# Poincaré-Birkhoff-Witt theorem

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 $n\in\mathbb{N}$, 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 $n\in\mathbb{N}$; 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

 $\displaystyle\pi(x_{1},\ldots,x_{n})$ $\displaystyle=\pi(x_{1},\ldots,x_{i-1},x_{i+1},x_{i},x_{i+2},\ldots,x_{n})$ $\displaystyle+x_{1}\cdots x_{i-1}[x_{i},x_{i+1}]x_{i+1}\cdots x_{n}.$

As $B$ is a basis of $\mathfrak{k}$, $[x_{i},x_{i+1}]$ is a linear combination of $B$. Using this to expand the second term above, we find that it is in $\sum_{i=0}^{n-1}\pi(M_{i})$ by the induction hypothesis. The argument of $\pi$ in the first term, on the other hand, is lexicographically smaller than $(x_{1},\ldots,x_{n})$, but contains the same entries. Clearly this rewriting proces must end, and this concludes the induction step.

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.

## References

• 1 N. Jacobson. . Dover Publications, New York, 1979
Title Poincaré-Birkhoff-Witt theorem PoincareBirkhoffWittTheorem 2013-03-22 13:03:38 2013-03-22 13:03:38 CWoo (3771) CWoo (3771) 7 CWoo (3771) Theorem msc 17B35 PBW-theorem LieAlgebra UniversalEnvelopingAlgebra FreeLieAlgebra