following a course on group theory is definitly a good idea if you want to continue in theoreticsl physics. i'm not quite sure if it is that important for technical physics though...
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We defined the Lie algebra as the {M¦matrix n x n so that exp (tm) belongs to G for all t belonging to R} where G is the lie group
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this matrix M is not really the algebra; it is the generator of the group. but it is crucial to the algebra.
an algebra -as you noted- is defined by the rules of multiplication. so the algebra of the lie group is defined by the algebra of the generators and these are the (anti)commutation relations of the matrices. (so e.g. the commutation relations of the pauli-spin matrices for SU(2))
for example: the group SO(2). (rotations in the 2d plane) in the 2d representation.
the symmetry transformation is simply:
(x' ; y' ) = (cos(f), -sin(f) ; sin(f), cos(f)) (x ; y) =R(f) (x; y)
now, since SO(2) is a Lie group, we can define a generator. (there is only 1 generator of SO(2)).
the claim is (which you can prove by taking a taylor expansion) is that the matrix R(f) = EXP(-i*f*I), with I = (0, -i ; i, 0).
The algebra of SO(2) is trivial, since it only has 1 generator. but for completeness: the algebra is given by:
[I, I] = 0 ; {I, I} = 2 I^2
some extra terminology (that you see very often): When the elements of the group (or equivalently the generators) commute, we call the group abelian; otherwise non-abelian. so SO(2) is abelian, SU(2) not.
btw: what kind of quantum course are you following that Lie groups are mentioned, but not explained?
Bo
