Classification of Simple Graded Lie Algebras
of Finite Growth
Olivier Mathieu *
D, M. I., ENS, 45, rue d'Ulm, F-75005 Paris, France, and
Rutgers University, Department of Mathematics, Hill Center
New Brunswick, NJ 08903, USA
Introduction
Conventions; The ground field is (C. By LA we mean Lie algebra.
Let us start with a few definitions.
• A LA 3? endowed with a decomposition
is called a graded LA if we have [J£n,J£m] £ jSf„+m. Moreover we will always
assume that dim if„ < oo for any n < co. With our convention any graded
LA is an ordinary LA and the notion should not be confused with super LA
which are often called graded LA as well.
• A subspace V of 3? is called homogeneous if we have
V = ®„ezVn
(where Vn = V n if n). The LA if is called simple graded if any homogenous
ideal is trivial (i.e. 0 or if) and if dim if > 2.
• Say that if has finite growth if
dim if „ <P(n)
for some polynomial P.
We have recently proved the following theorem [M2].
Theorem (1990). Let 3? be a simple graded LA of finite growth. Then S£ is
isomorphic to one of the following LA:
1. A simple finite dimensional LA
2. An affine LA
* Work done under the hospitality of IAS at Princeton. I thank IAS for its support (DMS
Grant 8610730).
Proceedings of the International Congress
of Mathematicians, Kyoto, Japan, 1990
800
Olivier Mathieu
3. A LA of Cartan typé
4. W (Virasoro-Witt LA).
The previous theorem has been conjectured by V.G. Kac. Alltogether there are
14 infinite series and 13 exceptional LA. In part 1 (zoology) we will give precise
definitions of the involved LA. Before we would like to make a few remarks. The
origin of Kac conjecture comes from the following result [K].
Theorem (V. G. Kac, 1967). Let if'be a simple graded LA of finite growth. Assume
(*) 3? is generated by its "local part" if_i © ifo © if i
(**) the J^Q-module if_i is irreducible.
Then ££ is isomorphic to
1. a finite dimensional LA,
2. an affine LA or
3. a Cartan type LA.
Moreover it follows from 1967 Kac paper the existence of "continous families"
of simple graded LA. Thus there are no hopes for a classification without the
growth hypothesis. Note also that in characteric p ^ 0 the classification of finite
dimensional simple L^4 is still open. *
Part 1: Zoology
In the section we will describe some species i.e. the LA involved in the Theorem.
Altough each of them admit infinitely many different gradings we can describe
all of them. For the simplicity of the exposition we will describe theses gradings
in one case only.
(1.1) Finite Dimensional Simple LA
Recall that finite dimensional simple Lie algebras have been classified around
1900 by Killing and Cartan. Four infinite series and five exceptional Lie algebras
occur in their classification. The LA of the four infinite series are called classical
LA. They are the following one.
An or %\(n + 1)
Bn or so(2n + 1)
Cn or sp(2n)
Dn or so (2n)
It is not easy to give a simple description of the five exceptional simple LA
E6,El9Es,F4a.ndG2
1
At ICM conference G. Seligman tells us that H. Strade and R. Wilson have rencently
announced the classification of finite dimensional simple LA over field of characteristic
p>7.
Classification of Simple Graded Lie Algebras of Finite Growth
801
(1.2) Affine (Kac-Moody) LA
Let g be a finite dimensional simple LA, let œ be an automorphism of g of
finite order £ and let r\ be /-root of unity. Set L(g) = g ® <L[t9t~x], Define the
automorphism œ of L(g) by:
œ(g®f) = rjncû(g)®t11.
Let L(Q,œ,rj) be the LA of fixed points under fà. A LA isomorphic to some
L(g, œ, r\) is called affine. The definition is not accurate because there are many
non-trivial isomorphisms between various L(g,co,?j). Fortunately V.G. Kac found
a one to one parametrization of these isomorphism classes [K]. Actually he proved
that affine LA are exactely parametrized by automorphisms of Dynkin diagrams.
All together there are 6 infinite series and 7 exceptional affine LA. With the usual
notation affine algebras are
AV) »W
r w /)W
A® DP) D(3) P (1) P (2) F ( 1 ) F (1) G{1) P (1)
Affine LA are also called loop algebras because any element of L(g) can be
identified with a g-valued map on Sl whose Fourrier decomposition is finite.
(1.3) Cartan Type LA
Let Wn be the LA of derivations of the ring of polynomials (C[Zi, •• •, Xn]. Thus
an element d of W„ is a vector field with polynomial coefficients, Note Lie(S) be
the Lie derative action on spaces of differential forms.
Set S„ = {d G W„|L/e(9) • v = 0} where v is the usual volume form
dXiA'"/\dXn.
For n = 2m let œ = ^T
dXi A dXm+i be the usual symplectic form and
set H„ = {de Vfn\L1e(d) • œ = 0}.
For n — 2m + 1 let a = dXn + V
XidXm+i — Xm+idXi be the usual
contact 1-form and set K„ = {d e W„|a A Lie(d) • a = 0} .
The LA W„,S,„H„,Kn are called Cartan type LA. These four infinite serie
have been discovered by Cartan around 1910 [C].
(1.4) The Virasoro-Witt LA
Let W be the LA of derivations of (C[T, T" 1 ].
Remarks: 1, Each of the previous LA admits infinitely many gradings but only
finitely many of them satisfy the hypotheses (*) (**) of Kac theorem.
Example for ^ — Ww. Let a = a\9 ' ' ' > an be a sequence of non zero integers of
same sign. There is a unique grading of !£ such that the element Z" n • • • X™nd/dXj
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Olivier Mathieu
is homogeneous of degree a\m\ -\
\- anmn — aj. For any G e Sn (where Sn is the
symétrie group) the gradings associated with a\9 • • •, an and with aa{\), • • •, aa{n) are
obviously isomorphic. It is easy to prove that the induced map from (Z^uZ!L)/S n
to the set of isomorphism classes of gradings of S£ is one to one and onto . But
the grading associated with ( 1 , . . . , 1) is the only one satisfying Kac's hypotheses
(*) (**) (n > 2).
Example: No grading of W,Wi satisfies (*).
2. The terminology "affine LA", "Virasoro LA" is often used for central
extension of LA considered here.
3. Affine LA are "simple graded" but not simple (because of evaluation maps
L(g) —> g). Conversely a simple graded LA which is not simple is affine (it is a
statement).
Table I. Simple graded LA of finite growth
dim if < oo
An nn Cn Dn
#6 Ei E^ F4 G2
A® 5 (1) CW Z)W
Affine
Cartan type
Virasoro-Witt
^n
"n
^n
*^n
Df E « Ef £<»
E « F® Gf
W„S„H„K„
W
Part 2: About the Proofs
In this section we will describe the general plan of the proof. The proof divides
into 3 Steps.
Step 1 : Define 4 abstract classes of graded LA.
Step 2: Lemma: Any simple graded LA belongs to one of the 4 previous classes.
Step 3: Show 4 classification theorems (i.e. one for each class).
Step 1. 4 Definitions: To meet step 1 we will define four abstract classes of graded
Lie algebras.
Let h be a finite dimensional nilpotent LA and let M be a finite dimensional
/z-module. Recall that M decomposes as
M = ®MX
where X runs over h* and Ml is the generalized eigenspace associated with X
(Engel Theorem). Let g be a finite dimensional LA. A Cartan subalgebra (or
CSA) is a nilpotent subalgebra equal to its normalizer. It is classical that CSA
Classification of Simple Graded Lie Algebras of Finite Growth
803
do exist. Moreover any two CSA are conjugated under a product of elementary
automorphisms. Let if be a graded LA. Pick a CSA h of S£0 and consider each
S^n as an /i-module. Thus we have:
cp __ (T\ cpk
m%
Set:A = {(n,X)/&*^0}.
Let Q be the subgroup of TL x h* generated by A .
Definition 1. S£ is called without roots iff A ^ TL x 0
Definition 2. if is called weakly integrable iff
1) z l ^ Z x O
2) p|ad s (if;j).if = 0
for any (n9X) e A,X ^ 0,
Set: JS?+ = ©s>o^s,if~ = ®s<o^sDefinition 3. Say that ^ is of type V iff dim ^
is infinite.
or dim S£~ is finite but dim ^
A subset X of Q is called quasi-order iff
Va G ß3JV £ OVm ^ NVJSi,.. ft„ e X we have 8 + f t + • • • + ßm e X.
Let a e Q. The LA if is called ä-deep if we have \y?x>&\ = ^ for any
quasi-order X such that X U {a} is still a quasi-order (by definition we set
&x = 0 ^).
(n,A)eX
Definition 4. if is called deep if if is a-deep for some a = (n9X) € A with
7Î^0,A^0.
Thus Definitions 1, 2, 3, 4 define 4 abstract classes of graded LA. Moreover
any two CSA of £t?o are conjugated under a degree 0 automorphism of ^. Hence
the definitions do not depend on a choice for h.
Step 2
Lemma 1. Any simple graded LA S£ satisfies exactely one of the following assertions.
1) if is without roots.
2) ^£ is weakly integrable.
3) ^ is of type #.
4) 3? is deep.
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Olivier Mathieu
Step 3. 4 Classification Theorems
The previous lemma splits the category of simple graded LA into four subcategories. Each of the following four theorem is a classification theorem for each
of the four classes. Thus The main theorem is an obvious corollary of these four
theorems.
Theorem 1. Let S£ be a simple graded LA without roots. Then S£ has infinite
growth.
Theorem 2. Let S£ be a weakly integrable simple graded LA. Then dim S£ < oo or
££ is affine.
Theorem 3. Let 3? be a simple graded LA of type C. The ££ is of Cartan type.
Theorem 4. Let S£ be a deep simple graded LA of finite growth. Then S£ is
isomorphic to W.
Theorem 1 has the following consequence. Any simple graded LA S£ with
S£o = 0 has infinite growth. The growth hypothesis in theorem 1 is crucial
because there are simple graded LA £^ with if 0 = 0 . However there are no
growth hypotheses for Theorems 2, 3.
Thus Theorems 1, 2, 3, 4 and the Lemma implies the classification of simple
graded LA of finite growth.
Some References:
Say that I£ has growth < 1 if we have dim S£n<C for some constant C. In a
previous paper [Ml] we classify simple graded LA of growth < 1.
1) The proof of Theorem 1 follows the same line as Theorem 1 in [Ml].
2) 90 % of Theorem 2 was already proved in [Ml].
3) The proof of Theorem 3 essentialy uses homological Kostant formula, Kac
theorem and a calculation of characteric variety (following a nice trick of V.
Guillemin).
4) The proof of Theorem 4 is the main difficulty. At some point we use GabberKac theorem. Otherwise it is elernentary.
Part 3: More About the Proofs
The proof of the theorem is quite long. The main "tools" are the following ones:
1) Basic Tool: We get informations from any "formal construction" of ideals.
Obvious examples are centers, derived algebras... We can also use the notion of
"quasi-order" for that purpose. Another typical example is the following. Let J^
be any graded LA.
Classification of Simple Graded Lie Algebras of Finite Growth
805
Lemma 2. Assume that S£ = sé + $ where \sé, j / J g r f and \sé, SS\ £ $. Then
a + [a90\ is an ideal.
The lemma is obvious but it is used many time to construct subalgebras which
behaves like si (2) or like an Heisenberg algebra.
2) Another Tool: Partial LA. Let a < 0, b > 0 be integers, A partial LA is a
graded vector space
r=©r (
endowed with partial brackets P/ X Py -» P?+^ (for a < /, j , i + j < b) satisfying
partial Jacobi identities , For a graded LA X, its partial part
Part &= ®a^i<b &\
is a partial LA. Conversely any partial LA P is the partial part of some graded
LA S£. Among such if's there is a minimal model ifminCOLemma 3. Let if be a graded LA and P be a partial LA, If P is a subquotient
of Part ^ then ifminCH is a subquotient of J^,
Especially if ifmin(P) does have infinite growth, J27 does. It allows us to
restrict the possible partial part of graded LA of finite growth because we prove
that for particular 6 series of partial LA P their models ifmin(f ) have infinite
growth.
3) Another Tool: Ranks. The rank of a graded LA ^£ is the dimension over Q of
Q ®z Q. In the proof of Theorem 4 we study two cases :
1) rank Se = 1
2) rank if > 2
Actually the rank > 2 case is by far easier.
4) Last Tool: Coadjoint Estimates. Let S£^ = ®£f?*n be the graded dual of the
graded LA 3^, For a homogenous Ç G J£* the space ^ • ( £ S£* is homogenous.
For a simple graded LA S£ it is easy to show that the growth of if • Ç is
indépendant of Ç =£ 0, The following lemma is very crucial in proving Theorem 4.
Lemma 4. Assume ^ deep, simple graded and of finite growth. Then S£ • Ç has
growth exactly 1.
Olivier Mathieu
806
Table li. Simple graded Lie algebras
Type
Without root
Weakly integrable
Type#
Deep
Finite growth
0
Infinite growth
A lot [Continous families]
Finite dim. or Affine
0
Cartan type
0
Virasoro-Witt
A lot [Continous families]
References
[C]
E. Cartan: Les groupes de transformations continus infinis simples. Ann. Sci. Ecole
Norm. Sup. 26 (1909) 93-161
[K] V. G. Kac: Simple graded Lie algebras of finite growth. Math. USSR Izv. 2 (1968)
1271-1311
[Ml] O. Mathieu: Classification des algebres de Lie graduées simples de croissance < 1.
Invent, math. 86 (1986) 371-426
[M2] O. Mathieu: Classification of simple graded Lie algebras of finite growth. Preprint
1990