scholarly journals Stationary Subalgebras in General Position for Tensor Product

2020 ◽  
pp. 4-15
Author(s):  
O.G. Styrt
Keyword(s):  
Tensor Product ◽  
Compact Group ◽  
Direct Summand ◽  
Lie Groups ◽  
Lie Algebras ◽  
Linear Algebra ◽  
General Position ◽  
Linear Groups ◽  
Group Representations ◽  
Stationary Subalgebra

The paper studies stationary subalgebras in general position of compact linear groups. We prove that, except for several specific cases, a stationary subalgebra in general position of a tensor product of real or complex compact group representations acts as a scalar on all tensor factors but possibly one. In the real case, it means that this stationary subalgebra in general position is contained in one of the direct summand subalgebras. We used the following concepts to solve this problem: conventional linear algebra arguments; theory of Lie groups, Lie algebras and their representations; and methods similar to those of solving similar problems for complex reductive linear groups.

Infinitely divisible projective representations of the Lie algebras

1972 ◽  
Vol 72 (3) ◽  
pp. 357-368 ◽  
Author(s):  
D. Mathon
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Algebraic Method ◽  
Group Cohomology ◽  
Continuity Condition ◽  
Group Representations ◽  
Infinitely Divisible ◽  
Projective Representations ◽  
One To One ◽  
Representations Of Lie Groups

Infinitely divisible group representations were first defined by Streater(1) as an important concept closely related to continuous tensor product. Araki(2) analysed the factorizable representations of Lie groups and obtained a generalization of the Levy–Khinchine formula. A similar concept for Lie algebras was defined and studied by Streater in (3). Although the definition is not strictly an infinitesimal analogue of infinitely divisible representations of Lie groups, the results of (3) in the cohomological formulation are very similar to Araki's main theorem. Parthasarathy and Schmidt(4) generalized the concept of infinite divisibifity to the projective representations of locally compact groups and obtained a one-to-one correspondence between infinitely divisible projective representations and 1-co-cycles in the group cohomology with coefficients in a Hubert space. A similar generalization for Lie algebras is studied in the present paper. Infinitely divisible projective representations of Lie algebras are studied by a purely algebraic method, independently of (4) (since not all our projective representations are necessarily integrable). As expected, a one-to-one relation is obtained between the infinitely divisible projective representations and 1-co-cycles in the cohomology on the corresponding enveloping algebra with coefficients in a Hilbert space. The present problem is simpler than the group case since there is no continuity condition on the multiplier in a Lie algebra. A similar algebraic method was used in a discussion of infinitely divisible representations of canonical anticommutation relations (9).

scholarly journals Note on Three-Dimensional Lie Groups

1955 ◽  
Vol 2 (3) ◽  
pp. 112-115 ◽  
Author(s):  
E. M. Patterson
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Three Dimensional ◽  
Linear Groups ◽  
Bilinear Forms ◽  
Canonical Forms ◽  
First Finding ◽  
Groups Of Automorphisms

The object of this note is to construct a set of real three-dimensional Lie groups such that every real three-dimensional Lie group is locally isomorphic with some group in the set. The construction is effected by first finding canonical forms for the constants of structure of real three-dimensional Lie algebras; these canonical forms give rise to certain bilinear forms, and the Lie groups are obtained as linear groups isomorphic with groups of automorphisms which leave these bilinear forms invariant.

scholarly journals Joseph Fourier 250thBirthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 250
Author(s):  
Frédéric Barbaresco ◽  
Jean-Pierre Gazeau
Keyword(s):  
Fourier Analysis ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Lie Groups ◽  
Coherent States ◽  
Geometric Theory ◽  
Plancherel Formula ◽  
Group Representations ◽  
Modern Development ◽  
Xxist Century

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics.

scholarly journals Representations of complex semisimple Lie groups and Lie algebras

1966 ◽  
Vol 72 (3) ◽  
pp. 522-526 ◽  
Author(s):  
K. R. Parthasarathy ◽  
R. Ranga Rao ◽  
V. S. Varadarajan
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Semisimple Lie Groups ◽  
Complex Semisimple

scholarly journals Completely bounded bimodule maps and spectral synthesis

2017 ◽  
Vol 28 (10) ◽  
pp. 1750067 ◽  
Author(s):  
M. Alaghmandan ◽  
I. G. Todorov ◽  
L. Turowska
Keyword(s):  
Tensor Product ◽  
Compact Group ◽  
Closed Subset ◽  
Spectral Synthesis ◽  
Locally Compact Group ◽  
Product Formula ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Multiplication Algebra

We initiate the study of the completely bounded multipliers of the Haagerup tensor product [Formula: see text] of two copies of the Fourier algebra [Formula: see text] of a locally compact group [Formula: see text]. If [Formula: see text] is a closed subset of [Formula: see text] we let [Formula: see text] and show that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] then [Formula: see text] is a set of local spectral synthesis for [Formula: see text]. Conversely, we prove that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] and [Formula: see text] is a Moore group then [Formula: see text] is a set of spectral synthesis for [Formula: see text]. Using the natural identification of the space of all completely bounded weak* continuous [Formula: see text]-bimodule maps with the dual of [Formula: see text], we show that, in the case [Formula: see text] is weakly amenable, such a map leaves the multiplication algebra of [Formula: see text] invariant if and only if its support is contained in the antidiagonal of [Formula: see text].

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