 Chiral decomposition of the complex Lorentz algebra (source code)

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= Chiral decomposition of the complex Lorentz algebra

Over the complex numbers, $A_i=(J_i+iK_i)/2$ and $B_i=(J_i-iK_i)/2$ generate commuting copies of $\mathfrak{sl}_2(\mathbb C)$, giving
$$
\mathfrak{so}(1,3)_\mathbb C\cong
\mathfrak{sl}_2(\mathbb C)\oplus\mathfrak{sl}_2(\mathbb C).
$$
Finite-dimensional irreducible representations are labelled $(j_L,j_R)$.

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