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Welcome to the language barrier between physicists and mathematicians I have a potentially simple question here, about the tangent space of the lie group so (n), the group of orthogonal $n\times n$ real matrices (i'm sure this can be. Physicists prefer to use hermitian operators, while mathematicians are not biased towards hermitian operators

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What is the fundamental group of the special orthogonal group $so (n)$, $n>2$ What is the lie algebra and lie bracket of the two groups? The answer usually given is

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I have known the data of $\\pi_m(so(n))$ from this table

To gain full voting privileges, 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 that an antisymmetric matrix has n(n−1) 2 n (n 1) 2 degrees of freedom, but i can't take this idea any further in the demonstration of the proof

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Instead, you can save this post to reference later. U (n) and so (n) are quite important groups in physics I thought i would find this with an easy google search