# Difference Group Written in Capital Letters and in Lower Case Letters

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?

• 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 $\backslash {\rm mathfrak}\{so\}(3)$. – Luboš Motl Jul 28 '16 at 5:41