How is Lie algebra named? Why is it $\mathfrak{su}(2)$ for group $SU(2)$, but $\mathfrak{o}(3)$ for group $SO(3)$? What does the "$\mathfrak{s}$" in algebra $\mathfrak{su}(2)$ mean?
Why is the Lie algebra corresponding to group SO(3) called o(3)?
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lie-algebras
lie-groups
1 Answers
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The "s" stands for special. Lie algebras are usually named after Lie groups. $\text{U}(2)$ and $\text{SU}(2)$ have different Lie algebras so they're named $\mathfrak{u}(2)$ and $\mathfrak{su}(2)$, but as it turns out, $\text{O}(3)$ and $\text{SO}(3)$ have the same Lie algebra, so $\mathfrak{o}(3) \cong \mathfrak{so}(3)$.
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1@KarsusRen ... A skew-hermitian complex $2\times 2$ matric is *not* necessarily traceless, e.g., consider the diagonal complex $2\times 2$ matrix whose diagonal entries are all equal to $i\in\mathbb{C}$. – 2012-06-26