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.
- 1 N. Jacobson. . Dover Publications, New York, 1979
|Date of creation||2013-03-22 13:03:38|
|Last modified on||2013-03-22 13:03:38|
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