= Lorentz algebra in three spacetime dimensions
{c}
{title2=$\mathfrak{so}(1,2)_{\mathbb C}\cong\mathfrak{sl}_2(\mathbb C)$}
With signature $(-,+,+)$, two <Lorentz boost> generators and one spatial rotation obey $[J,K_1]=K_2$, $[J,K_2]=-K_1$ and $[K_1,K_2]=-J$. Replacing the boosts by $iK_1,iK_2$ gives the compact cyclic brackets, proving the complexified isomorphism with the <SU(2) Lie algebra>. The real algebra has dimension three and rank one. Its finite-dimensional complex <irreducible representations> have dimensions $m+1$ for nonnegative integer <highest weight> $m$.
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