Standard Lyndon bases of Lie algebras and enveloping algebras

Arun Ram
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 C with integer entries αi,αj, 1i,jn. 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 C corresponding to a finite dimensional simple Lie algebra. Let A={x1,x2,,xn,h1,h2,,hn,y1,y2,,yn}, and let J be the ideal of Serre relations in Lie(A), i.e., the ideal generated by the elements [hi,hj] (1i,jn), (S1) [xi,yi]-hi, [xi,yi]if ij, (S2) [hi,xj]- αj,αi xj, [hi,yj]+ αj,αi yj, (S3) (adxi)-αj,αi+1 (xj)(ij), (Sij+) (adyi)-αj,αi+1 (yj)(ij). (Sij-) Here (ada)k(b)=[a,[a,[a,,[a,b]]]]. Let X={x1,x2,,xn} be ordered by x1<x2<<xn, and let J+ be the Lie ideal in Lie(X) generated by the relations (Sij+). Let Y={y1,y2,,yn}, y1<<yn, and let J- be the Lie ideal in Lie(Y) generated by the relations (Sij-). Define 𝔤=Lie(A)/J, 𝔫+=Lie(X)/J+, and𝔫-=Lie(Y) /J-.

Let αi be independent vectors. The αi are called the simple roots. The root lattice is the lattice Q=i=1nαi. Let Q+={α=i=1najαiQ|ai0}. The height of a root α=i=1naiαiQ+ is ht(α)=i=1nai. The weights of words w=xi1xikX* and w=yi1yikY* are defined by wt()=αi1++ αikandwt() =-αi1--αik, respectively. Note that the length of a word w such that wt(w)=α is equal to ht(α).

With notations for standard bracketings as in (1.1) we define 𝔤α=-span {b[]|X*,wt()=α} and 𝔤-α=-span {b[]|Y*,wt()=-α}, for each αQ+. The set Φ+= {αQ+|α0,dim(𝔤α)0} is the set of positive roots. Let 𝔥 be the linear span of the generators hi. The following facts about finite dimensional simple Lie algebras 𝔤 are standard ([Hum1972] Thms. 18.3, 14.2, 8.4) (3.1a) 𝔤𝔫-𝔥𝔫+. (3.1b) 𝔫+αΦ+𝔤α and 𝔫-αΦ+𝔤-α. (3.1c) Φ+ is finite. (3.1d) dim(𝔤α)=1 for all αΦ+. The following result follows easily from the above facts.

(a) For each αΦ+ there is a unique standard Lyndon word αX* with respect to the ideal J+ such that wt(α)=α. (b) The words α,α and the letters hi are the standard Lyndon words in A* with respect to the ideal J. These words form a basis of the finite dimensional simple Lie algebra 𝔤.


(a) Since the standard Lyndon words in X* with respect to the ideal J+ form a basis of 𝔫+, it follows that 𝔤α is the subset of 𝔫+ spanned by the bracketings of standard Lyndon words of weight α with letters in X. Similarly, 𝔤-α is the subspace of 𝔫- spanned by all b[w] such that w is a standard Lyndon word with letters in Y and such that wt(t)=-α. The statement in (a) now follows from (3.1a) and (3.1b).

(b) It follows from (3.1d) that for each αΦ+ there is a unique standard Lyndon word α of weight α and that 𝔤α=b[α]. Furthermore, it is clear from the form of the relations in J+ and J- that 𝔤-α=b[α], where if α=xi1xik, then α is the word in Y* given by α=yi1yik.

Our goal is to determine the standard Lyndon words α, for all αΦ+. 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.

Let α=i=1naiαiΦ+. By Proposition (2.9) we know that α is of the form α=β1 β2βk xe, (3.3) where (1) βj are standard Lyndon words for all 1jk, (2) βj is a left factor of βj-1 for all j>1, (3) xeX. In the following discussion we shall exclude the trivial case α=1 so that k>0. Because of (2), each of the factors βj begins with the same letter, say xbX, 1bn, and xbxe since α is Lyndon; in fact, the letter xb is the smallest letter in the word α. Thus, since wt(xb)=αb, b is the smallest integer in {1,,n} such that ab0. Since xb appears exactly ab times in α and it appears as the first letter of each of the factors βj, 1jk, it follows that kab. A scan of the root tables for the finite dimensional simple Lie algebras shows that ab3 and that ab=3 for only one positive root (this root is in type G2).

For each 1jk, let βj=wt(βj). Then the factorization in (3.3) must satisfy the following: (1) βj=wt(βj)Φ+ for all 1jk, since the factors βj are standard Lyndon words. (2) j=1kβj+αe=a where αe=wt(xe). (3) Since α is Lyndon, xe>xb, where xb is the first letter of the words βj and xe is the last letter of α. (4) k2, except for a single root in type G2. (5) If k=2, then β1-β2Q+, since β2 is a left factor of β1. (6) α is a Lyndon word. (7) All the Lyndon factors of α are standard Lyndon words of smaller length and thus correspond to roots γΦ+ such that ht(γ)<ht(α). Given these rules it is easy to construct the standard Lyndon words by induction. Let αΦ+, and assume that the standard Lyndon words β are known for all βΦ+ such that ht(β)<ht(α). There are very few choices of roots βjΦ+, 1jk, k2, such that β1-β2Q+ if k=2 and α-jβj=αe, where αe, 1en. The words βj are all known since ht(βj)<ht(α). Restricting to the cases where xe is greater than the first letter of the words βj leaves very few possibilities. In fact, one finds that for each root αΦ+ (except the root α=α1+2α2+4α3+2α4 in type F4) there is a unique word α which satisfies conditions (1)-(7) above. Since there is a unique standard Lyndon word corresponding to the root α this word must be α.

Consider the root α=α1+2α2+4α3+2α4 in F4. Applying (1)-(7) above, leaves two possibilities for the word α: w1=x1x2x3x4x3x2x3x4x3 and w2=x1x2x3x4x3x4x2x3x3. Modulo the ideal J+ we can write w1 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 x's in writing the words and at each step we have underlined the letters which are being changed modulo the defining relations for the ideal J+): w1 = 123432343_ = 12(123432433)+ 12(12343233_4) = 12 [ (123432433)+ 13(12342_3334)+ (123433234)- 13(123433324) ] = 12 [ (123432433)+ 13(1232_43334)+ (123433234)- 13(123433324) ] = 12 [ (123432433)+ 16 [ (132243334)+ (122_343334) ] + (123433234)- 13(123433324) ] = 12 [ (123432433)+ 16 [ (132243334)+ [ 2·(212343334)- (221343334) ] ] + (123433234)- 13(123433324) ] . Thus w1=x1x2x3x4x3x2x3x4x3 is not a standard Lyndon word and α=x1x2x3x4x3x4x2x3x3.

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 pw=(i1i2ik) in the tree consisting of a chain of successive vertices and edges moving away from the root determines a word w=xi1xikX*. 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.

An i i+1 n-1 n Bn i i+1 n-1 n n n-1 i+2 i+1 Cn i i+1 n-1 n n-1 i+2 i+1 i i+1 n-2 n-1 n Dn i i+1 n-2 n n-1 i+2 i+1 n-1 F4 1 2 3 4 3 2 1 2 3 4 3 4 2 3 3 2 3 2 2 3 2 3 4 3 4 3 4 3 4 G2 2 2 1 2 1 1 1 2 2
1 3 4 5 2 6 2 7 2 4 8 2 4 3 7 4 5 3 4 5 3 3 5 3 6 3 4 6 3 3 4 2 7 3 4 4 2 3 4 2 5 2 4 5 2 2 6 2 4 4 6 2 4 3 2 4 3 4 3 5 3 3 4 3 1 3 4 5 6 7 8 2 4 5 6 7 8 7 3 6 3 4 2 5 3 4 5 4 3 4 5 6 3 4 5 6 7 3 4 5 6 7 8 4 5 6 7 8 5 6 7 8 6 7 8 7 8 8
Figure 1. The trees giving the standard Lyndon paths. The root of each tree is the left most vertex. For An,Bn,Cn,Dn, we give a generic tree with root i, where i=1,2,,n (i=1,2,,n-1 for Dn). The standard paths for E7 (respectively E6) are the paths from the trees for E8 not containing 8 (respectively 7 and 8). 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 An,Bn,Cn,Dn are Chevalley bases. However, the standard Lyndon basis for type G2 is definitely not a Chevalley basis. In fact, in type F4 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.

page history