Fundamental group of the special orthogonal group SO(n)
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 …
lie groups - Lie Algebra of SO (n) - Mathematics Stack Exchange
Apr 24, 2017 · Where a, b, c, d ∈ 1, …, n a, b, c, d ∈ 1,, n. And so(n) s o (n) is the Lie algebra of SO (n). I'm unsure if it suffices to show that the generators of the ...
Dimension of SO (n) and its generators - Mathematics Stack Exchange
Nov 18, 2015 · 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 …
Prove that the manifold $SO(n)$ is connected
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 …
Homotopy groups O(N) and SO(N): $\\pi_m(O(N))$ v.s. $\\pi_m(SO(N))$
Oct 3, 2017 · 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 …
Distinguishing $SU(n)$ from $SO(n)$ - Mathematics Stack Exchange
May 24, 2017 · 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 …
What is the relationship between SL (n) and SO (n)?
To add some intuition to this, for vectors in Rn R n, SL(n) S L (n) is the space of all the transformations with determinant 1 1, or in other words, all transformations that keep the volume constant. This is …
orthogonal matrices - Irreducible representations of $SO (N ...
Sep 21, 2020 · I'm looking for a reference/proof where I can understand the irreps of $SO(N)$. I'm particularly interested in the case when $N=2M$ is even, and I'm really only ...
Homotopic type of $GL^+(n)$, $SL(n)$ and $SO(n)$
Question: Consider GL+(n) ⊃ SL(n) ⊃ SO(n) G L + (n) ⊃ S L (n) ⊃ S O (n) the groups of matrices n × n n × n with positive determinant, determinant 1 1 and orthogonal with positive determinant, …
Why $\\operatorname{Spin}(n)$ is the double cover of $SO(n)$?
Dec 16, 2024 · You can let $\text {Spin} (n)$ act on $\mathbb {S}^ {n-1}$ through $\text {SO} (n)$. Since $\text {Spin} (n-1)\subset\text {Spin} (n)$ maps to $\text {SO} (n-1)\subset\text {SO} (n)$, you could …