Poincaré-Birkhoff-Witt theorem
Let be a Lie algebra over a field , and let
be a -basis of equipped with a linear
order . The Poincaré-Birkhoff-Witt-theorem (often
abbreviated to PBW-theorem) states that the monomials
constitute a -basis of the universal enveloping algebra of . Such monomials are often called ordered monomials or PBW-monomials.
It is easy to see that they span : for all , let denote the set
and denote by the
multiplication map. Clearly it suffices to prove that
for all ; to this end, we proceed by induction. For
the statement is clear. Assume that it holds for , and consider a
list . If it is an element of , then we are
done. Otherwise, there exists an index such that .
Now we have
As is a basis of , is a linear
combination of . Using this to expand the second term above, we find
that it is in by the induction hypothesis.
The argument of in the first term, on the other hand, is
lexicographically smaller than , 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 |