Lie Algebras
3 Poincaré-Birkhoff-Witt
We would in general like to understand the structure of
To do this, we will use a general tool due to George Bergman for finding canonical forms for elements of associative
Say that we have a set
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Proof. Suppose
. is reduction finite, and if we have two reduction sequences of it to an irreducible, we can consider doing the same sequence on and extend them to one that makes each irreducible (note that this won’t change the result, only the sequence). Then by uniqueness for we see that the two reductions are the same, and that is -linear. □
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Proof. Note in particular that the hypotheses imply that if
is reduction unique, and that if the conclusion holds, . By linearity we only need to show this when are monomials, and when is a single reduction . But , so this follows since is reduction unique. □
We say that a overlap ambiguity is a pair
Say that a compatible partial ordering
Finally we arrive at this theorem, which can be considered a Diamond Lemma for rings:
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Theorem 3.3. Suppose we have
as above where is compatible with and satisfies the descending chain condition. Then the following are equivalent:-
1. Every ambiguity is resolvable.
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2. Every ambiguity is resolvable with respect to
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3. Every element is reduction unique.
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4. The natural quotient identifies the submodule spanned by irreducible monomials with
, which is .
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Proof. Clearly
. If is true, defines a projection onto . The kernel is contained in by definition of and contains , as for any , by Lemma 3.2. Thus as an -module, , giving . Conversely if is true, Then since reductions are equal in the quotient, they must be unique.Thus it suffices to prove
, and by linearity, we need only show this for a monomial . From the descending chain condition, we can assume that any smaller monomial is reduction unique. Now suppose that we have two reductions of a monomial .If there is no ambiguity, they commute, so we can create two more reductions to an irreducible where the first two steps are the first step of these reductions but in different orders. They will give the same element, and by induction will show that the two original reductions are also the same.
If there is ambiguity, then since it is resolvable relative to
by induction the difference can be resolved to , so by linearity the two resolutions must agree. □
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Proof. Construct an ordering on monomials in the basis where monomials of smaller degree are smaller, and if two monomials have the same degree, we compare them lexicographically using the order on our basis. This is a well-ordering on monomials, and a sufficient set of relations for
can be written as for in the basis. The order is compatible with this, and we can check that every ambiguity is resolvable with respect to our order. Here, the only nontrivial kind of ambiguity that can occur is an overlap ambiguity, when we have something of the form . The difference is:Where at the last step, we have used the Jacobi identity. This shows that the ambiguity is resolvable under the ordering, so that by Theorem 3.3 we are done. □
Note that as a corollary, any Lie algebra