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- Fundamental group of the special orthogonal group SO(n)
Also, if I'm not mistaken, Steenrod gives a more direct argument in "Topology of Fibre Bundles," but he might be using the long exact sequence of a fibration (which you mentioned)
- lie groups - Lie Algebra of SO (n) - Mathematics Stack Exchange
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 $\mathfrak {so} (n)$ consists of skew-adjoint matrices (with respect to the Euclidean inner product on $\mathbb {R}^n$), physicists prefer to multiply them
- Prove that the manifold $SO (n)$ is connected
The question really is that simple: Prove that the manifold $SO (n) \subset GL (n, \mathbb {R})$ is connected it is very easy to see that the elements of $SO (n
- Homotopy groups O(N) and SO(N): $\\pi_m(O(N))$ v. s. $\\pi_m(SO(N))$
I have known the data of $\\pi_m(SO(N))$ from this Table: $$\\overset{\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad\\textbf{Homotopy groups of
- Dimension of SO (n) and its generators - Mathematics Stack Exchange
The generators of $SO(n)$ are pure imaginary antisymmetric $n \\times n$ matrices How can this fact be used to show that the dimension of $SO(n)$ is $\\frac{n(n-1
- orthogonal matrices - Irreducible representations of $SO (N . . .
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
- Are $SO(n)\\times Z_2$ and $O(n)$ isomorphic as topological groups?
I am doing Exercise 4-16 in Armstrong's Basic Topology The question is : are $SO (n)\times Z_2$ and $O (n)$ isomorphic as topological groups? (I have proved the
- Why $\\operatorname{Spin}(n)$ is the double cover of $SO(n)$?
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 then use the argument directly for $\text {Spin} (n)$, using that $\text {Spin} (3)$ is simply connected because $\text {Spin} (3)\cong\mathbb {S}^3$ I'm not aware of another natural geometric object
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