What do SO(N) and U(N) mean in group theory?

[captain]

what do these groups mean? I think that SO(N) means rotation, but I am not sure? Also does the N mean the number of dimensions?

 

[Dick]

I think (and someone may correct me), but without a field specified U(N) is the group of complex unitary matrices of size NxN. (Unitary A meaning the product of A and it's hermitian conjugate is the identity). O(N) is the group of real unitary matrices (a subgroup of U(N)). SO(N) is the subgroup of O(N) with determinant 1. Yes, N is the dimension of the space the matrices operate on. But that's not necessarily the same as the dimension of the group as a manifold.

 

[CompuChip]

And SO(3) indeed are the rotations. This is not true for higher dimensions though, for example SO(4) contains -I, with I the 4x4 identity matrix (it's unitary and has determinant +1) which is not a rotation (rather, it's a reflection in all coordinates, space and time).

 

[HallsofIvy]

"SO" is "special othrogonal matrices". It is the orthogonal part that says "real unitary" and special means determinant 1. Yes, in 3 dimensions they correspond to rotations.

 

[captain]

what about U(N) and SU(N)? what are they and what is the difference between them including O(N) and SO(N)? is any group with a S in front of it special in some way?

 

[CompuChip]

The S does stand for "special", though the only thing special about the S-groups as opposed to the "non S"-groups, is that they have determinant 1.

 

[HallsofIvy]

"O", "orthogonal" matrices are real matrices whose inverse is equal to the transpose: row and columns swapped. If $AA^T= I$, then $det(A)det(A^T)= det(I)= 1$. Of course, the determinant the transpose of a matrix is the same as the determinant of the matrix itself so that says $(det(A))^2= 1$. That means the determinant is either 1 or -1. The "special" in SO requires that the determinant be 1.

Remember that "unitary" matrices over the complex numbers have their conjugate transpose (reverse rows and columns and take complex conjugate of each number) as their inverse.