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Fundamental group of the special orthogonal group SO(n) Question: What is the fundamental group of the special orthogonal group $SO (n)$, $n>2$? Clarification: The answer usually given is: $\mathbb {Z}_2$ But I would like
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
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
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?
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