Input: a semisimple Lie algebra L, together with a simple system A -- {c~1,... , c~l} of the root system of L.
Output: a canonical set of generators of L.
Step 1 For 1 _< i < 1 do the following Step 1.1 Choose a non-zero xi E La i.
Step 1.2 Determine an element Yi E L_ai Step 1.3 Set hi - [xi, yi].
such t h a t [[xi, yi], xi] - 2xi.
Step 2 R e t u r n {xi, yi, hi ] 1 < i < l}.
C o m m e n t s : By Root fact 12 we have that there is a Yi C L_a~ satisfying the requirement of Step 1.2. Furthermore such an element can be found by solving a single linear equation in one variable. Consequently the algorithm terminates and the o u t p u t consists of non-zero xi C L ~ , Yi C L_a~ and hi C H such t h a t [xi, yi] - hi and [hi,xi] - 2xi. We have to show that the other relations of (4.14) are satisfied as well. For this let 2i C La~, ~]i C
_
L_a~, hi E H be a canonical set of generators. Then since all root spaces are l-dimensional, there are s #i E F such that 2i - Aixi, ~]i - #iYi and consequently hi - Ai#ihi. By (5.7) we know that [hj,2i] - (ai, a j ) 2 j for 1 _< i, j _< l. But this is equivalent to
)~j#j[hj, xi] - (oli, o~j}xi.
Setting i - j we see that
)~j].tj
-- 1. But then all relations of (5.7) are au- tomatically satisfied by the xi, yi, hi. Hence the o u t p u t is a set of canonical generators.Now we have a lemma t h a t will imply that a set of canonical generators {hi, xi, yi ] 1 <_ i <_ l} generates the whole Lie algebra L.
5.11 C o n s t r u c t i n g isomorphisms 183 L e m m a h . 1 1 . 2 Let c~,fl E 9 be such that a + ~ E (~. Choose h , x , y C L
(where x e La, y e L - a , h E H ) such that [h,x] = 2x, [h,y] = - 2 y , [x, y] = h (cf. Root fact 12). Let x z C L z be a non-zero root vector. T h e n Ix, xZ] ~ O. Furthermore, let ~ - r a , . . . , ~ + qc~ be the c~-string containing 1~. T h e n [y, [x, x~]] = q(r + 1)x~ and Ix, [y, x~]] = (q + 1)rx~.
P r o o f . Let K a be the Lie algebra spanned by h, x, y. Let V be the space spanned by all L z + k a for - r < k _< q. T h e n as seen in the p r o o f of R o o t fact 17, V is an irreducible K a - m o d u l e . Choose a non-zero vo E L~+qa and set vi - (ady)ivo for i > 0. T h e n v0 is a highest weight vector of weight (~ + q(~)(h) - 2q + ~(h) = q + r (where the last equality follows from Root fact 17). Also Vq spans L~ and hence x~ is a non-zero multiple of Vq. By L e m m a 5.1.2 we see t h a t adx(vq) = q(r + 1)Vq_l, which is non-zero because q > 0. So also a d x ( x z ) ~ 0. F u r t h e r m o r e a d y ( a d x ( v q ) ) = q(r + 1)Vq and a d x ( a d y ( v q ) ) - a d x ( v q + l ) - (q + 1)rVq. Since xz is a multiple of Vq the
same holds with xz in place of Vq. [:]
Now we are in a position to construct our basis. Fix a set of canonical generators h i , x i , yi of L. Let ~ E ~ + be a positive root. By Corollary 5.5.6 together w i t h a straightforward induction on ht(~) we see t h a t ~ can be w r i t t e n as a sum of simple roots
- - O~ i 1 -~- " ' " - t - O~ i k
such t h a t ai~ + " " + a i m is a root for 1 _< m _< k. We abbreviate the repeated c o m m u t a t o r [xik, [xik_i, ["" [xi2, x i ~ ] ' " ]]] by [ x i k , . . . , xil]. T h e n by L e m m a 5.11.2, [ x i k , . . . , Xil] is non-zero and hence spans L B. Now for every positive root ~ E (I)+ we fix a sequence i l , . . . , ik such t h a t ~ - ai~ + " " + aik and c~i~ + . . . + aim is a root for 1 _< m _< k. T h e n it follows t h a t the elements
h i , . . . , hl , [xik , . . . , xil ] , [Yik , " " , Yil ]
span L. We say t h a t these elements form a canonical basis of L with respect to the simple system A and the choice of sequences i l , . . . , ik.
We introduce the following notation. If I = ( i l , . . . , ik) is a sequence of integers, t h e n we set
= xi ], -
Also we set c~i - c~il + " - + (~i~. Now suppose t h a t I - ( i l , . . . ,ik) and x e L commutes with xik, i.e., [x, xik] = 0. Write Ix, x1] = Ix, [xik,xj]]
where J - ( i l , . . . ,ik-1). T h e n by a straightforward application of the
184 T h e classification of the simple Lie algebras
5.11 Constructing isomorphisms 185 index in M equals s. Hence by the first case above xz -- #XM where # depends on C only. It follows that xz - )~xj with ~ - #/~,.
For Y1 and y j we can proceed with exactly the same arguments. E3
P r o p o s i t i o n 5 . 1 1 . 4 Let L be a semisimple Lie algebra with root system ~.
Let A - {(~1,... , (~l} be a simple system of 9 with Caftan matrix C. Let h i , . . . , hi, x i, Y1 be a canonical basis of L. Then the structure constants of L with respect to this basis are rational numbers depending on C only.
P r o o f . First [hi, hj] - 0 and for I - ( i l , . . . ,ik) we have [hj,xi] - Em_l(Olim,O~j)XI, k which is easily proved by induction on k. Similarly
[hj,y~] k
We consider the remaining commutators [xt, xj], [xt, y j] and [yI, Y J].
The first and the third are similar so we deal with the first two only. Since
a d x / - [adxik, [adxik_l,
[''" [adxi 2 ,adXil]""
]]]we see t h a t adxI is a linear combination of terms of the form adxsk " " adxs~.
Hence it is enough to prove that a d x s ( x j ) and a d x s ( y j ) are linear combi- nations of basis elements, where the coefficients are in Q and depend on C only.
Concerning the first commutator, we determine from C whether c~j § as is a root. If yes, then we set g - (J,s) and [ x s , x j ] -
Axg
where A C Q depends only on C by L e m m a 5.11.3. Otherwise [xs, xj] - O.For the second c o m m u t a t o r suppose J = ( j l , . . . , jn) and we use induc- tion on n. If n - 1, then we know from (5.7) t h a t [xs, yj~] - 5s,j~hs. Now suppose n > 1 and let r be such that jr - s and j r + l , . . . , jn ~ s (if there is no such r, then [xs, y j ] - 0). Set K - ( j l , . . . , j r - l ) , then using (5.8) and the Jacobi identity we see that
[~, y J] = [yj~,..., ~j~§ [~, [y~, y~]]]
: [yj~,. 9 9 yj~+~, [y~, [ ~ , yK]]] + [yj~,.. 9 yj~+~, [ [ ~ , y~], y~]].
The first term of this sum is dealt with by the induction hypothesis. For the second term we note that [x s, Y s ] - hs and [hs, YK] is a linear combination of YM with rational coefficients determined by the C a r t a n matrix. [3 A l g o r i t h m
IsomorphismOfSSLieAlgebras
Input:
semisimple Lie algebras L1, L2 with root systems ~1 and (I)2 respec- tively, and an isomorphism r (I)l ---+ (I)2.Output: an isomorphism r L1 --+ L2.
186 The classification of the simple Lie algebras Step 1 Compute a simple system A1 - {c~1,... , al} of ~ .
Step 2 For every positive root ~ C ~+ fix a sequence IZ - ( i l , . . . ,ik) such that c~i~ + ' - " + a i , ~ is a root for 1 <_ m <_ k and /3 - c~i.
Compute a set of canonical generators by CanonicalGenerators(L1, A1).
Let h i , . . . , hi, xi~, YIz for ~ C 9 + be the corresponding canonical basis.
Step 3 Set A2 = { r r Then A2 is a simple system of ~2.
For-~ C (I)+ set I~ - Ir By CanonicalGenerators(L2, A2)calculate a set of canonical generators of L2 Let 9 h~ .. , . , h~l x' , I ~ , Y I ~ , f ~
be the canonical basis of L2 corresponding to/k2.
Step 4 Return the linear map r L1 -+ L2 that sends hi to h~, xI~
and yI~ to yi~(z )
to x ~ I~(~)
C o m m e n t s : Since r is an isomorphism of root systems it maps a simple system A1 onto a simple system A2. Also the Cartan matrix of ~1 relative to A1 is exactly the same as the Cartan matrix of (I)2 relative to A2. Hence by Proposition 5.11.4, the structure constants of L1 with respect to the canonical basis are exactly the same as the structure constants of L2 with respect to the canonical basis. Hence r is an isomorphism.
Corollary 5.11.5 Let L1 and L2 be semisimple Lie algebras with root sys- tems ~1 and ~2 of rank 1. Let A1 and A2 be simple systems of ~21 and ~2 respectively. Suppose that the Cartan matrices of ~1 with respect to A1 and
~2 with respect to A2 are identical. Then there is a unique isomorphism ' a n d r 1 < i < l
r ~ L2 such that r - h~, r - x i _ _ ,
_ h ' ' '
where {hi,xi, yi I 1 < i <_ l} and { i,xi, yi I 1 < i < l} are canonical generators, respectively of L1 with respect to A1 and L2 with respect to A2.
P r o o f . Since the set {hi, xi, yi I 1 <_ i _ l} generates L1 there can only be one isomorphism satisfying the stated relations. The existence follows from
applying IsomorphismOfSSLieAlgebras. [3
Corollary 5.11.6 Lie algebras.
The root system is a complete invariant of semisimple Remark. For an example of the use of the algorithm IsomorphismOfSSLieAl- gebras we refer to Example 5.15.11.
5.12 Constructing the semisimple Lie algebras 187
5.12 Constructing the semisimple Lie algebras
In Sections 5.9 and 5.10 we showed that the indecomposable root systems are exactly those with a Dynkin diagram occurring in Table 5.2. In the previous section we proved that the root system of a simple Lie algebra is a complete structural invariant. The next question is whether for each indecomposable root system there exists a simple Lie algebra. For this several constructions have been proposed. First of all it is possible to argue case-by-case, i.e., to go through the list of Table 5.2 and construct a simple Lie algebra for each diagram. It is for example possible to prove that z[l+l (F) is a simple Lie algebra with a root system of type A1, and 021+l(F), sp2/(F), o21(F) are of types B1, Cl and Dl respectively. For proofs of these results we refer to [48].
A second idea is to give a uniform construction of all simple Lie algebras by generators and relations. This will be described in Section 7.11.
A third approach to the problem is to construct a Lie algebra starting from the root system. From the root system it is easy to see what the dimension of the simple Lie algebra L must be and many products of basis elements are determined by the root system. Root facts 1 and 11, for example, determine commutation relations in L. After collecting as many obvious commutation relations as possible one can try to complete this to a multiplication table of L. This is the approach that we take here.
The idea is straightforward: given a root system 9 of rank l we define a Lie algebra L that is the sum of a C a r t a n subalgebra H of dimension 1 and 1-dimensional root spaces. More precisely, let A = {c~1,... , c~l} be a simple system of ~. Let H be an/-dimensional vector space with basis { h i , . . . , hi}.
For c~ C 9 we set ha - ~-~i=1 kihi 1 if o~ = ~-~li= 1 kio~i. Furthermore, for c~ E let La be a 1-dimensional vector space spanned by xa. T h e n we set
L - H |
aEO
The multiplication table of L relative to the basis { h i , . . . , hl}U{xa I c~ E ~}
is given by
[hi, h i ] - 0 for 1 < i , j < l,
[hi,xa] - (c~,o~i)xa f o r l < i < l a n d a E ~ , [ x a , X - a ] - - ~ h a 2 for c~ E (I),
[xa, xZ] = 0 for c~, ~ E 9 such that c~ +/~ r 9 and ~ # -c~, [xa,xz] - Na,zxa+z for c~, fl E 9 such that c~ + ~ C ~.