 Poincare-Birkhoff-Witt theorem (source code)

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= Poincare-Birkhoff-Witt theorem
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{wiki=Poincaré–Birkhoff–Witt_theorem}

For an ordered basis $(x_i)$ of a <Lie algebra> $\mathfrak g$, the ordered monomials $x_1^{a_1}\cdots x_n^{a_n}$ form a basis of $U(\mathfrak g)$. In particular, a triangular decomposition $\mathfrak g=\mathfrak n^-\oplus\mathfrak t\oplus\mathfrak n^+$ gives $U(\mathfrak g)\cong U(\mathfrak n^-)\otimes U(\mathfrak t)\otimes U(\mathfrak n^+)$ as vector spaces.

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