On non-compact groups I. Classification of non-compact real simple Lie groups and groups containing the Lorentz group

1965 ◽  
Vol 287 (1411) ◽  
pp. 519-531 ◽  
Keyword(s):  
Lie Groups ◽  
Lorentz Group ◽  
Compact Groups ◽  
Bilinear Forms ◽  
Unimodular Group ◽  
Simple Lie Groups ◽  
Real Complex

The classification of all simple real Lie groups is discussed and given explicitly in the form of tables. In particular four classes of groups are identified which contain the Lorentz group. These are the groups which leave bilinear forms in real, complex and quaternionic spaces invariant, and the unimodular group in n -dimension.

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.

The Betti Numbers of the Simple Lie Groups

1958 ◽  
Vol 10 ◽  
pp. 349-356 ◽  
Author(s):  
A. J. Coleman
Keyword(s):  
Lie Groups ◽  
Orthogonal Group ◽  
Betti Numbers ◽  
Algebraic Methods ◽  
Classical Groups ◽  
Compact Lie Groups ◽  
Poincaré Polynomial ◽  
Unimodular Group ◽  
Simple Lie Groups

The purpose of the present paper1 is to simplify the calculation of the Betti numbers of the simple compact Lie groups. For the unimodular group and the orthogonal group on a space of odd dimension the form of the Poincaré polynomial was correctly guessed by E. Cartan in 1929 (5, p. 183). The proof of his conjecture and its extension to the four classes of classical groups was given by L. Pontrjagin (13) using topological arguments and then by R. Brauer (2) using algebraic methods.

scholarly journals Twisted 6d (2, 0) SCFTs on a circle

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Zhihao Duan ◽  
Kimyeong Lee ◽  
June Nahmgoong ◽  
Xin Wang
Keyword(s):  
Lie Groups ◽  
Point Of View ◽  
Partition Functions ◽  
Congruence Subgroups ◽  
Gauge Groups ◽  
Simple Lie Groups ◽  
Instanton Contribution ◽  
Elliptic Genera ◽  
Supersymmetric Gauge ◽  
The One

Abstract We study twisted circle compactification of 6d (2, 0) SCFTs to 5d $$ \mathcal{N} $$ N = 2 supersymmetric gauge theories with non-simply-laced gauge groups. We provide two complementary approaches towards the BPS partition functions, reflecting the 5d and 6d point of view respectively. The first is based on the blowup equations for the instanton partition function, from which in particular we determine explicitly the one-instanton contribution for all simple Lie groups. The second is based on the modular bootstrap program, and we propose a novel modular ansatz for the twisted elliptic genera that transform under the congruence subgroups Γ0(N) of SL(2, ℤ). We conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of the genus one fibered Calabi-Yau threefolds, upon which one can determine the twisted elliptic genera recursively. We use our results to obtain the 6d Cardy formulas and find universal behaviour for all simple Lie groups. In addition, the Cardy formulas remain invariant under the twist once the normalization of the compact circle is taken into account.

scholarly journals Simple Lie groups stabilizing G-invariant norms

2021 ◽  
Vol 609 ◽  
pp. 308-316
Author(s):  
Marcell Gaál ◽  
Robert M. Guralnick
Keyword(s):  
Lie Groups ◽  
Simple Lie Groups
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