The generators of a continuous group form a basis of a lie algebra. Properties:

- lie algebra == vector space with dim = dim of corresponding group.
- lie algebra closed under commutation [ , ].

[Xi,Xj]=ifijkXk

where fijk = structure constants of the lie algebra.

For a compact Lie group:

- fijk=-fjik always
- there exists a basis of generators s.t. fijk=eijk= antisymmetric tensor
- Jacobi identity => Sum(lm,fijlflkm+cyclic permutations)=0
- Xi's are hermitian

A lie algebra can correspond to multiple groups.
Examples:

SU(4) and SO(6)

SU(2) and SO(3)

SO(4) and SU(2)xSU(2)

SO(5) and SP(2)

source

jl@crush.caltech.edu index

group_generation

covering_group

GL