scholarly journals Another proof for the continuity of the Lipsman mapping

2020 ◽  
Vol 72 (7) ◽  
pp. 945-951
Author(s):  
A. Messaoud ◽  
A. Rahali
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Semidirect Product ◽  
Lie Group ◽  
Inner Product ◽  
Coadjoint Orbits ◽  
Compact Lie Group ◽  
Real Vector ◽  
Finite Dimensional ◽  
Unitary Dual

UDC 515.1 We consider the semidirect product G = K ⋉ V where K is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space V equipped with an inner product 〈 , 〉 . By G ^ we denote the unitary dual of G and by 𝔤 ‡ / G the space of admissible coadjoint orbits, where 𝔤 is the Lie algebra of G . It was pointed out by Lipsman that the correspondence between G ^ and 𝔤 ‡ / G is bijective. Under some assumption on G , we give another proof for the continuity of the orbit mapping (Lipsman mapping) Θ : 𝔤 ‡ / G - → G ^ .

scholarly journals Traces for Star Products on the Dual of a Lie Algebra

2003 ◽  
Vol 15 (05) ◽  
pp. 425-445 ◽  
Author(s):  
Pierre Bieliavsky ◽  
Simone Gutt ◽  
Martin Bordemann ◽  
Stefan Waldmann
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Star Product ◽  
Coadjoint Orbits ◽  
Star Products ◽  
Compact Lie Group ◽  
Regular Orbit ◽  
Elementary Argument

In this paper, we describe all traces for the BCH star-product on the dual of a Lie algebra. First we show by an elementary argument that the BCH as well as the Kontsevich star-product are strongly closed if and only if the Lie algebra is unimodular. In a next step we show that the traces of the BCH star-product are given by the ad-invariant functionals. Particular examples are the integration over coadjoint orbits. We show that for a compact Lie group and a regular orbit one can even achieve that this integration becomes a positive trace functional. In this case we explicitly describe the corresponding GNS representation. Finally we discuss how invariant deformations on a group can be used to induce deformations of spaces where the group acts on.

scholarly journals Invariant Whitney functions

2019 ◽  
Vol 30 (02) ◽  
pp. 1950009
Author(s):  
Hans-Christian Herbig ◽  
Markus J. Pflaum
Keyword(s):  
Lie Group ◽  
Invariant Function ◽  
Compact Lie Group ◽  
Similar Statement ◽  
Real Vector ◽  
Linear Action ◽  
Smooth Functions ◽  
Finite Dimensional ◽  
Subanalytic Set ◽  
Whitney Functions

Theorem 1 of [G. W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975) 63–68.] says that for a linear action of a compact Lie group [Formula: see text] on a finite dimensional real vector space [Formula: see text], any smooth [Formula: see text]-invariant function on [Formula: see text] can be written as a composite with the Hilbert map. We prove a similar statement for the case of Whitney functions along a subanalytic set [Formula: see text] fulfilling some regularity assumptions. In order to deal with the case when [Formula: see text] is not [Formula: see text]-stable, we use the language of groupoids.

scholarly journals Linear Imbeddings of Self-Dual Homogeneous Cones

1972 ◽  
Vol 46 ◽  
pp. 121-145 ◽  
Author(s):  
I. Satake
Keyword(s):  
Vector Space ◽  
Lie Group ◽  
Isotropy Subgroup ◽  
Real Vector Space ◽  
Real Vector ◽  
Finite Dimensional ◽  
Homogeneous Cones ◽  
Homogeneous Cone

Let G be a reductive algebraic Lie group acting linearly on a (finite-dimensional) real vector-space U with a maximal compact isotropy subgroup K and suppose that the quotient Ω = G/K is a self-dual homogeneous cone in U. Let (G′, K′) be another such pair corresponding to a self-dual homogeneous cone Ω′ in U′.

Topological equivalence and rigidity of flows on certain solvmanifolds

1999 ◽  
Vol 19 (3) ◽  
pp. 559-569
Author(s):  
D. BENARDETE ◽  
S. G. DANI
Keyword(s):  
Lie Group ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Real Vector ◽  
Orbit Equivalent ◽  
Finite Dimensional ◽  
Generic Class ◽  
Induced Flow ◽  
Simply Connected ◽  
Solvable Lie Group

Given a Lie group $G$ and a lattice $\Gamma$ in $G$, a one-parameter subgroup $\phi$ of $G$ is said to be rigid if for any other one-parameter subgroup $\psi$, the flows induced by $\phi$ and $\psi$ on $\Gamma\backslash G$ (by right translations) are topologically orbit-equivalent only if they are affinely orbit-equivalent. It was previously known that if $G$ is a simply connected solvable Lie group such that all the eigenvalues of $\mathrm{Ad} (g) $, $g\in G$, are real, then all one-parameter subgroups of $G$ are rigid for any lattice in $G$. Here we consider a complementary case, in which the eigenvalues of $\mathrm{Ad} (g)$, $g\in G$, form the unit circle of complex numbers.Let $G$ be the semidirect product $N \rtimes M$, where $M$ and $N$ are finite-dimensional real vector spaces and where the action of $M$ on the normal subgroup $N$ is such that the center of $G$ is a lattice in $M$. We prove that there is a generic class of abelian lattices $\Gamma$ in $G$ such that any semisimple one-parameter subgroup $\phi$ (namely $\phi$ such that $\mathrm{Ad} (\phi_t)$ is diagonalizable over the complex numbers for all $t$) is rigid for $\Gamma$ (see Theorem 1.4). We also show that, on the other hand, there are fairly high-dimensional spaces of abelian lattices for which some semisimple $\phi$ are not rigid (see Corollary 4.3); further, there are non-rigid semisimple $\phi$ for which the induced flow is ergodic.

On structure theory of pre-Hilbert algebras

2009 ◽  
Vol 139 (2) ◽  
pp. 303-319 ◽  
Author(s):  
José Antonio Cuenca Mira
Keyword(s):  
Vector Space ◽  
Associative Algebra ◽  
Real Vector Space ◽  
Inner Product ◽  
Structure Theory ◽  
Classical Theorem ◽  
Real Vector ◽  
Algebraic Identity ◽  
Hilbert Algebras

Let A be a real (non-associative) algebra which is normed as real vector space, with a norm ‖·‖ deriving from an inner product and satisfying ‖ac‖ ≤ ‖a‖‖c‖ for any a,c ∈ A. We prove that if the algebraic identity (a((ac)a))a = (a2c)a2 holds in A, then the existence of an idempotent e such that ‖e‖ = 1 and ‖ea‖ = ‖a‖ = ‖ae‖, a ∈ A, implies that A is isometrically isomorphic to ℝ, ℂ, ℍ, $\mathbb{O}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\CC}}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\mathbb{H}}}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\mathbb{O}}}$ or ℙ. This is a non-associative extension of a classical theorem by Ingelstam. Finally, we give some applications of our main result.

scholarly journals Analytic Extensions of Representations of *-Subsemigroups Without Polar Decomposition

2020 ◽  
Author(s):  
Daniel Oeh
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Polar Decomposition ◽  
Complex Hilbert Space ◽  
Analytic Extension ◽  
Involutive Automorphism ◽  
Finite Dimensional ◽  
Continuous Unitary Representation ◽  
Continuous Representations ◽  
Connected Lie Group

Abstract Let $(G,\tau )$ be a finite-dimensional Lie group with an involutive automorphism $\tau $ of $G$ and let ${{\mathfrak{g}}} = {{\mathfrak{h}}} \oplus{{\mathfrak{q}}}$ be its corresponding Lie algebra decomposition. We show that every nondegenerate strongly continuous representation on a complex Hilbert space ${\mathcal{H}}$ of an open $^\ast $-subsemigroup $S \subset G$, where $s^{\ast } = \tau (s)^{-1}$, has an analytic extension to a strongly continuous unitary representation of the 1-connected Lie group $G_1^c$ with Lie algebra $[{{\mathfrak{q}}},{{\mathfrak{q}}}] \oplus i{{\mathfrak{q}}}$. We further examine the minimal conditions under which an analytic extension to the 1-connected Lie group $G^c$ with Lie algebra ${{\mathfrak{h}}} \oplus i{{\mathfrak{q}}}$ exists. This result generalizes the Lüscher–Mack theorem and the extensions of the Lüscher–Mack theorem for $^\ast $-subsemigroups satisfying $S = S(G^\tau )_0$ by Merigon, Neeb, and Ólafsson. Finally, we prove that nondegenerate strongly continuous representations of certain $^\ast $-subsemigroups $S$ can even be extended to representations of a generalized version of an Olshanski semigroup.

scholarly journals The Fixed Point Method for Fuzzy Approximation of a Functional Equation Associated with Inner Product Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
M. Eshaghi Gordji ◽  
H. Khodaei
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Vector Space ◽  
Fixed Point Method ◽  
Inner Product ◽  
Product Spaces ◽  
Real Vector ◽  
Fixed Integer ◽  
Inner Product Spaces ◽  
Fuzzy Approximation

Th. M. Rassias (1984) proved that the norm defined over a real vector space is induced by an inner product if and only if for a fixed integer holds for all The aim of this paper is to extend the applications of the fixed point alternative method to provide a fuzzy stability for the functional equation which is said to be a functional equation associated with inner product spaces.

scholarly journals Ruang Fase Tereduksi Grup Lie Aff (1)

2021 ◽  
Vol 3 (2) ◽  
pp. 180-186
Author(s):  
Edi Kurniadi
Keyword(s):  
Phase Space ◽  
Lie Algebra ◽  
Group Action ◽  
Lie Group ◽  
Dual Space ◽  
Coadjoint Orbits ◽  
Coadjoint Representation

ABSTRAKDalam artikel ini dipelajari ruang fase tereduksi dari suatu grup Lie khususnya untuk grup Lie affine  berdimensi 2. Tujuannya adalah untuk mengidentifikasi ruang fase tereduksi dari  melalui orbit coadjoint buka di titik tertentu pada ruang dual  dari aljabar Lie . Aksi dari grup Lie    pada ruang dual  menggunakan representasi coadjoint. Hasil yang diperoleh adalah ruang Fase tereduksi  tiada lain adalah orbit coadjoint-nya yang buka di ruang dual . Selanjutnya, ditunjukkan pula bahwa grup Lie affine     tepat mempunyai dua buah orbit coadjoint buka.  Hasil yang diperoleh dalam penelitian ini dapat diperluas untuk kasus grup Lie affine  berdimensi  dan untuk kasus grup Lie lainnya.ABSTRACTIn this paper, we study a reduced phase space for a Lie group, particularly for the 2-dimensional affine Lie group which is denoted by Aff (1). The work aims to identify the reduced phase space for Aff (1) by open coadjoint orbits at certain points in the dual space aff(1)* of the Lie algebra aff(1). The group action of Aff(1) on the dual space aff(1)* is considered using coadjoint representation. We obtained that the reduced phase space for the affine Lie group Aff(1) is nothing but its open coadjoint orbits. Furthermore, we show that the affine Lie group Aff (1) exactly has two open coadjoint orbits in aff(1)*. Our result can be generalized for the n(n+1) dimensional affine Lie group Aff(n) and for another Lie group.

EQUIVARIANT HOLOMORPHIC HERMITIAN PRINCIPAL BUNDLES OVER A EUCLIDEAN SPACE

2013 ◽  
Vol 05 (03) ◽  
pp. 345-360
Author(s):  
INDRANIL BISWAS
Keyword(s):  
Euclidean Space ◽  
Vector Space ◽  
Algebraic Group ◽  
Inner Product ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Principal Bundles ◽  
Isomorphism Classes ◽  
Finite Dimensional

Let V be a finite dimensional complex vector space equipped with an inner product. Let G denote the group of all affine automorphisms of V preserving the metric defined by the inner product. Let H be a connected reductive affine algebraic group defined over ℂ. We give an explicit classification of the isomorphism classes of G-equivariant holomorphic hermitian principal H-bundles over V.

On the Cortex of the Groups 𝕋n ⋉ ℍn

2020 ◽  
pp. 2150149
Author(s):  
Aymen Rahali
Keyword(s):  
Heisenberg Group ◽  
Semidirect Product ◽  
Representation Formula ◽  
Irreducible Representations ◽  
Coadjoint Orbits ◽  
Dimensional Torus ◽  
Product Group ◽  
Unitary Dual ◽  
Identity Representation

Let [Formula: see text] be the [Formula: see text]-dimensional Heisenberg group and let [Formula: see text] be the [Formula: see text]-dimensional torus acting on [Formula: see text] by automorphisms. We consider the semidirect product group [Formula: see text] The cortex, [Formula: see text] of [Formula: see text] is the set of all unitary irreducible representations [Formula: see text] in the unitary dual [Formula: see text] of [Formula: see text] that cannot be Hausdorff separated from the identity representation [Formula: see text] of [Formula: see text] In this paper, we describe explicitly the cortex ([Formula: see text]) of [Formula: see text] using the coadjoint orbits of the group.

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