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a) To determine the number of generators needed for the group O(n) we write a rotation matrix as:

[tex] R=e^{-i\theta J} [/tex]

where [itex] J [/itex] is an n x n matrix, Hermitian and imaginary, and therefore anti-symmetric. The number of indepedent parameters [itex] \theta [/itex] (and hence the number of generators) is the number of independent matrices. This number can be found by counting the number of parameters required to make up any n x n antisymmetric matrix. This is n(n-1)/2- WHY?

b)Show for any n:

[tex] [J_{ij},J_{kl}]=\plusminus (\delta_{ij}J_{il}-\delta_{ik}J_{jl}-\delta_{jl}J_{ik}+\delta_{il}J_{ik}) [/tex]

where [itex] J_{ij} [/itex] are two index objects with matrix elements:

[tex] (J_{ij})_{kl} = -i(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk}) [/tex]

and

[tex] [J_{ij},J_{kl}] [/tex]

is the commutator

Ok...

So part a):

I am a little confused. I know that the matrix must be imaginary and hermitian, but I don't think that is enough to prove that only n(n-1)/2 parameters are required to make a n x n antisymmetric matrix. In fact I am not even sure what determines whether the parameters are independent. Is a complex number and its conjugate independent? If not, then I think I understand. But if not I am lost.

part b) No clue.

I have never taken a group theory class and this was thrown into a Quantum Mechanics homework set so I am pretty lost. Any help would really be appreciated.

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# Homework Help: Group Theory

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