance condition with respect to a homomorphism β: g \to V(M). As a consequence, we get a generalization of the Luscher--Mack Theorem which applies to a class of semigroups that need not have a polar decomposition. Our integrability results also apply naturally to local representations and representations arising in the context of reflection positivity.

math.RT↗
arXiv2014

On an invariance property of the space of smooth vectors

Let $(π, \mathcal H)$ be a continuous unitary representation of the (infinite dimensional) Lie group $G$ and $γ\: \mathbb R \to \mathrm{Aut}(G)$ define a continuous action of $\mathbb R$ on $G$. Suppose that $π^\#(g,t) = π(g) U_t$ defines a continuous unitary representation of the semidirect product group $G \rtimes_γ\mathbb R$. The first main theorem of the present note provides criteria for the invariance of the space $\mathcal H^\infty$ of smooth vectors of $π$ under the operators $U_f = \int_\mathbb R f(t)U_t\, dt$ for $f \in L^1(\mathbb R)$, resp., $f \in \mathcal S(\mathbb R)$. Using this theorem we show that, for suitably defined spectral subspaces $\mathfrak g_{\mathbb C}(E)$, $E \subseteq \mathbb R$, in the complexified Lie algebra $\mathfrak g_{\mathbb C}$, and $\mathcal H^\infty(F)$, $F\subseteq \mathbb R$, for $U$ in $\mathcal H^\infty$, we have \[ \mathsf{d}π(\mathfrak g_{\mathbb C}(E)) \mathcal H^\infty(F) \subseteq \mathcal H^\infty(E + F).\]

math.RT↗
arXiv2013

Unitary Representations of Unitary Groups

In this paper we review and streamline some results of Kirillov, Olshanski and Pickrell on unitary representations of the unitary group $\U(\cH)$ of a real, complex or quaternionic separable Hilbert space and the subgroup $\U_\infty(\cH)$, consisting of those unitary operators $g$ for which $g - \1$ is compact. The Kirillov--Olshanski theorem on the continuous unitary representations of the identity component $\U_\infty(\cH)_0$ asserts that they are direct sums of irreducible ones which can be realized in finite tensor products of a suitable complex Hilbert space. This is proved and generalized to inseparable spaces. These results are carried over to the full unitary group by Pickrell's Theorem, asserting that the separable unitary representations of $\U(\cH)$, for a separable Hilbert space $\cH$, are uniquely determined by their restriction to $\U_\infty(\cH)_0$. For the $10$ classical infinite rank symmetric pairs $(G,K)$ of non-unitary type, such as $(\GL(\cH),\U(\cH))$, we also show that all separable unitary representations are trivial.

math.RT↗
arXiv2013

Reflection Positivity and Conformal Symmetry

The concept of reflection positivity has its origins in the work of Osterwalder--Schrader on constructive quantum field theory and duality between unitary representations of the euclidean motion group and the Poincare group. On the mathematical side this duality can be made precise as follows. If $\g$ is a Lie algebra with an involutive automorphism $τ$. Decompose $\g = \fh \oplus \fq = \ker(τ- \1) \oplus \ker(τ+ \1)$ into $τ$-eigenspaces and let $\g^c := \fh \oplus i \fq$. At the core of the notion of reflection positivity is the idea that this duality can sometimes be implemented on the level of unitary representations. The idea is simple on the Lie algebra level: Let $(π,\cH^0)$ be a representation of $\g$ where $π$ acts by skew-symmetric operators. Assume that there exists a unitary operator $J$ of order two such that $J πJ = π\circ τ$ and a $\g$-invariant subspace $\cK^0$ which is {\it $J$-positive}. Then complex linear extension leads to a representation of $\g^c$ on $\cK^0$ by operators which are skew-symmetric with respect to $h_J$, so that we obtain a "unitary" representation of $\g^c$ on the pre-Hilbert space $\cK_J^0 := \cK_J/{v \: \cK^0 \: h_J(v,v)=0}$. The aim of this article is twofold. First we discuss reflection positivity in an abstract setting using {\it reflection positive distributions} on the Lie group $G_τ=G\rtimes {1,τ}$ and {\it reflection positive distribution vectors} of a unitary representation of $G_τ$. Then we apply these ideas to the conformal group $\OO_{1,n+1}^+(\R)$ of the sphere $\bS^n$ as well as the the half-space picture mostly used in physics.

math.RT↗
arXiv2013

Norm continuous unitary representations of Lie algebras of smooth sections

We give a complete description of the bounded (i.e. norm continuous) unitary representations of the Fréchet-Lie algebra of all smooth sections, as well as of the LF-Lie algebra of compactly supported smooth sections, of a smooth Lie algebra bundle whose typical fiber is a compact Lie algebra. For the Lie algebra of all sections, bounded unitary irreducible representations are finite tensor products of so-called evaluation representations, hence in particular finite-dimensional. For the Lie algebra of compactly supported sections, bounded unitary irreducible (factor) representations are possibly infinite tensor products of evaluation representations, which reduces the classification problem to results of Glimm and Powers on irreducible (factor) representations of UHF C*-algebras. The key part in our proof is the classification of irreducible bounded unitary representations of Lie algebras that are the tensor product of a compact Lie algebra and a unital real continuous inverse algebra: every such representation is a finite product of evaluation representations. On the group level, our results cover in particular the bounded unitary representations of the identity component of the group of smooth gauge transformations of a principal fiber bundle with compact base and structure group, and the connected component of the group of special unitary n times n matrices with values in an involutive commutative continuous inverse algebra.

math.RT↗
arXiv2012

Differentiable vectors and unitary representations of Frechet-Lie supergroups

For a locally convex Lie group with the Trotter property, we prove that the space of k-times differentiable vectors of a unitary representation is equal to the intersection of domains of k-fold products of the Lie algebra action. The result also holds for continuous representations of locally exponential groups on metrizable locally convex spaces. As an application, we extend a key stability theorem in the representation theory of Lie supergroups beyond the Banach case.

math.RT↗