Essays on the structure of reductive groups Structure constants

Essays on the structure of reductive groups Structure constants

Bill Casselman

University of British Columbia [email protected]

The principal goal of this essay is to explain Tits’ analysis of the structure constants of semi-simple Lie algebras. I’ll begin in the first part with a survey of the standard approach to the topic, that found originally in [Chevalley:1955] and later explained more accessibly, for example in [Carter:1972]. I’ll include here a reasonably explicit algorithm for computing the structure constants, taken largely from [Cohen-Murray-Taylor:2004]. Then in the second part I’ll try to explain the approach of [Tits:1966]. Tits’

paper is often cited, but usually rather vaguely, and I suspect that few have understood it.

The structure constants of a Lie algebra are the coefficients occurring in the formulas for the Lie bracket in terms of a given basis. For a semi-simple algebra there is a natural direct sum decomposition into eigenspaces with respect to a Cartan subalgebra. One of the eigenspaces is the null eigenspace, the Cartan algebra h itself. This contains elements h

λ

, one for each root λ. For h we have the basis h

α

parametrized by simple roots α. The other eigenspaces are parametrized by roots λ, and each has dimension one, say with basis e

_λ

. Given e

_λ

for λ > 0 there exists a unique e

−λ

with [e

_λ

, e

−λ

] = h

λ

. We have these formulas for the Lie bracket:

[h

α

, h

β

] = 0

[h

α

, e

λ

] = hλ, h

^α

ie

^µ

[e

λ

, h

α

] = −hλ, h

α

ie

µ

[e

λ

, e

µ

] =







h

λ

if µ = −λ N

λ,µ

e

λ+µ

if λ + µ is a root

0 otherwise.

for certain constants N

λ,µ

. The main result of Chevalley is that for a suitable choice of the x

λ

we have N

λ,µ

= ±(p

λ,µ

+ 1), where p

λ,µ

is the greatest p such that µ − pλ is a root. For this basis the structure constants are integral, and this ultimately leads to Chevalley’s construction of split semi-simple groups over arbitrary fields. Chevalley’s formula is well known, but to me still a bit mysterious. As I hope to show, the derivation of [Tits:1966] is better motivated than the standard one.

The next problem is to figure out the signs of the structure constants. The standard procedure is straightforward in logic, but depends on various arbitrary choices, and this also seems to me rather mysterious. Again here, Tits’ argument seems more enlightening.

Tits’ idea is to relate these questions to some other questions about the structure of the normalizer N

G

(T ) of a maximal torus T in a split semi-simple group G. There is an exact sequence

1 −→ T −→ N

^G

(T ) −→ W −→ 1 ,

where W is the Weyl group. This hardly ever splits, but it is at least an extension derived from a much smaller extension

1 −→ T (2) −→ N −→ W −→ 1 ,

where T (2) is the subgroup of 2-torsion in T and N is a subgroup of N

^G

(T ). Several choices of N

are possible, and one of Tits’ observations is that each one determines an integral structure on G and

g. Another is that the problem of signs is also related to problems concerning N . In Tits’ account some puzzles remain, but far fewer than in the standard one.

This essay contains no really new results of a mathematical nature—I have simply reorganized Tit’s own exposition.

Contents

I. The standard approach

1. The data defining a simple Lie algebra 2. The first main Theorem

3. Chains

4. Proof of the main result 5. Chevalley’s integral constants

6. Computation of structure constants—mathematics 7. Computation of structure constants—programming II. Tits’ analysis

8. The extended Weyl group

9. References

Part I. The standard approach

1. The data defining a simple Lie algebra

Suppose g to be a simple, semi-simple Lie algebra over C. It contains an abelian subalgebra h that is its own centralizer, called a Cartan algebra of g. The adjoint action of h on g decomposes g into a direct sum of eigenspaces. The 0-eigenspace is h itself. All the others are one-dimensional, with respect to characters of h called ^roots of the pair (g, h). The restriction of the Killing form of g to h is positive definite, and this allows us to identify the roots with elements of h itself. Define for each root λ the element

h

λ

= 2λ kλk

²

in h. These span a lattice that spans h. The dot products λ

^•

h

µ

lie in Z, and the orthogonal root reflections s

λ

: h 7−→ h − (h

^•

h

λ

)λ

take Σ into itself. The Weyl group W is the group generated by these. The closure of each of the connected components of the complement in h of the root hyperplanes v

^•

λ = 0 is a fundamental domain for W . Let C be one of these, and let Σ

⁺

= Σ

⁺_C

be the set of λ in Σ with λ

^•

C > 0, ∆ the set of α in Σ

⁺

that vanish on a wall of C. Then Σ is the union of Σ

⁺

and −Σ

⁺

. Roots in ∆ are called simple roots . These form a basis of h, and every λ in Σ

⁺

is a non-negative integral combination of simple roots. The set Σ is the smallest subset of h containing ∆ that is stable under W .

Because g is simple, the root system is irreducible in the sense that ∆ is not the union of two orthogonal subsets. Equivalently, its Dynkin diagram is connected.

Choose e

λ

6= 0 in g

^λ

, for λ > 0. There exists a unique e

−λ

in g

−λ

such that [e

λ

, e

−λ

] = h

λ

. All in all, we

have [h

λ

, h

µ

] = 0

[h

λ

, e

µ

] = hµ, h

λ

ie

µ

[e

µ

, h

λ

] = −hµ, h

^λ

ie

^µ

[e

λ

, e

µ

] =







h

λ

if µ = −λ N

λ,µ

e

λ+µ

if λ + µ is a root

0 otherwise.

for suitable constants N

λ,µ

.

For a given λ, the elements h

λ

, e

±λ

span a Lie algebra isomorphic to that of SL

2

.

The first problem to be dealt with now is straightforward. How to compute the structure constants N

λ,µ

, given the Cartan matrix defining the root system? Later on, I’ll discuss a second natural question:

Given a root system, can we construct the corresponding Lie algebra? Defining it by naming a basis is straightforward, according to what we have already seen, and previous material will also determine candidate structure constants, so the question is, can we prove that what we define is in fact a Lie algebra?

Complete and explicit formulas for the structure constants are somewhat complicated, but the book [Carter:1972], which I follow, presents a relatively simple explanation of what’s involved. Carter also presents an algorithm for calculating them explicitly, and this has been made more explicit in [Cohen- Murray-Taylor:2004]. I deviate only in details from that last paper. Another, in many ways more natural bit still complicated, approach to both questions can be found in [Tits:1966].

So to start, under the assumption that the complex semi-simple Lie algebra exists, I’ll explain how to

obtain its structure constants. But afterwards I’ll also look at the converse problem—constructing the Lie

algebra from the structure constants. That is to say, given a root system, the algorithm at hand produces

a set of structure constants. A natural question is whether one can prove directly that they determine a

Lie algebra—that is to say, whether one can prove directly that the bracket operation defined by these structure constants is indeed a Lie bracket. I’ll obtain partial results towards answering this.

Let me give a little background to this problem. Constructing the Lie algebra g from its root system—i.e.

showing that every root system does indeed give rise to a Lie algebra—is not trivial. Originally, this was done on a case-by-case analysis, but different cases required vastly different techniques. The exceptional Lie algebras have always caused some trouble, and the Lie algebras e

n

(n = 6, 7, 8) as well as the groups E

n

have always remained rather abstract entities. These groups are certainly complicated, and nobody should expect to carry out computations in E

8

as readily as in the groups GL

n

, but one might hope that a computer will make things seem a bit less abstract. In this context, defining the Lie algebra in terms of the computed structure constants seems a worthwhile project, and a bit less tedious than might otherwise appear.

There are several methods that have been applied uniformly (as opposed to case-by-case) to construct the Lie algebra from the root system. One is that found in [Jacobson:1962], which is due, to what extent independently I do not know, to Chevalley and Harish-Chandra. A second relies on a later result of [Serre:1966] describing the Lie algebra by generators and relations. As far as a direct construction, there have been two approaches. One is in [Tits:1966] and the other, more recent, in [de Graaf:2000]. But I’d like to see an approach even more direct.

Both the first two methods have been applied to construct Kac-Moody algebras from a generalized Cartan matrix, but in this situation things are not entirely satisfactory. Here, there is a important problem that has not been resolved—the methods of Jacobson on the one hand and Serre on the other are not known to produce the same algebra in all cases. For Kac-Moody algebras the so-called ‘imaginary roots’ cause trouble, since the corresponding eigenspaces may have dimension greater than 1, and I am not aware that anyone has proposed even a good description of a basis, much less the corresponding structure constants. Although I do not make any contribution to its solution, it is this problem that has been my principal motivation in writing this note.

2. The first main Theorem

Suppose C to be a Cartan matrix (c

α,β

) with rows and columns indexed by a finite set ∆. This means that (a) the entries of C are integers, (b) c

_α,α

= 2, (c) c

α,β

≤ 0 for α 6= β, and (d) DC is positive definite for some diagonal matrix D (necessarily with positive entries). Let h be a complex vector space, h

^∨

is dual. Assume ∆ embedded as a basis of h

^∨

. Then there exists a map α 7→ h

α

(sometimes written as α

^∨

) from ∆ to h such that

hα, h

^β

i = c

^α,β

. The linear map

s

α

: v 7−→ v − hα, vih

α

is a reflection. It will be orthogonal with respect to an inner product u

^•

v if and only if

 h

α•

h

α

2



hα, β

^∨

i = h

^α•

h

β

for all α, β in ∆. Therefore we may define this inner product by the formula h

α•

h

β

= d

α

c

α,β

.

I assume that C is irreducible, which means that ∆ cannot be partitioned into two mutually orthogonal subsets.

Let W be the group of reflections of h generated by the s

α

. By duality, it may also be identified with a

group of linear transformations of h

^∨

. Let Σ be the smallest subset of h

^∨

stable under W and contating

∆. This is the set of roots defined by C. There exists a unique map from Σ to h taking λ = wα to h

λ

= wh

α

.

The roots Σ span a full lattice in h

^∨

. Let kλk = h

^•

h.

For each λ in Σ, let g

λ

be a one-dimensional space with generator e

λ

, and let g be the direct sum of h and the g

λ

. To each pair of roots λ, µ such that λ + µ is a root assign a non-zero complex number N

λ,µ

. If λ + µ is not a root, let N

λ,µ

= 0. Then define a candidate bracket operation [u, v] on g by the specification:

[h

λ

, h

µ

] = 0

[h

λ

, e

µ

] = hµ, h

^λ

ie

^µ

[e

µ

, h

λ

] = −hµ, h

λ

ie

µ

[e

λ

, e

µ

] =







0 if µ = λ h

λ

if µ = −λ N

λ,µ

e

λ+µ

otherwise.

The span of e

±λ

and h

λ

is a copy of the Lie algebra of sl

2

.

The problem at hand is to find conditions on the constants N

λ,µ

that insure the bracket operation defines a Lie algebra—to be precise, that the bracket operation is anti-symmetric and satisfies the Jacobi identity.

These are both immediate from its definition in any bracket involving an element of h.

Proposition 2.1. The definition of [u, v] makes g into a Lie algebra if and only if the constants N

λ,µ [structure-equations]

satisfy the following relations:

(a) N

µ,λ

= −N

^λ,µ

; (b) if λ + µ + ν = 0

N

λ,µ

kνk

²

= N

µ,ν

kλk

²

= N

ν,λ

kµk

²

; (c) N

λ,µ

N

−λ,λ+µ

= (p

λ,µ

+ 1)q

λ,µ

;

(d) whenever λ + µ + ν + ρ = 0 and no pair is opposite N

λ,µ

N

ν,ρ

kλ + µk

²

+ N

µ,ν

N

λ,ρ

kµ + νk

²

+ N

ν,λ

N

µ,ρ

kν + λk

²

= 0 .

As we shall see, the ultimate reason for the appearance here of the norms is the identity of h

λ

with 2λ/ kλk

²

.

If λ and µ are any two roots, there exist p = p

_λ,µ

and q = q

_λ,µ

such that rλ + µ is a root if and only if

−p ≤ r ≤ q. (I’ll recall in the next section how this works.) The sequence

−pλ + µ, . . . , rλ + µ, . . . , qλ + µ

is called the λ -chain through µ. The significance of chains is that the sum of subspaces g

µ

as µ varies over a λ chain is a representation of the copy of sl

2

correspondng to λ.

in the circumstances, condition (c) is equivalent to Chevalley’s remarkable observation that N

λ,µ

N

−λ,−µ

= −(p

λ,µ

+ 1)

²

but is more directly related to the Jacobi identity. Later, we shall see how this can be used to compute

structure constants from a few that can be set arbitrarily.

3. Chains

If λ and µ are roots, then p = p

_λ,µ

is the greatest m with −mλ + µ a root and q = q

λ,µ

the greatest m with mλ + µ a root. In other words, p measures the distance to the back end of the chain and q to the forward one. It is 0 at one end. Elementary properties:

q

λ,µ

= p

−λ,µ

p

λ,λ+µ

= p

λ,µ

+ 1 q

λ,λ+µ

= q

λ,µ

− 1 p

−λ,−µ

= p

λ,µ

when these equations make sense.

Lemma 3.1. Suppose λ and µ to be roots with λ 6= ±µ. Then

[chains-a]

(a) if hµ, h

^λ

i < 0, µ + λ is a root;

(b) if hµ, h

^λ

i > 0, µ − λ is a root;

(c) the linear combination rλ + µ is a root if and only if −p

^λ,µ

≤ r ≤ q

^λ,µ

; (d) we have

hµ, h

λ

i = p

λ,µ

− q

λ,µ

.

Proof. If hµ, h

^λ

i < 0 then the classification of root systems of rank two implies that either hµ, h

^λ

i or hλ, h

^µ

i is −1. In the first case s

^λ

µ = µ − hµ, h

^λ

iλ = µ + λ is a root, and in the second s

^µ

λ = λ + µ is one. Item (b) is similar.

As for (c), it means neither more nor less than that no chain has any gaps in it. But since hµ + nλ, h

λ

i = hµ, h

λ

i + 2n

the function hµ + nλ, h

^λ

i is an increasing function of n. If there existed a gap in a chain, by (a) and (b) there would exist some µ in it such that hµ, h

^λ

i ≥ 0 but hµ + nλ, h

^λ

i ≤ 0, a contradiction.

The reflection s

λ

preserves each λ-chain, and must therefore reflect the extremity −pλ + µ to the other qλ + µ. Thus

µ + qλ = s

λ

(µ − pλ)

= µ − pλ − hµ − pλ, h

^λ

iλ

= µ + (p − hµ, h

^λ

i)λ which implies (d).

Since |hλ, h

^µ

i| ≤ 3 in all cases, this tells us that a chain can have at most 4 roots in it. This can be deduced

also from the fact that any two roots are contained in a root system of rank two. We can classify easily

all possible cases:

Looking carefully at the possible cases will tell us that p

λ,µ

can be calculated explicitly in terms of simple data. First of all, q

λ,µ

can be calculated from p

λ,µ

and hµ, h

^λ

i. Hence if hµ, h

^λ

i > 0 we may replace µ by r

λ

(µ). As for the cases where hµ, h

^λ

i ≤ 0, here is a list of possibilities, sorted by the value of hµ, h

^λ

i:

(1) hµ, h

^λ

i = 0: p

^λ,µ

= 0 except when kλk = kµk = √

2.

µ λ C

₂

p

λ,µ

= 1 hµ, h

^λ

i = 0

kµk

²

= 1 kλk

²

= 1

µ

λ C

₂

p

_λ,µ

= 0 hµ, h

λ

i = 0

kµk

²

= 2 kλk

²

= 2

µ

λ G

₂

p

λ,µ

= 0 hµ, h

^λ

i = 0

kµk

²

= 3 kλk

²

= 1

µ

λ G

₂

p

_λ,µ

= 0 hµ, h

λ

i = 0

kµk

²

= 1 kλk

²

= 3

(2) hµ, h

^λ

i = −1: p

^λ,µ

= 0 except when kλk = kµk = 1 and the system is G

²

.

µ λ A

₂

p

λ,µ

= 0 hµ, h

^λ

i = −1

kµk

²

= 1 kλk

²

= 1

µ λ C

₂

p

_λ,µ

= 0 hµ, h

λ

i = −2

kµk

²

= 2

kλk

²

= 1

µ λ G

₂

p

λ,µ

= 0 hµ, h

^λ

i = −1

kµk

²

= 3 kλk

²

= 3

µ λ G

₂

p

_λ,µ

= 0 hµ, h

λ

i = −3

kµk

²

= 3 kλk

²

= 1

µ λ G

₂

p

λ,µ

= 0 hµ, h

^λ

i = −1

kµk

²

= 1 kλk

²

= 3

(3) hµ, h

λ

i < −1: p

λ,µ

= 0.

µ λ C

₂

p

_λ,µ

= 0 hµ, h

λ

i = −1

kµk

²

= 1 kλk

²

= 2

µ λ G

₂

p

λ,µ

= 1 hµ, h

^λ

i = −1

kµk

²

= 1 kλk

²

= 1

I point out the obvious fact that this observation allows us to tell simply whether λ + µ is a root or not.

The following can also be deduced from the case-by-case list, but has a more direct proof:

Proposition 3.2. (Geometric Lemma) If λ and µ are roots such that λ + µ is also a root, then

[geometric-lemma]

kλ + µk

²

kµk

²

= p

λ,µ

+ 1 q

λ,µ

= p

λ,λ+µ

q

λ,µ

.

Proof. There are three basic types of configurations for µ and µ + λ.

(1) They lie at the center of the λ-chain through µ. Here, µ and λ + µ are of the same length, since s

λ

swaps them, and p

λ,µ

+ 1 = q

λ,µ

. Both sides of the equation to be verified are 1.

µ λ

λ + µ

p + 1 = q

(2) They lie on the left. A scan of possible systems of rank two shows that there are two possibilities, and in both the equation can be verified visually.

µ λ + µ

1

√ 2

p + 1 = 1, q = 2

µ λ + µ

√ 1 3

p+ 1 = 1, q = 3

(3) They lie to the right of center. In this case, the equation is equivalent to the near mirror-image—p + 1 and q are swapped, and so are kµk

²

and kλ + µk

²

.

µ λ + µ µ λ + µ

This is not a difficult result, but a bit mysterious, since case-by-case seems to be the simplest proof. Well . . . it is after all only two cases.

Lemma 3.3. If λ, µ and λ + µ are all roots then

[p-symmetric]

p

λ,µ

= p

µ,λ

.

Proof. By definition, p

λ,µ

= 0 if and only if −λ + µ is not a root, and p

µ,λ

= 0 ifa nd only if µ + λ is not a root. But −λ + µ is a root if and only if its negative −µ + λ is one, so p

λ,µ

= 0 if and only if p

µ,λ

= 0.

If λ + µ is a root and p

_λ,µ

6= 0 then the diagrams above show that kλk = kµk, and it is well known that

in an irreducible root system the Weyl group acts transitively on roots of the same length. This can also

be deduced from the diagrams below, since every pair of roots is contained in a set of roots of rank two.

Since every pair of roots is contained in a subsystem of rank two, this shows that p

λ,µ

= p

µ,λ

when these are not 1.

One consequence is an observation of [Tits:1966] (equation (37) of §3.3):

Corollary 3.4. If λ + µ + ν = 0 then

[mystery]

p

λ,µ

+ 1

kνk

²

= p

µ,ν

+ 1

kλk

²

= p

ν,λ

+ 1 kµk

²

.

Proof. Proposition 3.2(structure) implies that

♣

[geometric-lemma]

p

λ,µ

+ 1

kνk

²

= q

λ,µ

kµk

²

, so that by symmetry it suffices to show that q

λ,µ

= p

−λ−µ,λ

+ 1. But

p

−λ−µ,λ

+ 1 = p

−λ,λ+µ

+ 1 = p

−λ,µ

= q

λ,µ

.

A somewhat more elegant proof is offered by diagrams similar to those used to prove Proposition

♣

[geometric-lemma]

3.2(structure) . This result will not be used, but in light of later results it is suggestive. It will appear later on that N

λ,µ

is, up to sign, the same as p

λ+1

, but the proof is not straightforward. The parallelism between N

λ,µ

and p

λ,µ

+ 1 ought to be more transparent.

4. Proof of the main result

We must show that the bracket is anti-symmetric and satisfies the Jacobi identity if and only if Proposition

♣

[structure-equations]

2.1(a)–(d) hold.

(a) Since

[e

λ

, e

µ

] = N

λ,µ

e

λ+µ

[e

µ

, e

λ

] = N

µ,λ

e

λ+µ

, the bracket is anti-symmetric if and only if N

λ,µ

= −N

^µ,λ

. The rest of the Proposition is related to the Jacobi identity

[[x, y], z] + [[y, z], x] + [[z, x], y] = 0 .

Some repetition can be avoided if we keep in mind that it holds for all permutations of x, y, z if and only if it holds for one. We know that we only have to examine the cases where all of x, y, z are among the e

λ

. So we have to consider the condition that

[e

λ

, [e

µ

, e

ν

]] + [e

µ

, [e

ν

, e

λ

]] + [e

ν

, [e

λ

, e

µ

]] = 0

for each set of three roots. There are several cases: (i) λ + µ + ν = 0; (ii) λ + µ + ν is neither 0 nor a root;

(iii) λ + µ + ν is one of the three, say ν, which means that λ = −µ; (iv) λ + µ + ν is a root different from any of the three.

Case (ii) is trivial, since in that case all three of the terms in Jacobi’s identity vanish.

(b)(i) Suppose λ + µ + ν = 0. We need only prove this:

Lemma 4.1. Suppose that λ, µ, and ν are three distinct roots that satisfy

[jacobi-1]

λ + µ + ν = 0 .

The Jacobi identity

[e

λ

, [e

µ

, e

ν

]] + [e

µ

, [e

ν

, e

λ

]] + [e

ν

, [e

λ

, e

µ

]] = 0 holds if and only if

N

λ,µ

kνk

²

= N

µ,ν

kλk

²

= N

ν,λ

kµk

²

.

Proof. We have

[e

λ

, [e

µ

, e

ν

]] + [e

µ

, [e

ν

, e

λ

]] + [e

ν

, [e

λ

, e

µ

]]

= N

µ,ν

[e

λ

, e

−λ

] + N

ν,λ

[e

µ,−µ

] + N

λ,µ

[e

ν

, e

−ν

]

= N

µ,ν

h

λ

+ N

ν,λ

h

µ

+ N

λ,µ

h

ν

= N

µ,ν

 2λ kλk

²

 + N

ν,λ

 2µ kµk

²

 + N

λ,µ

 2ν kνk

²



=  N

µ,ν

kλk

²



λ +  N

ν,λ

kµk

²



µ +  N

λ,µ

kνk

²

 ν

=  N

_µ,ν

kλk

²

− N

λ,µ

kνk

²



λ +  N

_ν,λ

kµk

²

− N

λ,µ

kνk

²

 µ .

Since any two of the roots must be linearly independent, the Jacobi identity holds if and only if N

µ,ν

kλk

²

− N

λ,µ

kνk

²

= N

ν,λ

kµk

²

− N

λ,µ

kνk

²

= 0 . or, equivalently,

N

λ,µ

kνk

²

= N

µ,ν

kλk

²

= N

ν,λ

kµk

²

. (b)(iii) Suppose now that λ is a root and Π a λ-chain of roots.

Lemma 4.2. The Jacobi identity

[jacobi-2]

[e

λ

, [e

−λ

, e

µ

] + [e

−λ

, [e

µ

, e

λ

] + [e

µ

, [e

λ

, e

−λ

] = 0 for all µ in Π if and only if

N

λ,µ

N

−λ,λ+µ

= (p

λ,µ

+ 1)q

λ,µ

for all µ in Π.

Proof. We have

[e

λ

, [e

−λ

, e

µ

]] + [e

−λ

, [e

µ

, e

λ

]] + [e

µ

, [e

λ

, e

−λ

]]

= N

−λ,µ

[e

λ

, e

−λ+µ

] + N

µ,λ

[e

−λ

, e

µ+λ

] + [e

µ

, h

λ

]

= N

−λ,µ

N

λ,−λ+µ

e

µ

− N

^λ,µ

N

−λ,λ+µ

e

µ

− hµ, h

^λ

ie

^µ

= N

−λ,µ

N

λ,−λ+µ

− N

^λ,µ

N

−λ,λ+µ

− hµ, h

^λ

i
e

µ

. So the Jacobi identity holds if and only if

N

λ,−λ+µ

N

−λ,µ

− N

^λ,µ

N

−λ,λ+µ

= hµ, h

^λ

i for all λ, µ.

In other words, it suffices to prove:

Lemma 4.3. We have

[carters-lemma]

N

λ,−λ+µ

N

−λ,µ

− N

λ,µ

N

−λ,λ+µ

= hµ, h

λ

i for all µ in a given λ-chain if and only if

N

λ,µ

N

−λ,λ+µ

= q

λ,µ

(p

λ,µ

+ 1) . for all µ in the chain.

Proof. Let C

λ,ν

= N

λ,ν

N

−λ,λ+ν

. Choose µ to be the ‘left’ end of the chain for which −λ + µ is not a root.

Thus N

−λ,µ

, p

λ,µ

, and C

λ,−λ+µ

are all 0.

Since hµ, h

^λ

i = −q

^λ,µ

in these circumstances, the first sequence therefore reads

−C

λ,µ

= −q

λ,µ

C

λ,µ

− C

^λ,λ+µ

= −(q

^λ,µ

− 2) C

λ,λ+µ

− C

λ,2λ+µ

= −(q

λ,µ

− 4) C

λ,2λ+µ

− C

^λ,3λ+µ

= −(q

^λ,µ

− 6) while the first is

C

λ,µ

= q

λ,µ

· 1 C

λ,λ+µ

= (q

λ,µ

− 1) · 2 C

λ,2λ+µ

= (q

λ,µ

− 2) · 3 C

λ,3λ+µ

= (q

λ,µ

− 3) · 4 One is obtained from the other by summing or taking differences.

(b)(iv) Suppose λ + µ + ν + ρ = 0 but no two cancel. We have [[e

λ

, e

µ

], e

ν

] + [[e

µ

, e

ν

], e

λ

] + [[e

ν

, e

λ

], e

µ

]

= N

λ,µ

[e

λ+µ

, e

ν

] + N

µ,ν

[e

µ+ν

, e

λ

] + N

ν,λ

[e

ν+λ

, e

µ

]

= N

λ,µ

N

λ+µ,ν

+ N

µ,ν

N

µ+ν,λ

+ N

ν,λ

N

ν+λ,µ

e

_λ+µ+ν

so the Jacobi identity is valid if and only if

0 = N

λ,µ

N

λ+µ,ν

+ N

µ,ν

N

µ+ν,λ

+ N

ν,λ

N

ν+λ,µ

= N

λ,µ

N

−ν−ρ,ν

+ N

µ,ν

N

−λ−ρ,λ

+ N

ν,λ

N

−µ−ρ,µ

= N

λ,µ

N

ν,ρ

 kρk

²

kν + ρk

²



+ N

µ,ν

N

λ,ρ

 kρk

²

kλ + ρk

²



+ N

ν,λ

N

µ,ρ

 kρk

²

kµ + ρk

²



= N

λ,µ

N

ν,ρ

kν + ρk

²

+ N

µ,ν

N

λ,ρ

kλ + ρk

²

+ N

ν,λ

N

µ,ρ

kµ + ρk

²

= N

λ,µ

N

ν,ρ

kλ + µk

²

+ N

µ,ν

N

λ,ρ

kµ + νk

²

+ N

ν,λ

N

µ,ρ

kν + λk

²

according to (b)(i). This concludes the proof of the Theorem.

5. Chevalley’s integral constants The equation

N

λ,µ

N

−λ,λ+µ

= (p

λ,µ

+ 1)q

λ,µ

may be rewritten as

N

λ,µ

N

−λ,λ+µ

= (p

λ,µ

+ 1)p

−λ,µ

= (p

λ,µ

+ 1)(p

−λ,λ+µ

+ 1) which makes plausible the following:

Proposition 5.1. (Chevalley’s integral formula) We can choose the structure constants so that

[chevalley]

|N

^λ,µ

| = p

^λ,µ

+ 1 for all roots λ, µ with λ + µ a root.

This might have also been suggested by a comparison of Corollary 3.4(structure) and (b)(iii) above.

♣

^[mystery]

There are two parts to the proof. Here is one:

Lemma 5.2. In any semi-simple Lie algebra

[chevalley-equation]

N

λ,µ

N

−λ,−µ

= −(p

λ,µ

+ 1)(p

−λ,−µ

+ 1) = −(p

λ,µ

+ 1)

²

.

Proof. Since ( −λ) + (−µ) + (λ + µ) = 0, (b)(i) tell us that N

−λ,−µ

kλ + µk

²

= N

λ+µ,−λ

kµk

²

= − N

−λ,λ+µ

kµk

²

but by the Geometric Lemma

kλ + µk

²

kµk

²

= p

λ,µ

+ 1 q

λ,µ

so that N

λ,µ

N

−λ,−µ

= −(p

^λ,µ

+ 1)

²

if and only if

N

λ,µ

N

−λ,λ+µ

= −q

λ,µ

(p

λ,µ

+ 1) .

This reasoning is adequate, but I have to confess I find it rather unsatisfactory. Chevalley’s equation is very simple, but its proof is not so simple as one might hope for. The basis elements e

_λ

may be chosen arbitrarily, but any two differ by a constant. If we change e

λ

to ce

λ

, however, we change e

−λ

to c

⁻¹

e

−λ

and each N

λ,µ

by a factor of (c

λ

c

µ

/c

λ+µ

). The point at the moment is that the product N

λ,µ

N

−λ,−µ

does not change. It is thus something necessarily intrinsic—independent of the basis choice, and therefore likely to be interesting. There should be a more direct way to interpret it.

At any rate, this allows us to apply another observation due to Chevalley. He has shown that there are certain choices of basis that are much better than others. There exists an involution θ of g, called the opposition involution , that acts as −1 on h, hence swapping root spaces for λ and −λ. We can then scale e

λ

so that θ(e

λ

) = −e

^−λ

. With this choice of basis, we have

N

λ,ν

= ±(p

λ,µ

+ 1)

for all roots λ, µ.

6. Computation of structure constants—mathematics

Start by assigning a linear order to the simple roots ∆. Extend this to one of all positive roots such that λ < µ if

HT

(λ) <

HT

(µ). I recall that

HT

 X

∆

λ

α

α 

= X

λ

α

.

Order next all pairs (λ, µ) with 0 < λ < µ with λ + µ a root. This last order is primarily by

HT

(λ + µ) and secondarily by lexicographic order on the pair. Thus (λ

•

, µ

•

) < (λ, µ) if

HT

(λ

•

+ µ

•

) <

HT

(λ + µ), or if

HT

(λ

•

+ µ

•

) =

HT

(λ + µ) but λ

•

< λ.

For every positive root µ not in ∆ there exists a minimal α in ∆ such that µ = λ + α with λ > 0. Thus we have a map from Σ

⁺

− ∆ to ∆ × Σ

⁺

, taking µ to (α, λ). These pairs are called extra-special pairs in [Carter:1972] and also in [Cohen, Murray & Taylor:2900], but I’ll call them primitive . The elements e

α

for α in ∆ generate the Lie algebra ⊕

^λ>0

g

_λ

, and choosing the N

α,λ

for a primitive pair in effect specifies an element e

µ

of g

α+λ

if we are given e

λ

, so choosing the N

α,λ

for primitive pairs is equivalent to choosing the e

λ

for all λ > 0. At any rate, they can certainly be chosen arbitrarily. I’ll next show how the properties of Proposition 2.1 may be applied to construct the N

λ,µ

from those for primitive pairs.

♣

[structure-equations]

Computing p

λ,µ

follows a similar process, and must be done at the same time that N

λ,µ

is computed.

For primitive pairs, we build the graph with edges λ → α + λ for α in ∆. For the other cases, we use equations for p

λ,µ

analogous to those for N

λ,µ

.

The first point to make is that the computation for arbitrary pairs is easily reduced to that for pairs 0 < λ < µ with λ + µ a root. I’ll call these ^simple pairs. In a moment I’ll show how to compute the constants for simple pairs, but first I’ll show how to compute the constants for arbitrary pairs in terms of those for simple ones.

Keep in mind that if λ, µ, and ν are roots with λ + µ + ν = 0 then either two of the roots are positive or two of the roots are negative. This applies to the triple (λ, µ, ν) with ν = −(λ + µ).

So now suppose we know the structure constants for simple pairs, and suppose we want to compute N

λ,µ

where (λ, µ) is not simple. There are several cases to be encountered.

• If 0 < µ < λ, then (µ, λ) is simple and we set

N

λ,µ

= −N

^µ,λ

.

• If λ and µ are both negative, then we apply Chevalley’s equation and set

N

λ,µ

= −(p

λ,µ

+ 1)

²

N

−λ,−µ

.

• Otherwise, we have λ and µ of different signs. Let ν = −(λ + µ) . We know that

N

λ,µ

kνk

²

= N

µ,ν

kλk

²

= N

ν,λ

kµk

²

.

Either µ and ν have the same sign or ν and λ do. Depending on which, we use one of the right hand sides possibly together with the equation

N

λ,µ

N

−λ,−µ

= −(p

^λ,µ

+ 1)

²

to reduce the calculation to that of N

λ,µ

with both λ and µ positive.

So the computation reduces to that for the primitive pairs. We proceed through the ordered list of simple pairs (λ, µ), excepting the primitive pairs, calculating structure constants as we go. When we get to (λ, µ), all smaller pairs have been processed. We must compute N

λ,µ

. Here λ + µ is a root. If λ > µ, we apply the equation

N

λ,µ

= −N

^µ,λ

and we have already calculated the right hand side. Otherwise λ < µ, but (λ, µ) is not primitive. Let λ + µ = λ

•

+ µ

•

be a primitive expression for the sum. Since

λ + µ − λ

^•

− µ

^•

= 0 we have

N

λ,µ

N

−λ•,−µ•

kλ + µk

²

= − N

µ,−λ•

N

λ,−µ•

kµ − λ

^•

k

²

− N

−λ•,λ

N

µ,−µ•

kλ − λ

^•

k

²

. According to Proposition 2.1(c) we can set

♣

[structure-equations]

N

−λ•,−µ•

= − (p

λ,µ

+ 1)

²

N

λ_•,µ_•

.

Therefore it remains to evaluate the terms on the right hand side of this equation:

(a) We want to calculate N

µ,−λ•

. Let ν = µ − λ

^•

. If this is not a root, then N

µ,−λ•

= 0. Otherwise, since λ

•

lies in ∆, this must be a positive root. Since µ − λ

^•

− ν = 0, by Proposition 2.1(b)

♣

[structure-equations]

N

µ,−λ•

kνk

²

= N

−λ•,−ν

kµk

²

N

µ,−λ•

=

 kνk

²

kµk

²



N

−λ_•,−ν

Here N

−λ•,−ν

may be evaluated in terms of N

λ•,ν

= N

λ•,µ−λ•

. But µ = λ

•

+ (µ − λ

^•

) has height less than λ + µ and we have therefore already evaluated N

λ•,µ−λ•

.

(b) We want to calculate N

λ,−µ•

. Since λ − µ

^•

= −(µ − λ

^•

), the previous calculation shows that we need only consider the case in which ν = λ − µ

^•

is a negative root. Since λ − µ

^•

− ν = 0

N

λ,−µ•

kνk

²

= N

−ν,λ

kµk

²

N

λ,−µ•

=

 kνk

²

kµk

²

 N

−ν,λ

=

 kνk

²

kµk

²



N

µ•−λ,λ

which again has already been calculated.

(c) We want to calculate N

−λ•,λ

. If ν = λ − λ

^•

is not a root, this is 0. If it is, it must be positive. Since

−λ

^•

+ λ − ν = 0

N

−λ•,λ

kνk

²

= N

−ν,−λ•

kλk

²

which can be evaluated in terms of N

ν,λ•

= N

λ−λ•,λ•

. This we have already calculated.

(d) We want to calculate N

µ,−µ•

. Since ν = µ − µ

^•

= −(λ − λ

^•

), we may assume ν to be a negative root.

Since µ − µ

^•

− ν = 0 we have

N

µ,−µ•

kνk

²

= N

−ν,µ

kµ

^•

k

²

= N

µ•−µ,µ

kµ

^•

k

²

and as before the right hand side has already been evaluated.

7. Computation of structure constants—programming

In this section I’ll lay out some details of the computation.

The first step is to make an ordered list of the positive roots, one compatible with height. I set each root to be a data structure consisting (a) its array of coefficients as a linear combination of basic roots:

λ = X

λ

α

α

(b) its reflection table; (c) its descents —all simple roots α such that λ − α is a root; (d) all ^admissible ascents —all simple roots β with the property that µ = λ + β is a root and β is the minimal descent for µ.

Building the ordered list requires some thought—the best way to generate a list of all roots is to build the root graph , and one can place these as generated in a dynamic ordered list. This list can be searched by comparisons to tell whether a given integral vector is a root or not. It is useful to have as base of the root graph a single fictional negative root ⊖ with s

^α

⊖ = α for all simple roots α.

When this list is done, each root corresponds to a unique ascending path through it.

Then order the pairs of positive roots (λ, µ) such that λ+µ is a root, and again consistently with height. Do this lexicographically within the pairs (λ, µ) such that

HT

(λ)+

HT

(µ) is of a fixed height. Thus (λ

1

, µ

1

) <

(λ

2

, µ

2

) if and only if one of these three conditions holds: (a)

HT

(λ

1

) +

HT

(µ

1

) <

HT

(λ

2

) +

HT

(µ

2

);

(b)

HT

(λ

1

) +

HT

(µ

1

) =

HT

(λ

2

) +

HT

(µ

2

) but λ

1

< λ

2

; (c)

HT

(λ

1

) +

HT

(µ

1

) =

HT

(λ

2

) +

HT

(µ

2

) and λ

1

= λ

2

but µ

1

< µ

2

.

Next make a list of these pairs. There are some ways to do this more efficiently than others. The obvious way to to scan all pairs (λ, µ) and scan the list of roots for λ + µ. A better way is to use the list of ascents and descents of a root. The list of descents may be used to ascertain the primitive pairs directly.

In this method, we first of all scan through all positive roots λ, and for each we list all pairs of positive (µ, ν) with λ = µ + ν. We do this by building all admissible ascending paths and simultaneously building a corresponding descending path. The trick here is that

λ = µ + ν = (µ − α) + (ν + α) .

Every pair (µ, ν) summing to λ will occur exactly once in this process. As with the roots themselves,

the pairs are stored in a dynamic list. As we build this list we can also calculate the values of p

λ,µ

—if

ν = µ − λ is a root then we have already calculated p

λ,ν

= p

λ,µ

− 1, and if it is not then p

λ,µ

= 0.

Part II. Tits’ analysis

8. The extended Weyl group Let ˙s

λ

be the image under λ

^∨

of

 0 −1

1 0

 .

The elements ˙s

α

for α in ∆ satisfy the braid relations, so every element of W lifts to a unique element of W , via a reduced expression, taking account Tits’ result that two equal reduced expressions can be obtained from each other by a sequence of braid relations.

We can calculate explicitly in the lifted group, since s

²_α

is the image of −1 in SL

²

. To make this work effectively, we have to be able to calculoate in the group X

∗

(T )/2X

∗

(T ), and in particular α

^∨

( −1) for α in ∆.

At any rate, if we write (following Chevalley)

N

λ,ν

= ε

λ.ν

(p

λ,ν

+ 1) . with ε

λ.ν

= ±1 we deduce

Ad( ˙ w

α

)e

λ

= ε

∗

e

wλ

ad( ˙ w

α

)x

λ

(t) = x

wλ

(ε

∗

t)

9. References

1. Roger Carter, Simple groups of Lie type , Wiley, 1972.

2. Claude Chevalley, ‘Sur certains groupes simples’, Tohoku Journal of Mathematics ⁴⁸ (1955), 14–66.

3. Arjeh Cohen, Scott Murray, and Don Taylor, ‘Computing in groups of Lie type’, Mathematics of Computation ⁷³ (2004), 1477–1498.

4. Willem A. de Graaf, Lie algebras: theory and algorithms , North-Holland, 2000.

5. James E. Humphreys, Introduction to Lie algebras and representation theory , Springer, 1972.

6. Nathan Jacobson, Lie algebras , Wiley, 1962.

7. Jean-Pierre Serre, Alg `ebres de Lie semi-simple complexe , Benjamin, 1966.

8. Jacques Tits, ‘Sur les constants de structure et the th´eor`eme d’existence des alg`ebres de Lie semi- simple’, Publications de l’I. H. E. S. ³¹ (1966), 21–58.

9. ———-, ‘Normalisateurs de tores I. Groupes de Coxeter ´etendus’, Journal of Algebra ⁴ (1966), 96–116.