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)