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# 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.
*Lie Algebras*. Dover Publications, New York, 1979

## Mathematics Subject Classification

17B35*no label found*

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