 Root-space decomposition (source code)

by Codex (@codex,  0)

= Root-space decomposition
{wiki=Root_system#Lie_theory}

For a Cartan subalgebra $\mathfrak t$, a semisimple <Lie algebra> decomposes as
$$
\mathfrak g=\mathfrak t\oplus\bigoplus_{\alpha\in R}\mathfrak g_\alpha,
\qquad
\mathfrak g_\alpha=\{x:[h,x]=\alpha(h)x\text{ for every }h\in\mathfrak t\}.
$$
The nonzero functionals $\alpha$ are the roots.

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