Standard Lyndon bases of Lie algebras and enveloping algebras
Last updated: 19 May 2015
This is an excerpt of the paper Standard Lyndon bases of Lie algebras and enveloping algebras by Pierre Lalonde Arun Ram.
The first author's research was supported by FCAR and NSERC grants. The second author's research was partially supported by a National
Science Foundation Postdoctoral Fellowship.
Order the words in by setting
This is a total order on words with the additional property that there are a finite number of words less than any given word.
Suppose that is a Lie ideal of
and that is the ideal in
generated by Let
It follows from (1.6) and (1.8), that is the enveloping algebra of
Define a Lyndon word to be Lie-standard with respect to if its bracketing
cannot be written as a sum of bracketings of strictly smaller Lyndon words
modulo the ideal of with respect to the ordering
Define a word to be standard with respect to
if cannot be written as a sum of strictly smaller words modulo the ideal again with
respect to the ordering Make the following notations:
is the set of Lyndon words,
is the set of Lie-standard Lyndon words,
is the set of standard words.
The standard words that we have defined are essentially a Gröbner basis. The following two theorems are the standard results from the Gröbner basis context.
The set of elements where
is a basis of
The set of all where
is not Lie-standard then can be written as a linear combination of
bracketings of Lyndon words modulo which are smaller than If any of these
words is not standard, express it as a sum of smaller words. Continue this process until all the words in the expansion are standard. The process must stop as
the number of words smaller than any given word is finite. Thus the elements
We now show that the set of Lie-standard Lyndon words is linearly independent. Suppose that there was a nontrivial relation among them. Then this relation
expresses the maximal word as a linear combination of lower words modulo a
contradiction to the standardness of the maximal word.
The set of words in is a basis for
The proof is exactly analogous to the proof of Theorem (2.1).
We shall show that i.e. that the set of Lie-standard
Lyndon words is the same as the set of standard Lyndon words (this is not a priori obvious).
for some Using (1.2) on each side,
for some integers
Subtracting from both sides,
for some integers Since
we have that
Any factor of a standard word is a standard word.
Suppose that is not standard so that we have
where Since is an ideal
we have that is not standard.
If then has a unique factorization
By (1.5) and the definition of the ordering
Expanding the sum in terms of standard words,
The result now follows from Corollary (2.5) since each appearing in the sum has a unique factorization
of the form
Each of the following sets is a basis of
The set of standard words with respect to
The set of products
The set of products where
Statement (B1) is Theorem (2.2). (B2) is a basis by (2.1) and the Poincaré-Birkhoff-Witt theorem. Theorem (2.6) gives a triangular relation between
the elements of the set (B3) and the elements of the set (B2) which proves that (B3) is a basis.
With notations as in Theorem (2.7),
(a) Corollary (2.5) gives that Since these are both bases we must have
(express the basis (B3) in terms of the basis (B1)).)
Combining this with Lemma (2.3) we have that
The following proposition will help us to compute the standard Lyndon words by induction on the length of the words.
Let be a standard Lyndon word. Then is of the form
are standard Lyndon words for all
is a left factor of
Let be the word with the last letter removed. By (1.4) has a factorization
into Lyndon words
for all By Proposition (2.4), each of
the factors is standard since they are factors of the standard word
It remains to prove that is a left factor of
for all Consider the following chain of inequalities. For
where the last inequality follows since the right hand side
is a right factor of the Lyndon word
It follows easily from
that is a left factor of
(consider these as words in a dictionary).