# Poincaré-Birkhoff-Witt theorem

Let $\U0001d524$ be a Lie algebra^{} over a field $k$, and let
$B$ be a $k$-basis of $\U0001d524$ equipped with a linear
order $\le $. The Poincaré-Birkhoff-Witt-theorem (often
abbreviated to PBW-theorem) states that the monomials

$${x}_{1}{x}_{2}\mathrm{\cdots}{x}_{n}\text{with}{x}_{1}\le {x}_{2}\le \mathrm{\cdots}\le {x}_{n}\text{elements of}B$$ |

constitute a $k$-basis of the universal enveloping algebra $U(\U0001d524)$ of $\U0001d524$. Such monomials are often called ordered monomials or PBW-monomials.

It is easy to see that they span $U(\U0001d524)$: for all $n\in \mathbb{N}$, let ${M}_{n}$ denote the set

$${M}_{n}=\{({x}_{1},\mathrm{\dots},{x}_{n})\mid {x}_{1}\le \mathrm{\cdots}\le {x}_{n}\}\subset {B}^{n},$$ |

and denote by $\pi :{\bigcup}_{n=0}^{\mathrm{\infty}}{B}^{n}\to U(\U0001d524)$ 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 $n-1\ge 0$, and consider a
list $({x}_{1},\mathrm{\dots},{x}_{n})\in {B}^{n}$. 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

$\pi ({x}_{1},\mathrm{\dots},{x}_{n})$ | $=\pi ({x}_{1},\mathrm{\dots},{x}_{i-1},{x}_{i+1},{x}_{i},{x}_{i+2},\mathrm{\dots},{x}_{n})$ | ||

$+{x}_{1}\mathrm{\cdots}{x}_{i-1}[{x}_{i},{x}_{i+1}]{x}_{i+1}\mathrm{\cdots}{x}_{n}.$ |

As $B$ is a basis of $\U0001d528$, $[{x}_{i},{x}_{i+1}]$ 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 $({x}_{1},\mathrm{\dots},{x}_{n})$, 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 |
---|---|

Canonical name | PoincareBirkhoffWittTheorem |

Date of creation | 2013-03-22 13:03:38 |

Last modified on | 2013-03-22 13:03:38 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 7 |

Author | CWoo (3771) |

Entry type | Theorem^{} |

Classification | msc 17B35 |

Synonym | PBW-theorem |

Related topic | LieAlgebra |

Related topic | UniversalEnvelopingAlgebra |

Related topic | FreeLieAlgebra |