scholarly journals LIE ALGEBRA REPRESENTATIONS AND RIGGED HILBERT SPACES: THE SO(2) CASE

2017 ◽  
Vol 57 (6) ◽  
pp. 379 ◽  
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano A Del Olmo
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Hilbert Spaces ◽  
Irreducible Representations ◽  
Rigged Hilbert Spaces ◽  
Suitable Framework ◽  
Lie Algebra Representations

It is well known that related with the irreducible representations of the Lie group <em>SO</em>(2) we find a discrete basis as well a continuous one. In this paper we revisited this situation under the light of  Rigged Hilbert spaces, which are the suitable framework to deal  with both discrete and bases in the same context and in  relation with physical applications.

CONTRACTIONS OF LIE ALGEBRA REPRESENTATIONS

1999 ◽  
Vol 11 (10) ◽  
pp. 1179-1207 ◽  
Author(s):  
U. CATTANEO ◽  
W. F. WRESZINSKI
Keyword(s):  
General Theory ◽  
Banach Spaces ◽  
Lie Algebra ◽  
Hilbert Spaces ◽  
3 Dimensional ◽  
Algebra Representation ◽  
Carrier Space ◽  
Definition Of ◽  
Lie Algebra Representations ◽  
Main Distinguishing Feature

A theory of contractions of Lie algebra representations on complex Hilbert spaces is proposed, based on Trotter's theory of approximating sequences of Banach spaces. Its main distinguishing feature is a careful definition of the carrier space of the limit Lie algebra representation. A set of quite general conditions on the contracting representations, satisfied in all known examples, is proven to be sufficient for the existence of such a representation. In order to show how natural the suggested framework is, the general theory is applied to the contraction of [Formula: see text] into the Lie algebra [Formula: see text] of the 3-dimensional Heisenberg group and to the related study of the limit N→∞ of a quantum system of N identical two-level particles.

scholarly journals One more method to construct the irreducible unitary representations of nipotent Lie groups

1986 ◽  
Vol 40 (1) ◽  
pp. 89-94
Author(s):  
A. A. Astaneh
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Nilpotent Lie Group ◽  
Irreducible Unitary Representations ◽  
Simply Connected

AbstractIn this paper one more canonical method to construct the irreducible unitary representations of a connected, simply connected nilpotent Lie group is introduced. Although we used Kirillov' analysis to deduce this procedure, the method obtained differs from that of Kirillov's, in that one does not need to consider the codjoint representation of the group in the dual of its Lie algebra (in fact, neither does one need to consider the Lie algebra of the group, provided one knows certain connected subgroups and their characters). The method also differs from that of Mackey's as one only needs to induce characters to obtain all irreducible representations of the group.

scholarly journals Groups, Special Functions and Rigged Hilbert Spaces

Axioms ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 89 ◽  
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano A. del Olmo
Keyword(s):  
Hilbert Space ◽  
Lie Algebra ◽  
Hilbert Spaces ◽  
Special Functions ◽  
Hermite Functions ◽  
Laguerre Functions ◽  
Rigged Hilbert Spaces ◽  
Infinite Dimensional ◽  
Common Framework ◽  
Test Vectors

We show that Lie groups and their respective algebras, special functions and rigged Hilbert spaces are complementary concepts that coexist together in a common framework and that they are aspects of the same mathematical reality. Special functions serve as bases for infinite dimensional Hilbert spaces supporting linear unitary irreducible representations of a given Lie group. These representations are explicitly given by operators on the Hilbert space H and the generators of the Lie algebra are represented by unbounded self-adjoint operators. The action of these operators on elements of continuous bases is often considered. These continuous bases do not make sense as vectors in the Hilbert space; instead, they are functionals on the dual space, Φ × , of a rigged Hilbert space, Φ ⊂ H ⊂ Φ × . In fact, rigged Hilbert spaces are the structures in which both, discrete orthonormal and continuous bases may coexist. We define the space of test vectors Φ and a topology on it at our convenience, depending on the studied group. The generators of the Lie algebra can often be continuous operators on Φ with its own topology, so that they admit continuous extensions to the dual Φ × and, therefore, act on the elements of the continuous basis. We investigate this formalism for various examples of interest in quantum mechanics. In particular, we consider S O ( 2 ) and functions on the unit circle, S U ( 2 ) and associated Laguerre functions, Weyl–Heisenberg group and Hermite functions, S O ( 3 , 2 ) and spherical harmonics, s u ( 1 , 1 ) and Laguerre functions, s u ( 2 , 2 ) and algebraic Jacobi functions and, finally, s u ( 1 , 1 ) ⊕ s u ( 1 , 1 ) and Zernike functions on a circle.

scholarly journals INDECOMPOSABLE REPRESENTATIONS OF THE EUCLIDEAN ALGEBRA 𝔢(3) FROM IRREDUCIBLE REPRESENTATIONS OF

2011 ◽  
Vol 83 (3) ◽  
pp. 439-449 ◽  
Author(s):  
ANDREW DOUGLAS ◽  
JOE REPKA
Keyword(s):  
Euclidean Space ◽  
Lie Algebra ◽  
Semidirect Product ◽  
Lie Group ◽  
Three Dimensional ◽  
Irreducible Representations ◽  
Dimensional Euclidean Space ◽  
Euclidean Group ◽  
New Class ◽  
Euclidean Algebra

AbstractThe Euclidean group E(3) is the noncompact, semidirect product group E(3)≅ℝ3⋊SO(3). It is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. The Euclidean algebra 𝔢(3) is the complexification of the Lie algebra of E(3). We embed the Euclidean algebra 𝔢(3) into the simple Lie algebra $\mathfrak {sl}(4,\mathbb {C})$ and show that the irreducible representations V (m,0,0) and V (0,0,m) of $\mathfrak {sl}(4,\mathbb {C})$ are 𝔢(3)-indecomposable, thus creating a new class of indecomposable 𝔢(3) -modules. We then show that V (0,m,0) may decompose.

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.

Deformations of Lie group and Lie algebra representations

1993 ◽  
Vol 34 (9) ◽  
pp. 4251-4272 ◽  
Author(s):  
Marc Lesimple ◽  
Georges Pinczon
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Lie Algebra Representations

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 ∇̃.

scholarly journals COMMUTATORS OF SKEW-SYMMETRIC MATRICES

2005 ◽  
Vol 15 (03) ◽  
pp. 793-801 ◽  
Author(s):  
ANTHONY M. BLOCH ◽  
ARIEH ISERLES
Keyword(s):  
Optimal Control ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Geometric Integration ◽  
Symmetric Matrices ◽  
Frobenius Norm

In this paper we develop a theory for analysing the "radius" of the Lie algebra of a matrix Lie group, which is a measure of the size of its commutators. Complete details are given for the Lie algebra 𝔰𝔬(n) of skew symmetric matrices where we prove [Formula: see text], X, Y ∈ 𝔰𝔬(n), for the Frobenius norm. We indicate how these ideas might be extended to other matrix Lie algebras. We discuss why these ideas are of interest in applications such as geometric integration and optimal control.

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