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I sometimes see that people write group names in capital letters and sometimes in lowercase letters. Like the 3 dimensional special orthogonal group is sometimes written as $SO(3)$ and sometimes as $so(3)$. Is there any difference?

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  • $\begingroup$ The lowercase letters are not just lowercase letters. They're Gothic fonts, the fracture, and represent Lie algebra, the tangent space of the Lie group near the identity with the commutator defined on it. A Lie group is e.g. ${\mathfrak so}(3)$. In $\rm\LaTeX$, it is $\backslash {\rm mathfrak}\{so\}(3)$. $\endgroup$ – Luboš Motl Jul 28 '16 at 5:41
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Usually the Lie groups are denoted by capital letters and the Lie algebra, which contains the generators of the Lie group, by lowercase letters. For instance, the Lie group SO(3) is associated with the Lie algebra so(3), which contains, for example in the case of the spin-1/2 representation, the Pauli matrices.

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  • $\begingroup$ You shouldn't say contains the Pauli matrices because that is after a particular representation and complexifying. It's a common abuse of language. $\endgroup$ – AHusain Jul 28 '16 at 4:52
  • $\begingroup$ @AHusain, OK so how should one say it then? $\endgroup$ – flippiefanus Jul 28 '16 at 5:09
  • $\begingroup$ That they provide a basis for this representation. (The spin 1/2 one) $\endgroup$ – AHusain Jul 28 '16 at 5:14
  • $\begingroup$ @AHusain, Yes, you're right, I forgot about that. See my edit. Is it better now? $\endgroup$ – flippiefanus Jul 28 '16 at 6:16

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