arXiv subjects

Karl-Hermann Neeb

Publications and source records attributed to Karl-Hermann Neeb.

At least 109 records · Page 6Linked to original sources

arXiv2009

Semibounded representations and invariant cones in infinite dimensional Lie algebras

A unitary representation of a, possibly infinite dimensional, Lie group $G$ is called semi-bounded if the corresponding operators $i\ddπ(x)$ from the derived representations are uniformly bounded from above on some non-empty open subset of the Lie algebra $\g$. In the first part of the present paper we explain how this concept leads to a fruitful interaction between the areas of infinite dimensional convexity, Lie theory, symplectic geometry (momentum maps) and complex analysis. Here open invariant cones in Lie algebras play a central role and semibounded representations have interesting connections to $C^*$-algebras which are quite different from the classical use of the group $C^*$-algebra of a finite dimensional Lie group. The second half is devoted to a detailed discussion of semibounded representations of the diffeomorphism group of the circle, the Virasoro group, the metaplectic representation on the bosonic Fock space and the spin representation on fermionic Fock space.

math.RT↗

arXiv2009

Borel--Weil Theory for Groups over Commutative Banach Algebras

Let $\cA$ be a commutative unital Banach algebra, $\g$ be a semisimple complex Lie algebra and $G(\cA)$ be the 1-connected Banach--Lie group with Lie algebra $\g \otimes \cA$. Then there is a natural concept of a parabolic subgroup $P(\cA)$ of $G(\cA)$ and we obtain generalizations $X(\cA) := G(\cA)/P(\cA)$ of the generalized flag manifolds. In this note we provide an explicit description of all homogeneous holomorphic line bundles over $X(\cA)$ with non-zero holomorphic sections. In particular, we show that all these line bundles are tensor products of pullbacks of line bundles over $X(\C)$ by evaluation maps. For the special case where $\cA$ is a $C^*$-algebra, our results lead to a complete classification of all irreducible involutive holomorphic representations of $G(\cA)$ on Hilbert spaces.

math.RT↗

arXiv2009

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.

math.DG↗

arXiv2009

Unitary highest weight modules of locally affine Lie algebras

Locally affine Lie algebras are generalizations of affine Kac--Moody algebras with Cartan subalgebras of infinite rank whose root system is locally affine. In this note we study a class of representations of locally affine algebras generalizing integrable highest weight modules. In particular, we construct such an integrable representation for each integral weight not vanishing on the center and show that, over the complex numbers, we thus obtain unitary representations w.r.t. a unitary real form. We also use Yoshii's recent classification of locally affine root systems to derive a classification of so-called minimal locally affine Lie algebras and give realizations as twisted loop algebras.

math.RT↗

arXiv2009

Geometric characterization of hermitian algebras with continuous inversion

A hermitian algebra is a unital associative ${\mathbb C}$-algebra endowed with an involution such that the spectra of self-adjoint elements are contained in ${\mathbb R}$. In the case of an algebra ${\mathcal A}$ endowed with a Mackey-complete, locally convex topology such that the set of invertible elements is open and the inversion mapping is continuous, we construct the smooth structures on the appropriate versions of flag manifolds. Then we prove that if such a locally convex algebra ${\mathcal A}$ is endowed with a continuous involution, then it is a hermitian algebra if and only if the natural action of all unitary groups $U_n({\mathcal A})$ on each flag manifold is transitive.

math.FA↗

arXiv2009

Borel-Weil Theory for Root Graded Banach-Lie groups

In this paper we introduce (weakly) root graded Banach--Lie algebras and corresponding Lie groups as natural generalizations of group like $\GL_n(A)$ for a Banach algebra $A$ or groups like $C(X,K)$ of continuous maps of a compact space $X$ into a complex semisimple Lie group $K$. We study holomorphic induction from holomorphic Banach representations of so-called parabolic subgroups $P$ to representations of $G$ on holomorphic sections of homogeneous vector bundles over $G/P$. One of our main results is an algebraic characterization of the space of sections which is used to show that this space actually carries a natural Banach structure, a result generalizing the finite dimensionality of spaces of sections of holomorphic bundles over compact complex manifolds. We also give a geometric realization of any irreducible holomorphic representation of a (weakly) root graded Banach--Lie group $G$ and show that all holomorphic functions on the spaces $G/P$ are constant.

math.RT↗

arXiv2008

A non-smooth continuous unitary representation of a Banach-Lie group

In this note we show that the representation of the additive group of the Hilbert space $L^2([0,1],\R)$ on $L^2([0,1],\C)$ given by the multiplication operators $π(f) := e^{if}$ is continuous but its space of smooth vectors is trivial. This example shows that a continuous unitary representation of an infinite dimensional Lie group need not be smooth.

math.RT↗

arXiv2008

Closedness of the tangent spaces to the orbits of proper actions

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 $\g \to T_x(M)$ have closed range for each $x \in M$, 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.

math.DG↗

arXiv2008

The second cohomology of current algebras of general Lie algebras

Let $A$ be a unital commutative associative algebra over a field of characteristic zero, $\k$ be a Lie algebra, and $\z$ a vector space, considered as a trivial module of the Lie algebra $\g := A \otimes \k$. In this paper we give a description of the cohomology space $H^2(\g,\z)$ in terms of well accessible data associated to $A$ and $\k$. We also discuss the topological situation, where $A$ and $\k$ are locally convex algebras.

math.RA↗

arXiv2008

Semi-bounded unitary representations of infinite-dimensional Lie groups

In this note we introduce the concept of a semi-bounded unitary representations of an infinite-dimensional Lie group $G$. Semi-boundedness is defined in terms of the corresponding momentum set in the dual $\g'$ of the Lie algebra $\g$ of $G$. After dealing with some functional analytic issues concerning certain weak-$*$-locally compact subsets of dual spaces, called semi-equicontinuous, we characterize unitary representations which are bounded in the sense that their momentum set is equicontinuous, we characterize semi-bounded representations of locally convex spaces in terms of spectral measures, and we also describe a method to compute momentum sets of unitary representations of reproducing kernel Hilbert spaces of holomorphic functions.

math.RT↗

arXiv2008

An abstract setting for hamiltonian actions

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 $sp(h,ω) \supeq ham(h,ω)$ 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 $g \to ham(h,ω)$, i.e. abstract hamiltonian actions, corresponding central and abelian extensions of $G$ and momentum maps $J : g \to V$.

math.SG↗

arXiv2008

Finite-dimensional Lie subalgebras of algebras with continuous inversion

We investigate the finite-dimensional Lie groups whose points are separated by the continuous homomorphisms into groups of invertible elements of locally convex algebras with continuous inversion that satisfy an appropriate completeness condition. We find that these are precisely the linear Lie groups, that is, the Lie groups which can be faithfully represented as matrix groups. Our method relies on proving that certain finite-dimensional Lie subalgebras of algebras with continuous inversion commute modulo the Jacobson radical.

math.FA↗

arXiv2008

Lie group extensions associated to projective modules of continuous inverse algebras

We call a unital locally convex algebra $A$ a continuous inverse algebra if its unit group $A^\times$ is open and inversion is a continuous map. For any smooth action of a, possibly infinite-dimensional, connected Lie group $G$ on a continuous inverse algebra $A$ by automorphisms and any finitely generated projective right $A$-module $E$, we construct a Lie group extension $\hat G$ of $G$ by the group $\GL_A(E)$ of automorphisms of the $A$-module $E$. This Lie group extension is a ``non-commutative'' version of the group $\Aut(\V)$ of automorphism of a vector bundle over a compact manifold $M$, which arises for $G = \Diff(M)$, $A = C^\infty(M,\C)$ and $E = Γ\V$. We also identify the Lie algebra $\hat\g$ of $\hat G$ and explain how it is related to connections of the $A$-module $E$.

math.OA↗

arXiv2008

Solvable Subgroups of Locally Compact Groups

It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are discussed as far as they carry.

math.GR↗

arXiv2007

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 αon 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.

math.OA↗

arXiv2007

Lie groups of bundle automorphisms and their extensions

We describe natural abelian extensions of the Lie algebra $\aut(P)$ of infinitesimal automorphisms of a principal bundle over a compact manifold $M$ and discuss their integrability to corresponding Lie group extensions. Already the case of a trivial bundle $P = M \times K$ is quite interesting. In this case, we show that essentially all central extensions of the gauge algebra $C^\infty(M,\fk)$ can be obtained from three fundamental types of cocycles with values in one of the spaces $\fz := C^\infty(M,V)$, $Ω^1(M,V)$ and $Ω^1(M,V)/\dd C^\infty(M,V)$. These cocycles extend to $\aut(P)$, and, under the assumption that $TM$ is trivial, we also describe the space $H^2({\cal V}(M),\fz)$ classifying the twists of these extensions. We then show that all fundamental types have natural generalizations to non-trivial bundles and explain under which conditions they extend to $\aut(P)$ and integrate to global Lie group extensions.

math.DG↗

arXiv2007

On the cohomology of vector fields on parallelizable manifolds

In the present paper we determine for each parallelizable smooth compact manifold $M$ the cohomology spaces $H^2(V_M,\barΩ^p_M)$ of the Lie algebra $V_M$ of smooth vector fields on $M$ with values in the module $\barΩ^p_M = Ω^p_M/dΩ^{p-1}_M$. The case of $p=1$ is of particular interest since the gauge algebra $C^\infty (M,k)$ has the universal central extension with center $\barΩ^1_M$, generalizing affine Kac-Moody algebras. The second cohomology $H^2(V_M, \barΩ^1_M)$ classifies twists of the semidirect product of $V_M$ with the universal central extension $C^\infty (M,k) \oplus \barΩ^1_M$.

math.RT↗

arXiv2007

An implicit function theorem for Banach spaces and some applications

We prove a generalized implicit function theorem for Banach spaces, without the usual assumption that the subspaces involved being complemented. Then we apply it to the problem of parametrization of fibers of differentiable maps, the Lie subgroup problem for Banach-Lie groups, as well as Weil's local rigidity for homomorphisms from finitely generated groups to Banach-Lie groups.

math.GR↗