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 ...
Tangent space of Lie group SO(n) - Mathematics Stack Exchange
May 9, 2024 · @FrancescoPolizzi that was easy thanks! So the two ways to look at the tangent space are indeed equivalent, which can be seen using the construction you showed. Should that be an …
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 …