Simple Lie group

The classification of Lie groups that are also simple groups depends on the prior classification of the complex simple Lie algebras: for which see the page on root systems. It is shown that a simple Lie group has a simple Lie algebra that will occur on the list given there, once it is complexified (that is, made into a complex vector space rather than a real one. This reduces the classification to two further matters

Firstly, for example, the SO(p,q,R) and SO(p+q,R) give rise to different Lie algebras with the same Dynkin diagram. In general there may be different real forms of the same complex Lie algebra.

Secondly the Lie algebra only determines uniquely the simply connected (universal) cover G* of the component containing the identity of a Lie group G. It may well happen that G* isn't actually a simple group, for example having a non-trivial center. We have therefore to worry about the global topology), by computing the fundamental group of G. This was done by Cartan.

For an example, the special orthogonal groups in even dimension: with -I a scalar matrix in the center these aren't actually simple groups, and having a two-fold spin cover. They aren't simply-connected either: they lie 'between' G* and G, in the notation above.

Table of contents
1 Classification by Lie algebra and Dynkin diagram
2 Infinite series
3 A series
4 B series
5 C series
6 D series
7 Exceptional algebras
8 G2
9 F4
10 E6
11 E7
12 E8

Classification by Lie algebra and Dynkin diagram

(duplicates root system)

According to his classification, we have

Infinite series

A series

A1, A2, ...

Ar corresponds to the special unitary group, SU(r+1).

B series

B1, B2, ...

Br corresponds to the special orthogonal group, SO(2r+1).

C series

C1, C2, ...

Cr corresponds to the symplectic group, Sp(2r).

D series

D1, D2, ...

Dr corresponds to the special orthogonal group, SO(2r).

Exceptional algebras

G2

See
G2.

F4

See F4.

E6

See E6.

E7

See E7.

E8

See E8.

See also Cartan matrix, Coxeter matrix, Dynkin diagram, Weyl group, Coxeter group, Kac-Moody algebras.


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