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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)
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
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
Help with a proof that SO (n) is path-connected.
I've found lots of different proofs that SO (n) is path connected, but I'm trying to understand one I found on Stillwell's book "Naive Lie Theory" It's fairly informal and talks about paths in a very
Homotopic type of $GL^+ (n)$, $SL (n)$ and $SO (n)$
I don't believe that the tag homotopy-type-theory is warranted, unless you are looking for a solution in the new foundational framework of homotopy type theory It sure would be an interesting question in this framework, although a question of a vastly different spirit
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
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
Representation theory of $SO (n)$ - Mathematics Stack Exchange
Regarding the downvote: I am really sorry if this answer sounds too harsh, but math SE is not the correct place to ask this kind of questions which amounts to «please explain the represnetation theory of SO (n) to me» and to which not even a whole seminar would provide a complete answer The book by Fulton and Harris is a 500-page answer to this question, and it is an amazingly good answer
general topology - proving that $SO (n)$ is path connected . . .
Alternatively, you can show that it is connected and locally path connected
Lie Algebra of U(N) and SO(N) - Mathematics Stack Exchange
U(N) and SO(N) are quite important groups in physics I thought I would find this with an easy google search Apparently NOT! What is the Lie algebra and Lie bracket of the two groups?