Standard Lyndon bases of Lie algebras and enveloping algebras
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
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.
Finite-dimensional simple Lie algebras
In this section we shall compute the standard Lyndon words corresponding to the finite dimensional simple Lie algebras over
Each such algebra is determined by a Cartan matrix with integer entries
A list of these Cartan matrices
can be found in [Bou1981] pp. 250-275. We shall use the Bourbaki conventions for numbering.
Fix a Cartan matrix corresponding to a finite dimensional simple Lie algebra. Let
and let be the ideal of Serre relations in
i.e., the ideal generated by the elements
be ordered by
and let be the Lie ideal in
generated by the relations
be the Lie ideal in generated by the relations
Let be independent vectors. The are called the simple roots.
The root lattice is the lattice
The height of a root
The weights of words
are defined by
respectively. Note that the length of a word such that
is equal to
With notations for standard bracketings as in (1.1) we define
for each The set
is the set of positive roots. Let be the linear span of the generators
The following facts about finite dimensional simple Lie algebras are standard ([Hum1972] Thms. 18.3, 14.2, 8.4)
The following result follows easily from the above facts.
For each there is a unique standard Lyndon word
with respect to the ideal
and the letters are the standard Lyndon words in
with respect to the ideal These words form a basis of the finite dimensional simple Lie algebra
(a) Since the standard Lyndon words in with respect to the ideal
form a basis of it follows that
is the subset of spanned by the bracketings of standard Lyndon words of weight
with letters in Similarly,
is the subspace of spanned by all
such that is a standard Lyndon word with letters in and such that
The statement in (a) now follows from (3.1a) and (3.1b).
(b) It follows from (3.1d) that for each
there is a unique standard Lyndon word of weight and that
Furthermore, it is clear from the form of the relations in and
then is the word in
Our goal is to determine the standard Lyndon words for all
This is done by induction on the
lengths of the words (heights of the roots). The main tools are Propositions (2.4) and (2.9) from section 2 and the tables of the positive roots for the finite
dimensional simple Lie algebras, [Bou1981] pp. 250-275.
By Proposition (2.9) we know that is of the form
are standard Lyndon words for all
is a left factor of
In the following discussion we shall exclude the trivial case
so that Because of (2), each of the factors
begins with the same letter, say
since is Lyndon; in fact, the letter is the smallest letter in
the word Thus, since
is the smallest integer in such that
Since appears exactly
times in and it appears as the first letter of each of the
it follows that A scan of the root tables for the finite dimensional
simple Lie algebras shows that and that
for only one positive root (this root is in type
For each let
Then the factorization in (3.3) must satisfy the following:
for all since the factors
are standard Lyndon words.
Since is Lyndon,
where is the first letter of the words
and is the last letter of
except for a single root in type
since is a left factor of
is a Lyndon word.
All the Lyndon factors of are standard Lyndon words of smaller length and thus correspond to roots
Given these rules it is easy to construct the standard Lyndon words by induction. Let
and assume that the standard Lyndon words are known for all
There are very few choices of roots
such that if
The words are all known since
Restricting to the cases where is greater than the first letter of the words
leaves very few possibilities. In fact, one finds that for each root
(except the root
in type there is a unique word which
satisfies conditions (1)-(7) above. Since there is a unique standard Lyndon word corresponding to the root this word must be
Consider the root
in Applying (1)-(7) above, leaves two possibilities for the word
Modulo the ideal we can write as a linear combination of standard Lyndon words which are smaller
in the order (greater in lexicographic order). This computation is as follows (we have suppressed the
in writing the words and at each step we have underlined the letters which are being changed modulo the defining relations for the ideal
is not a standard Lyndon word and
Figure 1 gives the standard Lyndon words corresponding for each of the finite dimensional simple Lie algebras. Let us explain how to read these diagrams. Each
tree is rooted. A path
in the tree consisting of a chain of successive vertices and edges moving away from the root determines a word
The trees are constructed (by applying the procedure described above) so that this word is always a standard word with respect to the ideal of Serre relations
determined by the corresponding Cartan matrix. If the word is Lyndon then we say that the path is Lyndon. In the discussion following Proposition (3.2)
we have described how one proves (case by case) the following theorem.
For each of the trees in Figure 1 the set of words determined by the Lyndon paths in the tree is the complete
set of standard Lyndon words for the corresponding finite dimensional simple Lie algebra.
Figure 1. The trees giving the standard Lyndon paths. The root of each tree is the left most vertex. For
we give a generic tree with root where
for The standard paths for (respectively
are the paths from the trees for not containing
(respectively and The white vertices end Lyndon paths, while black
vertices end non-Lyndon paths. The trees are designed so that for each root system all the ends of Lyndon paths corresponding to roots of the same height lie on a
common vertical line.
Remark. We have made some effort to compute the bracketing rule for the finite dimensional simple Lie algebras in terms of the basis of standard Lyndon words.
We have not yet succeeded in learning much from this exercise. We make only the following remarks, with a bit of reservation as the computations are complicated and
difficult to check precisely. It seems that the standard Lyndon bases for Types
are Chevalley bases. However, the standard Lyndon basis for type is definitely not a Chevalley basis. In fact, in type
there are even some structure coefficients that are not integral.
Remark. It is clear that all of the results in section 2 are valid for any Lie algebra given by generators and relations. Preliminary computations seem
to indicate that it will be very instructive to study root multiplicities for Kac-Moody Lie algebras by way of standard Lyndon words.