JGSP 28 (2012) 1–45
RELATIONS AMONG LOW-DIMENSIONAL SIMPLE LIE GROUPS
ROBERT GILMORE
Communicated by Gregory L. Naber
Abstract. The compact classical Lie groups can be regarded as groups of n × n
matrices over the real, complex, and quaternion fields R, C, and Q that satisfy
metric- and volume-conserving conditions. These groups, SO(n, R), SU(n, C), and
Sp(n, Q), are not all independent. Homomorphisms exist among some of these
groups for small dimension. We review these relations by describing the Lie algebras of the compact forms and their complex extensions. Other noncompact real
forms of these Lie algebras are constructed by systematic methods. The relations
among all distinct real forms is presented.
Contents
1
Introduction
2
2
Classical Results
3
3
Relations Among Low-Dimensional Algebras
5
4
Real Forms
4.1 Block Matrix Decomposition . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Subfield Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Field Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
7
7
8
5
Equivalences of Real Forms
9
6
Fundamental Representations
9
7
Dimensions of Irreducible Representations
10
8
Unitary Irreducible Representations
12
9
Isomorphisms and Homomorphisms
13
10 A3 and D3
10.1 Equivalence Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Nonzero Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Fundamental Representations of A3 . . . . . . . . . . . . . . . . . . . .
doi: 10.7546/jgsp-28-2012-1-45
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