The Origins of Infinite Dimensional Unitary Representations of Lie Groups

1994 ◽  
pp. 231-240
Author(s):  
Sugiura Mitsuo
Keyword(s):  
Lie Groups ◽  
Unitary Representations ◽  
Infinite Dimensional ◽  
Representations Of Lie Groups

Elements of Group Theory

2021 ◽  
pp. 51-110
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Group Theory ◽  
Lie Groups ◽  
Lorentz Group ◽  
Unitary Representations ◽  
Compact Groups ◽  
Mathematical Language ◽  
Group Representations ◽  
Infinite Groups ◽  
Infinite Dimensional ◽  
Symmetry Properties

The mathematical language which encodes the symmetry properties in physics is group theory. In this chapter we recall the main results. We introduce the concepts of finite and infinite groups, that of group representations and the Clebsch–Gordan decomposition. We study, in particular, Lie groups and Lie algebras and give the Cartan classification. Some simple examples include the groups U(1), SU(2) – and its connection to O(3) – and SU(3). We use the method of Young tableaux in order to find the properties of products of irreducible representations. Among the non-compact groups we focus on the Lorentz group, its relation with O(4) and SL(2,C), and its representations. We construct the space of physical states using the infinite-dimensional unitary representations of the Poincaré group.

scholarly journals Unitary representations of Lie groups and Gårding’s inequality

1989 ◽  
Vol 107 (3) ◽  
pp. 627-627
Author(s):  
Ola Bratteli ◽  
Fred Goodman ◽  
Palle Jorgensen ◽  
Derek W. Robinson
Keyword(s):  
Lie Groups ◽  
Unitary Representations ◽  
Representations Of Lie Groups

scholarly journals Contractions of Representations and Algebraic Families of Harish-Chandra Modules

2018 ◽  
Vol 2020 (11) ◽  
pp. 3494-3520 ◽  
Author(s):  
Joseph Bernstein ◽  
Nigel Higson ◽  
Eyal Subag
Keyword(s):  
Lie Groups ◽  
Unitary Group ◽  
Reductive Group ◽  
Mathematical Physics ◽  
Point Of View ◽  
Unitary Representations ◽  
Group Representations ◽  
Algebraic Point ◽  
Infinite Dimensional ◽  
Real Forms

Abstract We examine from an algebraic point of view some families of unitary group representations that arise in mathematical physics and are associated to contraction families of Lie groups. The contraction families of groups relate different real forms of a reductive group and are continuously parametrized, but the unitary representations are defined over a parameter subspace that includes both discrete and continuous parts. Both finite- and infinite-dimensional representations can occur, even within the same family. We shall study the simplest nontrivial examples and use the concepts of algebraic families of Harish-Chandra pairs and Harish-Chandra modules, introduced in a previous paper, together with the Jantzen filtration, to construct these families of unitary representations algebraically.

Unitary Representations of Lie Groups and Garding's Inequality

1989 ◽  
Vol 107 (3) ◽  
pp. 627 ◽  
Author(s):  
Ola Bratteli ◽  
Fred M. Goodman ◽  
Palle E. T. Jorgensen ◽  
Derek W. Robinson
Keyword(s):  
Lie Groups ◽  
Unitary Representations ◽  
Representations Of Lie Groups
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