The question really is that simple: Prove that the manifold SO(n) ⊂ GL(n,R) S O (n) ⊂ G L (n, R) is connected. it is very easy to see that the elements of SO(n) S O (n) are in one-to-one correspondence with the set of orthonormal basis of Rn R n (the set of rows of the matrix of an element of SO(n) S O (n) is such a basis). My idea was to show that given any orthonormal basis (ai)n1 (a i ...
The generators of SO(n) S O (n) are pure imaginary antisymmetric n × n n × n matrices. How can this fact be used to show that the dimension of SO(n) S O (n) is n(n−1) 2 n (n 1) 2? I know that an antisymmetric matrix has n(n−1) 2 n (n 1) 2 degrees of freedom, but I can't take this idea any further in the demonstration of the proof. Thoughts?
Question: What is the fundamental group of the special orthogonal group SO(n) S O (n), n> 2 n> 2? Clarification: The answer usually given is: Z2 Z 2. But I would like to see a proof of that and an isomorphism π1(SO(n),En) → Z2 π 1 (S O (n), E n) → Z 2 that is as explicit as possible. I require a neat criterion to check, if a path in SO(n) S O (n) is null-homotopic or not. Idea 1: Maybe ...
As pointed out in the comments, O(N) O (N) consists of two connected components which are both diffeomorphic to SO(N) S O (N). So π0(O(N)) =Z2 π 0 (O (N)) = Z 2, π0(SO(N)) = 0 π 0 (S O (N)) = 0, and for m ≥ 1 m ≥ 1, πm(O(N)) =πm(SO(N)) π m (O (N)) = π m (S O (N)). As for Spin(N) Spin (N), note that is it a double cover of SO(N) S O (N). When N = 1 N = 1, we see that Spin(1) =Z2 ...
Welcome to the language barrier between physicists and mathematicians. Physicists prefer to use hermitian operators, while mathematicians are not biased towards hermitian operators. So for instance, while for mathematicians, the Lie algebra so(n) consists of skew-adjoint matrices (with respect to the Euclidean inner product on Rn), physicists prefer to multiply them by I think − i (or maybe ...
Suppose that I have a group G G that is either SU(n) S U (n) (special unitary group) or SO(n) S O (n) (special orthogonal group) for some n n that I don't know. Which "questions" should I ask to determine which one it is? e.g. which structural differences are there between these groups? Of course a question like "Is G G isomorphic to SU(n) S U (n) for some n n?" doesn't count. I'm hoping for ...
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