Asymptotic Decomposition of Differential Systems with Small Parameter in the Representation Space of Finite-dimensional Lie Group

1995 ◽  
pp. 219-258
Author(s):  
Yu. A. Mitropolsky ◽  
A. K. Lopatin
Keyword(s):  
Small Parameter ◽  
Lie Group ◽  
Representation Space ◽  
Asymptotic Decomposition ◽  
Differential Systems ◽  
Finite Dimensional

Asymptotic decomposition of differential systems with a small parameter

1987 ◽  
Vol 39 (2) ◽  
pp. 162-170
Author(s):  
Yu. A. Mitropol'skii ◽  
A. K. Lopatin
Keyword(s):  
Small Parameter ◽  
Asymptotic Decomposition ◽  
Differential Systems

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.

scholarly journals Invariant connections and invariant holomorphic bundles on homogeneous manifolds

2014 ◽  
Vol 12 (1) ◽  
pp. 1-13
Author(s):  
Indranil Biswas ◽  
Andrei Teleman
Keyword(s):  
Lie Group ◽  
Complex Structure ◽  
Differentiable Manifold ◽  
Equivalence Classes ◽  
Transitive Action ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Group A ◽  
Holomorphic Bundles ◽  
Invariant Gauge

AbstractLet X be a differentiable manifold endowed with a transitive action α: A×X→X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects:equivalence classes of α-invariant K-connections on X α-invariant gauge classes of K-connections on X, andα-invariant isomorphism classes of pairs (Q,P) consisting of a holomorphic Kℂ-bundle Q → X and a K-reduction P of Q (when X has an α-invariant complex structure).

A New Form of the Segal-Bargmann Transform for Lie Groups of Compact Type

1999 ◽  
Vol 51 (4) ◽  
pp. 816-834 ◽  
Author(s):  
Brian C. Hall
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Joint Work ◽  
Compact Type ◽  
Bargmann Transform ◽  
Infinite Dimensional ◽  
Infinite Dimensional Analysis ◽  
Finite Dimensional ◽  
New Form ◽  
Two Parameter

AbstractI consider a two-parameter family Bs,t of unitary transforms mapping an L2-space over a Lie group of compact type onto a holomorphic L2-space over the complexified group. These were studied using infinite-dimensional analysis in joint work with B. Driver, but are treated here by finite-dimensional means. These transforms interpolate between two previously known transforms, and all should be thought of as generalizations of the classical Segal-Bargmann transform. I consider also the limiting cases s → ∞ and s → t/2.

scholarly journals ON STABILITY CRITERIA OF FRACTAL DIFFERENTIAL SYSTEMS OF CONFORMABLE TYPE

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040009
Author(s):  
AWAIS YOUNUS ◽  
THABET ABDELJAWAD ◽  
TAZEEN GUL
Keyword(s):  
Asymptotic Stability ◽  
Control Theory ◽  
Stability Criteria ◽  
Differential Systems ◽  
Central Concern ◽  
Finite Dimensional ◽  
Linear And Nonlinear ◽  
Stability Results

In this paper, stability results of central concern for control theory are given for finite-dimensional linear and nonlinear local fractional or fractal differential systems. The main purpose of this paper is to provide some results on stability and asymptotic stability of conformable order systems, together with some illustrating examples.

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 Cylindrical representations of some infinite dimensional nuclear Lie groups

1992 ◽  
Vol 46 (2) ◽  
pp. 295-310 ◽  
Author(s):  
Jean Marion
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Additive Group ◽  
Test Functions ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations ◽  
Direct Product Group ◽  
Necessary And Sufficient

Let Γ.𝒜 be the semi-direct product group of a nuclear Lie group Γ with the additive group 𝒜 of a real nuclear vector space. We give an explicit description of all the continuous representations of Γ.𝒜 the restriction of which to 𝒜 is a cyclic unitary representation, and a necessary and sufficient condition for the unitarity of such cylindrical representations is stated. This general result is successfully used to obtain irreducible unitary representations of the nuclear Lie groups of Riemannian motions, and, in the setting of the theory of multiplicative distributions initiated by I.M. Gelfand, it is proved that for any connected real finite dimensional Lie groupGand for any strictly positive integerkthere exist non located and non trivially decomposable representations of orderkof the nuclear Lie group(M;G) of all theG-valued test-functions on a given paracompact manifoldM.

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