PlanetMath: Poincaré-Birkhoff-Witt theorem

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Poincaré-Birkhoff-Witt theorem

(Theorem)

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

$\displaystyle x_1 x_2 \cdots x_n$ with $\displaystyle x_1 \leq x_2 \leq \cdots \leq x_n$ elements of $\displaystyle B$

constitute a

-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 denote the set

$\displaystyle M_n=\{(x_1,\ldots,x_n)\mid x_1 \leq \cdots \leq x_n\} \subset B^n,$

and denote by $\pi:\bigcup_{n=0}^\infty B^n \rightarrow U(\mathfrak{g})$ the multiplication map. Clearly it suffices to prove that

$\displaystyle \pi(B^n) \subseteq \sum_{i=0}^n \pi(M_i)$

for all $n \in \mathbb{N}$ ; to this end, we proceed by induction. For

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

, then we are done. Otherwise, there exists an index

such that $x_i>x_{i+1}$ . Now we have

$\displaystyle \pi(x_1,\ldots,x_n)$	$\displaystyle =\pi(x_1,\ldots,x_{i-1},x_{i+1},x_i,x_{i+2},\ldots,x_n)$
	$\displaystyle +x_1\cdots x_{i-1}[x_i,x_{i+1}]x_{i+1}\cdots x_n.$

is a basis of $\mathfrak{k}$ , $[x_i,x_{i+1}]$ is a linear combination of

. 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.

Bibliography

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

"Poincaré-Birkhoff-Witt theorem" is owned by CWoo. [ full author list (3) | owner history (2) ]

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Other names:

PBW-theorem

Keywords:

PBW, ordered monomials

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Cross-references: linear independence, proof, contains, argument, induction hypothesis, term, expand, linear combination, basis, index, induction, map, multiplication, span, easy to see, universal enveloping algebra, monomials, linear order, field, Lie algebra
There are 2 references to this entry.

This is version 4 of Poincaré-Birkhoff-Witt theorem, born on 2002-09-19, modified 2006-10-31.
Object id is 3467, canonical name is PoincareBirkhoffWittTheorem.
Accessed 4387 times total.

Classification:

AMS MSC:

17B35 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Universal enveloping algebras)

Pending Errata and Addenda

None.[ View all 2 ]

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