Number of generators of SU(n) group

[arroy_0205]

How do I calculate the number of generators of SU(n) group (which is extremely important in particle physics)? In the case of SO(n), I can do that using the physical interpretation of the group, i.e., it is related to rotations in n-dimensional Euclidean plane. What do I do in the case of SU(n)? I know the answer is n^2-1 but can not prove it.

Also if possible please indicate how to calculate the number of generators of O(n) and U(n) groups.

 

[DrDu]

A unitary matrix can be obtained from a hermitian matrix e.g. by exponentiation. A hermitian matrix has n(n+1)/2 real (symmetric part of the matrix) and n(n-1)/2 imaginary (anti-symmetric part of the matrix) entries giving n^2 independent elements (and thus generators) in total.
In a special unitary matrix, there is one further condition, hence there are only n^2-1 operations. Analogously an orthogonal matrix can be obtained by the exponential construction from an anti-symmetric hermitian matrix and thus has n(n-1)/2 generators. The condition of speciality only fixes the sign of the real matrices and thus doesn't restrict the total number of generators.

 

[thoms2543]

How to get \frac{n(n+1)}{2}?

 

[DrDu]

a real symmetric matrix has n element on the diagonal and n(n-1)/2 elements on the upper triangle (which are equal to the elements on the lower triangle. Taken together, there are n(n+1)/2 distinct elements.