Let $\mathfrak{g}$ be a Lie algebra over a field $k$ , and let $B$ be a $k$ -basis of $\mathfrak{g}$ equipped with a linear order $\leq$ . The Poincaré-Birkhoff-Witt-theorem (often abbreviated to PBW-theorem) states that the monomials$$ x_1 x_2 \cdots x_n \text{ with } x_1 \leq x_2 \leq \cdots \leq x_n \text{ elements of } B$$ constitute a $k$ -basis of the universal enveloping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$ . Such monomials are often called ordered monomials or PBW-monomials.

It is easy to see that they span $U(\mathfrak{g})$ : for all $n \in \NN$ , let $M_n$ denote the set$$ M_n=\{(x_1,\ldots,x_n)\mid x_1 \leq \cdots \leq x_n\} \subset B^n,$$ and denote by $\pi:\bigcup_{n=0}^\infty B^n \rightarrow U(\mathfrak{g})$ the multiplication map. Clearly it suffices to prove that$$ \pi(B^n) \subseteq \sum_{i=0}^n \pi(M_i)$$ for all $n \in \NN$ ; to this end, we proceed by induction. For $n=0$ the statement is clear. Assume that it holds for $n-1\geq 0$ , and consider a list $(x_1,\ldots,x_n) \in B^n$ . If it is an element of $M_n$ , then we are done. Otherwise, there exists an index $i$ such that $x_i>x_{i+1}$ . Now we have

The proof of linear independence of the PBW-monomials is slightly more difficult, but can be found in most introductory texts on Lie algebras, such as the classic below.

Bibliography

1

N. Jacobson. Lie Algebras. Dover Publications, New York, 1979