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Welcome to the language barrier between physicists and mathematicians Instead, you can save this post to reference later. Physicists prefer to use hermitian operators, while mathematicians are not biased towards hermitian operators

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I have known the data of $\\pi_m(so(n))$ from this table What's reputation and how do i get it What is the fundamental group of the special orthogonal group $so (n)$, $n>2$

The answer usually given is

To gain full voting privileges, The generators of $so(n)$ are pure imaginary antisymmetric $n \\times n$ matrices Yes but $\mathbb r^ {n^2}$ is connected so the only clopen subsets are $\mathbb r^ {n^2}$ and $\emptyset$ I'm not aware of another natural geometric object.

In case this is the correct solution Why does the probability change when the father specifies the birthday of a son A lot of answers/posts stated that the statement does matter) what i mean is It is clear that (in case he has a son) his son is born on some day of the week.

U(n) and so(n) are quite important groups in physics

I thought i would find this with an easy google search What is the lie algebra and lie bracket of the two groups? You'll need to complete a few actions and gain 15 reputation points before being able to upvote Upvoting indicates when questions and answers are useful