scholarly journals L1-DETERMINED PRIMITIVE IDEALS IN THE C∗-ALGEBRA OF AN EXPONENTIAL LIE GROUP WITH CLOSED NON-∗-REGULAR ORBITS

2020 ◽  
Vol 74 (1) ◽  
pp. 127-148
Author(s):  
Junko INOUE ◽  
Jean LUDWIG
Keyword(s):  
Lie Group ◽  
Primitive Ideals ◽  
Exponential Lie Group

Primitive ideals with bounded approximate units in L1-algebras of exponential lie groups

1986 ◽  
Vol 41 (3) ◽  
pp. 411-420 ◽  
Author(s):  
Mohammed El Bachir Bekka
Keyword(s):  
Lie Groups ◽  
Solvable Group ◽  
Lie Group ◽  
Irreducible Representations ◽  
Primitive Ideals ◽  
Exponential Lie Group ◽  
Exponential Lie Groups

AbstractLet G be an exponential Lie group. We study primitive ideals (i.e. kernels of irreducible *-representations of L1(G)), with bounded approximate units (b.a.u.). We prove a result relating the existence of b.a.u. in certain primitive ideals with the geometry of the corresponding Kirillov orbits. This yields for a solvable group of class 2, a characterization of the primitive ideals with b.a.u.

Représentations irréductibles bornées des groupes de Lie exponentiels

2001 ◽  
Vol 53 (5) ◽  
pp. 944-978 ◽  
Author(s):  
J. Ludwig ◽  
C. Molitor-Braun
Keyword(s):  
Banach Space ◽  
Irreducible Representation ◽  
Lie Group ◽  
Simple Extension ◽  
Induced Representations ◽  
New Type ◽  
Exponential Lie Group

AbstractLet G be a solvable exponential Lie group. We characterize all the continuous topologically irreducible bounded representations (T, ) of G on a Banach space by giving a G-orbit in n* (n being the nilradical of g), a topologically irreducible representation of L1(ℝn, ω), for a certain weight ω and a certain n ∈ ℕ, and a topologically simple extension norm. If G is not symmetric, i.e., if the weight ω is exponential, we get a new type of representations which are fundamentally different from the induced representations.

Topology on the unitary dual of completely solvable Lie groups

2015 ◽  
Vol 6 (4) ◽  
Author(s):  
Detlev Poguntke
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Orbit Space ◽  
Orbit Method ◽  
Solvable Lie Groups ◽  
Bijective Correspondence ◽  
Unitary Dual ◽  
Exponential Lie Group ◽  
Solvable Lie Group

AbstractIt was one of great successes of Kirillov's orbit method to see that the unitary dual of an exponential Lie group is in bijective correspondence with the orbit space associated with the linear dual of the Lie algebra of the group in question. To show that this correspondence is an homeomorphism turned out to be unexpectedly difficult. Only in 1994 H. Leptin and J. Ludwig gave a proof using the notion of variable groups. In this article their proof in the case of completely solvable Lie group is reorganized, some “philosophy” and some new arguments are added. The purpose is to contribute to a better understanding of this proof.

scholarly journals OPERATOR KERNELS FOR IRREDUCIBLE REPRESENTATIONS OF EXPONENTIAL LIE GROUPS

2008 ◽  
Vol 78 (2) ◽  
pp. 301-316
Author(s):  
DETLEV POGUNTKE
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Unitary Representation ◽  
Kernel Function ◽  
Lie Group ◽  
Linear Form ◽  
Irreducible Representations ◽  
Compactly Supported ◽  
Exponential Lie Group ◽  
Exponential Lie Groups

AbstractA nine-dimensional exponential Lie group G and a linear form ℓ on the Lie algebra of G are presented such that for all Pukanszky polarizations 𝔭 at ℓ the canonically associated unitary representation ρ=ρ(ℓ,𝔭) of G has the property that ρ(ℒ1(G)) does not contain any nonzero operator given by a compactly supported kernel function. This example shows that one of Leptin’s results is wrong, and it cannot be repaired.

Symmetry properties of fractional order transport equations

2012 ◽  
Vol 9 (1) ◽  
pp. 59-64
Author(s):  
R.K. Gazizov ◽  
A.A. Kasatkin ◽  
S.Yu. Lukashchuk
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lie Group ◽  
Initial Conditions ◽  
Group Analysis ◽  
Equivalence Transformation ◽  
Point Change ◽  
Infinitesimal Operator ◽  
Symmetry Properties ◽  
Fractional Differential

In the paper some features of applying Lie group analysis methods to fractional differential equations are considered. The problem related to point change of variables in the fractional differentiation operator is discussed and some general form of transformation that conserves the form of Riemann-Liouville fractional operator is obtained. The prolongation formula for extending an infinitesimal operator of a group to fractional derivative with respect to arbitrary function is presented. Provided simple example illustrates the necessity of considering both local and non-local symmetries for fractional differential equations in particular cases including the initial conditions. The equivalence transformation forms for some fractional differential equations are discussed and results of group classification of the wave-diffusion equation are presented. Some examples of constructing particular exact solutions of fractional transport equation are given, based on the Lie group methods and the method of invariant subspaces.

ε‎-Invariance

2018 ◽  
Author(s):  
Ercüment H. Ortaçgil
Keyword(s):  
Lie Algebra ◽  
Lie Group

The discussions up to Chapter 4 have been concerned with the Lie group. In this chapter, the Lie algebra is constructed by defining the operators ∇ and ∇̃.

The Centralizer

2018 ◽  
Author(s):  
Ercüment H. Ortaçgil
Keyword(s):  
Lie Group ◽  
Group Structure ◽  
Diffeomorphism Group ◽  
Group Diff ◽  
Local Solutions

The pseudogroup of local solutions in Chapter 3 defines another pseudogroup by taking its centralizer inside the diffeomorphism group Diff(M) of a manifold M. These two pseudogroups define a Lie group structure on M.

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

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