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 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 ^ .

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 Extension theorems for smooth functions on real analytic spaces and quotients by Lie groups and smooth stability

1977 ◽  
Vol 24 (4) ◽  
pp. 440-457
Author(s):  
G. S. Wells
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Analytic Space ◽  
Main Step ◽  
Compact Lie Group ◽  
Orbit Type ◽  
Smooth Functions ◽  
Extension Theorems ◽  
Real Analytic

AbstractExtension theorems are proved for smooth functions on a coherent real analytic space for which local defining functions exist which are finitely determined in the sense of J. Mather, (1968), and for smooth functions invariant under the action of a compact lie groupG. thus providing the main step in the proof that smooth infinitesimal stability implies smooth stability in the appropriate categories. In addition the remaining step for the category ofCxG-manifolds of finite orbit type is filled in.

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′.

scholarly journals Lifting Quasianalytic Mappings over Invariants

2012 ◽  
Vol 64 (2) ◽  
pp. 409-428
Author(s):  
Armin Rainer
Keyword(s):  
Lie Group ◽  
Special Functions ◽  
Linear Algebraic Group ◽  
Functions Of Bounded Variation ◽  
Local Power ◽  
Compact Lie Group ◽  
Real Representation ◽  
Finite Dimensional ◽  
Closed Orbits ◽  
Real Analytic

Abstract Let be a rational finite dimensional complex representation of a reductive linear algebraic group G, and let be a system of generators of the algebra of invariant poly- nomials ℂ[V]G. We study the problem of lifting mappings over the mapping of invariants . Note that ¾(V) can be identified with the cate- gorical quotientV//G and its points correspond bijectively to the closed orbits inV. We prove that if f belongs to a quasianalytic subclass satisfying some mild closedness properties that guarantee resolution of singularities in 𝒞 e.g., the real analytic class, then f admits a lift of the same class 𝒞 after desingularization by local blow-ups and local power substitutions. As a consequence we show that f itself allows for a lift that belongs to SBVloc, i.e., special functions of bounded variation. If ρ is a real representation of a compact Lie group, we obtain stronger versions.

Construction of Levi processes on path spaces of Lie groups

2016 ◽  
Vol 19 (01) ◽  
pp. 1650002
Author(s):  
A. A. Kalinichenko
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Weak Limit ◽  
Convolution Semigroup ◽  
Path Space ◽  
Probability Measures ◽  
Compact Lie Group ◽  
Piecewise Constant ◽  
Finite Dimensional

Given a compact Lie group and a conjugate-invariant Levi process on it, generated by the operator [Formula: see text], we construct the Levi process on the path space of [Formula: see text], associated with the convolution semigroup [Formula: see text] of probability measures, where [Formula: see text] is the distribution of the Levi process on [Formula: see text] generated by [Formula: see text]. The constructed process is obtained as the weak limit of piecewise constant paths, which, as well as proving its existence and properties, provides finite-dimensional approximations of Chernoff type to the integrals with respect to its distribution.

scholarly journals The action of a compact Lie group on nilpotent Lie algebras of type {{n,2}}

2016 ◽  
Vol 28 (4) ◽  
Author(s):  
Giovanni Falcone ◽  
Ágota Figula
Keyword(s):  
Lie Algebras ◽  
Lie Group ◽  
Compact Lie Group ◽  
Nilpotent Lie Algebras ◽  
Group Of Automorphisms ◽  
Finite Dimensional

AbstractWe classify finite-dimensional real nilpotent Lie algebras with 2-dimensional central commutator ideals admitting a Lie group of automorphisms isomorphic to

A SYMMETRY PRESERVING REDUCTION SCHEME FOR PERIODIC POINTS NEAR A FIXED POINT OF FAMILIES OF DIFFEOMORPHISMS WITH A COMPACT SYMMETRY GROUP

2006 ◽  
Vol 16 (09) ◽  
pp. 2545-2557
Author(s):  
MARIA-CRISTINA CIOCCI ◽  
JOHAN NOLDUS
Keyword(s):  
Fixed Point ◽  
Normal Form ◽  
Vector Space ◽  
Periodic Points ◽  
Reduction Scheme ◽  
Real Vector ◽  
Linear Action ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Normal Form Theorem

We present a generalized Lyapunov Schmidt (LS) reduction scheme for diffeomorphisms on a finite dimensional real vector space V which transform under real one-dimensional characters χ of an arbitrary compact group with linear action on V. Moreover we prove a normal form theorem, such that the normal form still has the desirable transformation properties with respect to χ.

scholarly journals ON THE CHOW RING OF CERTAIN LEHN–LEHN–SORGER–VAN STRATEN EIGHTFOLDS

2021 ◽  
pp. 1-24
Author(s):  
CHIARA CAMERE ◽  
ALBERTO CATTANEO ◽  
ROBERT LATERVEER
Keyword(s):  
Chow Group ◽  
Chow Ring ◽  
Similar Statement ◽  
Intersection Product ◽  
Finite Dimensional

Abstract We consider a 10-dimensional family of Lehn–Lehn–Sorger–van Straten hyperkähler eightfolds, which have a non-symplectic automorphism of order 3. Using the theory of finite-dimensional motives, we show that the action of this automorphism on the Chow group of 0-cycles is as predicted by the Bloch–Beilinson conjectures. We prove a similar statement for the anti-symplectic involution on varieties in this family. This has interesting consequences for the intersection product of the Chow ring of these varieties.

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