nLab classical Lie group

Idea

The classification of simple Lie groups contains four infinite series of “classical” simple Lie groups:

• $A_n = SU(n+1)$, the special unitary group

• $B_n = SO(2n+1)$, the special orthogonal group

• $C_n = Sp(n)$, the symplectic group

• $D_n = SO(2n)$, the special orthogonal group

Apart from this there are five exceptional Lie groups.

graphics grabbed from Schwichtenberg

More generally, other matrix Lie groups, such as the general linear group $GL(n)$, may be called “classical”.

Related concepts

• exceptional isomorphism

References

• Glen Bredon, Section 0.2 of: Introduction to compact transformation groups, Academic Press 1972 (ISBN 9780080873596, pdf)

• Howard Georgi, §19 in: Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]

  with an eye towards application to (the standard model of) particle physics and grand unified theory