Simple Lie algebras of small characteristic V. The non-Melikian case

0.36 MB·English

by  Alexander Premet

SIMPLE LIE ALGEBRAS OF SMALL CHARACTERISTIC V.
THE NON-MELIKIAN CASE
6
ALEXANDER PREMET AND HELMUT STRADE
0
0
2 Abstract. Let L be a finite-dimensional simple Lie algebra over an alge-

n braically closed field F of characteristic p > 3. We prove in this paper that
a if for every torus T of maximal dimension in the p-envelope of adL in DerL
J
the centralizer of T in adL acts triangulably on L, then L is either classical

3
or of Cartan type. As a consequence we obtain that any finite-dimensional
1
simple Lie algebra over an algebraically closed field of characteristic p > 5 is

] either classical or of Cartan type. This settles the last remaining case of the
A generalized Kostrikin-Shafarevichconjecture (the case where p=7).
R
.
h
t
a 1. Introduction and preliminaries
m
[ Let F be an algebraically closed field of characteristic p > 3. In this note

we go through the relevant parts of the second author’s classification of finite-
1
v dimensional simple Lie algebras of characteristic p > 7 to check whether the
3
results there need additions, modifications or supplementary proofs in order to
2
3 apply in the present case where p > 3. It turned out that only few changes are
1 necessary.
0
In what follows L will always denote a finite-dimensional simple Lie algebra
6
0 over F. We identify L with the subalgebra adL of the derivation algebra DerL
/
h and let L be the p-envelope of L in the restricted Lie algebra DerL. We denote
p
t by T a torus of maximal dimension in L and set
a p
m
H := c (T) = {x ∈ L | [t,x] = 0 for all t ∈ T}, H := c (T) = H ∩L.
: Lp p L
v e e
i A torus T is called standard if H(1) consists of nilpotent derivations of L. We
X
denote by Γ(L,T) the set of roots of L relative to T (roots are nonzero linear
r
a functionsγ ∈ T∗ suchthatL := {x ∈ L| [t,x] = γ(t)x for all t ∈ T}isnonzero).
γ
We have root space decompositions
L = H ⊕ M Lγ, Lp = H ⊕ M Lγ.
e
γ∈Γ(L,T) γ∈Γ(L,T)
By [P-St 04, Corollary 3.7], only four types of roots can occur in simple Lie
algebras of characteristic p > 3: solvable, classical, Witt, and Hamiltonian roots.
Inotherwords, foranyγ ∈ Γ(L,T)thesemisimple quotientL[γ] = L(γ)/radL(γ)
of the 1-section L(γ) := H⊕ L is either (0) or sl(2) or the Witt algebra or
Li∈Fp iγ
Mathematics Subject Classification (2000 Revision). Primary 17B20, 17B50.
1
contains an isomorphic copy of the Hamiltonian algebra H(2;1)(2) as an ideal of
codimension ≤ 1. The main result of [P-St 04] states that if L contains a torus
p
T′ of maximal dimension such that all roots in Γ(L,T′) are solvable or classical,
then L is either a classical Lie algebra or a filtered Lie algebra of type S or H;
see [P-St 04, Theorems C and D].
In this note we impose the following two assumptions on L:
• all tori of maximal dimension in L are standard;
p
• the set of roots of any torus of maximal dimension in L contains either
p
a Witt root ar a Hamiltonian root.
Theorem 1.1. If a finite dimensional simple Lie algebra L over F satisfies the
above assumptions, then L is isomorphic to a Lie algebra of Cartan type.
Combining Theorem 1.1 with [P-St 04, Theorems C and D] we derive:
Theorem 1.2. Let L be a finite dimensional simple Lie algebra over an alge-
braically closed field of characteristic p > 3 such that all tori of maximal dimen-
sion in L are standard. Then L is either classical or of Cartan type.
p
Due to [Wil 77, St 89a, P 94] the assumption on tori in Theorem 1.2 is fulfilled
automatically when p > 5. In this case Theorem 1.2 can be rephrased as follows:
Theorem 1.3. Any finite dimensional simple Lie algebra over an algebraically
closed field of characteristic p > 5 is either classical or of Cartan type.
Theorem 1.3 settles the last remaining case p = 7 of the Kostrikin–Shafarevich
conjecture on the structure of finite-dimensional restricted simple Lie algebras
over algebraically closed fields of characteristic p > 5; see [Ko-S 66]. In the
early 80s, G.M. Melikian discovered a restricted simple Lie algebra of charac-
teristic 5 which was neither classical nor of Cartan type, thereby showing that
the restriction on p in the Kostrikin–Shafarevich conjecture was necessary. In
1984, R.E. Block and R.L. Wilson succeeded to prove the Kostrikin–Shafarevich
conjecture for algebraically closed fields of characteristic p > 7; see [B-W 88].
As far as the general classification problem for p > 3 is concerned, Theorem 1.2
leaves open the case where p = 5 and L contains nonstandard tori of maximal
p
dimension. This isolated case will be treated in [P-St 06], the last paper of the
series.
2. Two-sections of L
Inthenexttwosectionsourstandinghypothesis isthatLisafinitedimensional
simple Lie algebra such that all tori of maximal dimension in L are standard.
p
The second assumption of Sect. 1 will come into force in Sect. 4. We retain the
notation introduced in [P-St 97], [P-St 99], [P-St 01], [P-St 04] with the following
two exceptions: to match the notation of [St 04] we will denote the divided power
algebra A(m;n) by O(m;n) and the Melikian algebra g(m,n) by M(m,n).
2
Our first result extends [St 89b, Theorems 3.1, 1.7, 1.8] and [B-O-St, Corollary
1.9] which hold for p > 7.
Theorem 2.1. Let T be any torus of maximal dimension in L .
p
(i) The subalgebra H = c (T) acts triangulably on L.
Lp
e
(ii) For every γ ∈ Γ(L,T) the radical radL(γ) is T-invariant and the factor
algebra L[γ] = L(γ)/radL(γ) is either zero or isomorphic to one of sl(2),
W(1;1), H(2;1)(2), H(2;1)(1).
Proof. Since all tori of maximal dimension in L are assumed to be standard, the
p
firststatementisnothingbut[P-St 04,Theorem3.12],whilethesecondstatement
(cid:3)
is immediate from [P-St 04, Corollary 3.7].
Given a filtered Cartan type Lie algebra g (not necessarily simple) we de-
note by g the standard maximal subalgebra of g. When g = W(1;1), we have
(0)
dim(g/g ) = 1,whilewheng ∼= H(2;1)(ǫ) withǫ = 1,2,wehavedim(g/g ) = 2.
(0) (0)
Theorem 2.1 shows that every 1-section L(γ) with γ ∈ Γ(L,T) contains a distin-
guished subalgebra Q(γ) such that Q(γ)/radQ(γ) is either zero or isomorphic to
sl(2). More precisely, the following holds:
(a) Q(γ) = L(γ) if L(γ) is solvable or L[γ] ∼= sl(2);
(b) (Q(γ)+radL(γ))/radL(γ) = L[γ] if L[γ] is of Cartan type.
(0)
This is analogous to [St 89b, Proposition 1.9] and [B-O-St, Proposition 1.11].
Recall that a root γ ∈ Γ(L,T) is called proper, if Q(γ) is T-invariant, and
improper otherwise; see [P-St 04]. This definition differs from that introduced
by Block–Wilson. However, it agrees with the Block–Wilson definition when
p > 7 and reflects better the desired properties of γ when p ∈ {5,7} (the formal
extension of the Block–Wilson definition to the case where p = 5 would imply
that all Hamiltonian roots are improper, which is not what we want).
The following result is very important for the classification:
Theorem 2.2. Let L(α,β) be any 2-section of L relative to a torus T of maximal
dimension in L , and I(α,β) the maximal solvable ideal of T + L(α,β). Let ψ
p
denote the canonical homomorphism T + L(α,β) → T + L(α,β)/I(αβ) , and
(cid:0) (cid:1)
put K := ψ(L(α,β)) and T := ψ(T). Then one of the following holds:
(1) L(α,β) is solvable and K = (0);
(2) S = T + K where S is one of sl(2), W(1;1), H(2;1)(2) or else S =
H(2;1)(2) and S ⊂ T + K ⊂ H(2;1)(1). Moreover, there exists a root
µ ∈ F α+F β such that K = ψ(L(µ));
p p
(3) S ⊕ S ⊂ T + K ⊂ (DerS )(1) ⊕ (DerS )(1) where S is one of sl(2),
1 2 1 2 i
W(1;1), H(2;1)(2) for i = 1,2;
(4) K = FD ⊕H(2;1)(2) where either D ∈ {0,D (xp−1xp−1)} or p = 5 and
H 1 2
D = xp−1∂ , and there is an automorphism σ of DerH(2;1)(2) such that
1 2
σ(T) = Fz ∂ ⊕Fz ∂ where z ∈ {x ,1+x }, i = 1,2;
1 1 2 2 i i i
3
(5) S ⊗ O(m;1) ⊂ T + K ⊂ Der(S ⊗ O(m;1)) where S is one of sl(2),
W(1;1), H(2;1)(2) and m = 1,2. Let π be the canonical projection from
2
Der(S⊗O(m;1)) ∼= (DerS)⊗O(m;1) ⋊Id ⊗W(m;1) onto W(m;1).
(cid:0) (cid:1) S
Then π (K) ∼= W(1;1) if m = 1 and π (K) ∼= H(2;1)(ǫ) if m = 2, where
2 2
ǫ = 1,2;
(6) S⊗O(1;1) ⊂ K ⊂ S⊗O(1;1), where S is one of sl(2), W(1;1), H(2;1)(2)
b
and either S = S or S = H(2;1)(2) and S = H(2;1)(1). Moreover,
b b
T = F(h ⊗1)⊕F(Id ⊗(1+x )∂ ) for some toral element h ∈ S;
0 S 1 1 0
b
(7) S ⊂ T + K ⊂ S where S is one of the nonrestricted Cartan type Lie
p
algebras W(1;2), H(2;1,Φ(τ))(1), H(2;1;∆) or else
H(2;(2,1))(2) ⊂ T +K ⊂ H(2;(2,1)) ;
p
(8) K is either a classical Lie algebra of type A , B or G or one of the
2 2 2
restricted Cartan type Lie algebras W(2;1), S(3;1)(1), H(4;1)(1), K(3;1).
Proof. 1) If L(α,β) is solvable, then we are in case (1). So assume from now that
L(α,β) is nonsolvable.
Recall from [P-St 04] that rad L(α,β) denotes the maximal T-invariant solv-
T
able ideal of L(α,β), and L[α,β] = L(α,β)/rad L(α,β). Since rad L(α,β) =
T T
I(α,β)∩L(α,β), we have that
∼
L[α,β] = ψ(L(α,β)) = K ֒→ T +K.
As in [P-St 04] we denote by S = ⊕r S the sum of all minimal T-invariant
i=1 i
e e
ideals of K = L[α,β] 6= (0). Since the Lie algebra T +K is semisimple, it acts
faithfully on S. We will identify T +K with a Lie subalgebra of DerS. As shown
e e
in [P-St 04, Sect. 4], we have that r ∈ {1,2}. Moreover, if r = 2, then we are in
case (3); see [P-St 04, Theorem 4.1].
2) From now on assume that S = S is the unique minimal T-invariant ideal of
1
e e
K = L[α,β]. Recall from [P-St 04] that TR(S) ≤ TR(L[α,β]) ≤ 2.
e
Suppose TR(S) = 2. If the Lie algebra S is restrictable, then [P-St 04,
e e
Theorem 4.2] shows that S = L[α,β]. Therefore, if S ia a classical Lie alge-
e e
bra or a restricted Lie algebra of Cartan type, then we are in case (8). As
explained in [P-St 04, Sect. 4] there is a natural restricted homomorphism
Ψ : T +L(α,β) → DerS which maps T +L(α,β) ⊂ T +L(α,β) onto T +K.
α,β p p
∼ e
Suppose S = M(1,1). Then S = DerS. Choose a two-dimensional non-
standard toreus T′ in S. There exeists a toreus T′ in the restricted Lie algebra
e ′
T +L(α,β) such that kerα∩kerβ ⊂ T′ and Ψ (T′) = T . By construction, T′
p α,β
is then a nonstandard torus of maximal dimension in L . Since no such tori can
p
∼
exist by our general assumption, we derive that S 6= M(1,1).
4 e
If S is a nonrestricted Cartan type Lie algebra, then [P-St 01, Theorem 1.1]
e
yields that S is one of W(1,2), H(2;1;Φ(τ))(1), H(2;1;∆), H(2;(2,1))(2). Ap-
e
plying [P-St 04, Theorem 4.2] shows that we are in case (7).
3) It remains to consider the situation where r = 1 and TR(S) = 1, which is
e
ruled by [P-St 04, Theorem 4.4]. Due of Theorem 2.1, case (1) of that theorem
is our case (2). Assuming case (2) of Theorem 4.4 in [P-St 04], part (b) of the
proof in loc. cit. shows that there exists an automorphism σ of the Lie algebra
DerH(2;1)(2) such that σ(K) = H(2;1)(2) ⊕FD and σ(T) = Fz ∂ ⊕ Fz ∂ ,
1 1 2 2
where D is as in case (4) of the present theorem and z ∈ {x ,1+x }, i = 1,2.
i i i
Now assume we are in case (3) of [P-St 04, Theorem 4.4]. Then S = S⊗O(1;1)
where S is one of sl(2), W(1;1), H(2;1)(2), and Ψ (T) is spannedeby h ⊗1 and
α,β 0
Id⊗(1+x )∂ for some nonzero toral element h ∈ S. Moreover, K ⊂ (DerS)⊗
1 1 0
O(1;1). To show that this is our case (6) we can assume that S = H(2;1)(2) (in
the other two cases DerS = adS and there is nothing to prove).
If K 6⊂ H(2;1)(2)⊗O(1;1), then there is a root µ ∈ Γ(L,T)∩(F α+F β) such
p p
that K(µ) 6⊂ H(2;1)(2) ⊗O(1;1). Restricting the composite
ψ¯: L(α,β) −ψ→ K ֒→ (DerS)⊗O(1;1) ։ DerS
to the 1-section L(µ) and using the above description of Ψ (T) it is easy to
α,β
∼ ¯
observe that K(µ) = ψ(L(µ)) is sandwiched between S and DerS. Consequently,
∼ ∼
µ is a Hamiltonian root and K(µ) = L[µ]. Theorem 2.1 now yields K(µ) =
H(2;1)(ǫ) where ǫ ∈ {1,2}. If ǫ = 2, then K(µ) = K(µ)(∞) ⊂ S contrary to our
choice of µ. Hence ǫ = 1. Since K(µ) 6⊂ H(2;1)(2) ⊗ O(1;1),eit must be that
ψ¯(L(µ)) = H(2;1)(1) by Theorem 2.1. This proves that K ⊂ H(2;1)(1)⊗O(1;1),
as claimed.
In case (4) of [P-St 04, Theorem 4.4] there exist µ,ν ∈ Γ(L,T) such that
∼
L[µ,ν] = M(1,1). But then, as before, L contains a nonstandard torus of max-
p
imal dimension, violating our general assumption. Case (5) of [P-St 04, Theorem
4.4] is included into case (5) of the present theorem.
Finally, consider case (6) of [P-St 04, Theorem 4.4]. Let h be a nonzero toral
0
element of S = H(2;1)(2) and choose a torus T′ in T + L(α,β) with dimT′ =
p
dimT and h ⊗ 1 ∈ Ψ (T′), Note that kerα ∩ kerβ ⊂ T′ and T′ is standard
0 α,β
by our general assumption. Choose µ ∈ Γ(K,Ψ (T′)) with µ(h ⊗1) = 0 and
α,β 0
regard it as a linear function on T′ vanishing on kerα∩kerβ. Then
c (h )⊗O(2;1) ⊂ K(µ) ⊂ c (h )⊗O(2;1) ⋊Id⊗π (K).
S 0 (cid:0) DerS 0 (cid:1) 2
Let ̺ be the canonical projection c (h ) ⊗O(2;1) ⋊Id⊗π (K) ։ π (K).
(cid:0) DerS 0 (cid:1) 2 2
Note that ̺ is T′-equivariant and maps the 1-section K(µ) onto π (K). By
2
[P-St 04, Theorem 4.4(6)], π (K) is sandwiched between H(2;1)(2) and H(2;1).
2
This implies that ̺(K(µ)) is semisimple. As ̺(K(µ)) is a homomorphic image
∼
of the 1-section L(µ), it must be that ̺(K(µ)) = L[µ]. Theorem 2.1 now yields
π (K) = ̺(K(µ)) ∼= H(2;1)(ǫ) where ǫ ∈ {1,2}, completing the proof. (cid:3)
2
5
We have now established (for p > 3) a refined version of [St 92, Theorem
II.2] which is the corrected version of [St 89b, Theorem 6.3] (or, equivalently, of
[B-O-St, Theorem 1.15]). In what follows we will need a description of T in the
respective cases.
Proposition 2.3. With the assumptions and notations of Theorem 2.2, T has
the following properties in the respective cases:
(1) T = (0);
(2) T ⊂ S and dimT = 1;
(3) T = Fh ⊕Fh where h ∈ S , i = 1,2;
1 2 i i
(4) T is conjugate to Fz ∂ ⊕Fz ∂ where z ∈ {x ,1+x }, i = 1,2;
1 1 2 2 i 1 i
(5) T = F(h⊗1)⊕F(d⊗1+Id⊗t) where h ∈ S and t ∈ π (K) are nonzero
2
toral elements, d ∈ DerS is toral, and [d,h] = 0;
(6) T = F(h⊗1)⊕FId⊗(1+x )∂ where h ∈ S \(0); and h[p] = h;
1 1
(7) T ⊂ S and, moreover, T ∩ S = (0) when S ∼= H(2;1;Φ(τ))(1) and
p
dimT ∩S = 1 otherwise;
(8) T ⊂ K.
Proof. (a) If K = (0), then T +L(α,β) is solvable, hence coincides with I(α,β).
Therefore, T = (0).
(b) From now on suppose K 6= (0). Then (0) 6= T +K = (T +L(α,β))/I(α,β)
is semisimple, hence acts faithfully on its socle S. Note that
e
S ⊂ (T +K)(1) ⊂ K ⊂ DerS.
e e
We regard T + K as a Lie subalgebra of DerS. The restricted homomorphism
e e
Ψ : T + L(α,β) → DerS introduced in [P-St 04, Sect. 4] then maps T onto
α,β p
e
T. It also maps L(α,β) ⊂ L onto the p-envelope of K in DerS. The latter
p p
e
p-envelope will be denoted by K . It contains the p-envelope S of S in DerS.
p p
e e e
Since S ⊂ T + S ⊂ DerS, the restricted Lie algebra T + S , is centerless.
p p
e e e e
In conjunction with [St 04, Theorem 1.2.6(3)] this shows that if T is a torus of
maximal dimension in T+K , then T ∩S is a torus of maximal dimension in S .
p p p
e e
In view of [St 04, Theorem 1.2.8(3) and Theorem 1.3.11(3)] we also have that
1 ≤ TR(T +K ) ≤ TR(T +L(α,β) ) ≤ TR(L (α,β)) ≤ 2.
p p p
(c) Let T + K be as in case (2) of Theorem 2.2. Then T + K = S where
S = S = S isoneofsl(2),W(1;1), H(2;1)(1) orelseT+K issandwiched between
p
S = He(2;e1)(2) and H(2;1)(1). In any event, T +K is semisimple, restricted, and
has absolute toral rank 1. It follows that dimT = 1 and T is a torus of maximal
dimension in T +K = T +K . But then S ∩T is a torus of maximal dimension
p
in S = S , by our concluding remark in part (b). Hence T ⊂ S, by dimension
p
e
reasons.
6
(d) In case (4) of Theorem 2.2 we have dimT = 2. In cases (3) and (5)–(8),
we have that either TR(S) = 2 or the Lie algebra S is not simple. From this it
e e
follows that in the remaining cases of Theorem 2.2 the torus T = Ψ (T) acts
α,β
on S as a two-dimensional torus of derivations. Indeed, otherwise there would
e
exist a root γ ∈ Γ(L,T) such that T +K = T +K(γ) and this would imply that
S = K(γ)(∞) is simple; see Theorem 2.1. By our concluding remark in part (b),
e
T ∩S is a torus of maximal dimension in S .
p
e e
(e)Suppose weareincase(3)ofTheorem 2.2. ThenS = S ⊕S , TR(S) = 2, and
1 2
e
S = S . Since dim(T ∩S ) = TR(S) due to our concluding remark in part (b),
p p
e e e e
it must be that T ⊂ S ⊕S .
1 2
(f) If T + K is as in case (4), then T already has the required form in view of
Theorem 2.2.
(g) Suppose we are in case (5) of Theorem 2.2. Because S ⊗ O(m;1) is a
(1)
[p]-nilpotent ideal of S = S ⊗O(m;1), it follows that
e
S = S ⊗O(m;1)+S ⊗F = S
p p
e e
˜
and TR(S) = TR(S) = 1. The discussion in part (b) now yields T ∩S = Fh for
p
e ˜ e
some nonzero toral element h ∈ S. According to [P-St 99, Theorem 2.6], there
e
is an automorphism ϕ of DerS such that ϕ(T) ⊂ (DerS)⊗F ⋊Id⊗W(m;1).
e
Since ϕ preserves the socle of DerS, it must be that
e
ϕ(h˜) ∈ S ⊗O(m;1) ∩ (DerS)⊗F = S ⊗F.
(cid:0) (cid:1) (cid:0) (cid:1)
˜
In other words, ϕ(h) = h⊗1 for some nonzero toral element h ∈ S.
Now let t˜ be any (nonzero) toral element such that T = Fh˜ ⊕ Ft˜. Write
ϕ(t˜) = d⊗1+Id⊗t with d ∈ DerS and t ∈ W(m;1). It is straightforward to
see that d and t are both toral, and [d,h] = 0.
It remains to show that t is a nonzero element of π (K). If t = 0, then T
2
lies in (DerS) ⊗ O(m;1), that is π (T) = (0). Since T is a torus of maximal
2
dimension in T +K , we apply [St 04, Theorem 1.2.8(4)] to get TR(π (K)) = 0.
p 2
But Theorem 2.2 says that in the present case π (K) is a semisimple restricted
2
Lie algebra of absolute toral rank one. Hence Ft is a torus of maximal dimension
in π (T +K ) = Ft+π (K). As π (K) is a restricted ideal of π (T +K ), we
2 p 2 2 2 p
conclude that t ∈ π (K); see [St 04, Theorem 1.2.6(3)].
2
(h) If T + K is as in case (6), then T already has the required form in view of
Theorem 2.2.
(i) Suppose T + K is as in case (7). If S is one of W(1;2), H(2;1;Φ(τ))(1),
H(2;1,∆), then Theorem 2.2 says that T ⊂ S . Suppose S = W(1;2). As
p
W(1;2) has codimension 1 in W(1;2) , it must be that T ∩W(1;2) 6= (0). Since
p
all weight spaces of the T-module W(1;2) are one-dimensional, by [St 92, Theo-
rem V.2(2)], we have that dim(T ∩S) = 1 in this case. If S = H(2;1;Φ(τ))(1),
7
then [St 92, Theorem VII.3(3)] yields T∩S = (0). Suppose S = H(2;1;∆). Then
H(2;1;∆) = F(x ∂ + x ∂ ) ⊕ H(2;1;∆). Hence S has codimension 1 in S ,
p 1 1 2 2 p
implying T∩S 6= (0). Suppose T ⊂ S. As explained in [P-St 01, Lemma 4.14(1)],
for example, T has at least p2−2 weights on S. Since dimS = p2, we then have
S = T ⊕ M Sγ, |Γ(S,T)| = p2 −2, dimSγ = 1 (∀γ ∈ Γ(S,T)).
γ∈Γ(S,T)
[p]
But then the subset T ∪ S spans a diagonalizable Lie subalgebra
(cid:0)Sγ∈Γ(S,T) γ (cid:1)
of S containing T. Since T is a torus of maximal dimension in S , we get
p p
[p]
S ⊂ T. However, this would imply that S is a restrictable Lie algebra,
Sγ∈Γ(S,T) γ
which is false. We conclude that dim(T ∩S) = 1 in the present case.
Now consider the case where S = H(2;(2,1))(2). It is well-known (and eas-
ily seen) that S = F∂p ⊕ S and H(2;(2,1)) = F∂p ⊕ H(2;(2,1)). But then
p 1 p 1
it is clear that the restricted Lie algebra H(2;(2,1)) /S is [p]-nilpotent. As
p p
T ⊂ H(2;(2,1)) by Theorem 2.2, we now derive that T ⊂ S . Since S has
p p
codimension 1 in S , we have T ∩S 6= (0), while part (f) of [B-W 88, (10.1.1)]
p
yields T 6⊂ S. Therefore, dim(T ∩S) = 1.
(j) Finally, suppose T +K is as in case (8) of Theorem 2.2. Then K = S = S
p
e e
is a restricted simple Lie algebra of absolute toral rank 2. Hence dim(T ∩S) = 2
by our discussion at the end of part (b). In other words, T ⊂ K. e (cid:3)
3. Optimal tori
Given a torus T of maximal dimension in L we denote by Γ (L,T) the subset
p p
of all proper roots in Γ(L,T). Note that if γ ∈ Γ(L,T) is proper, then Γ(L,T)∩
F∗γ ⊂ Γ (L,T). We say that T is an optimal torus if the number
p p
r(T) := |Γ(L,T)\Γ (L,T)|
p
is minimal possible; see [P-St 01, p. 242].
From now on we fix an F -linear map ξ: F → F such that ξp −ξ = Id (such
p F
a map is unique up to an F -valued additive function on F). The process of toral
p
switching (based on the ideas of [Win 69], [Wil 83], [P 86]) has been described in
detail in [P-St 99, Sect. 2]. Given x ∈ L , where α ∈ Γ(L,T), we denote by T
α x
thelinear spanof allt := t−α(t)(x+xp+···+xpm−1) with t ∈ T (here m = m(x)
x
is the smallest nonnegative integer with the property that xpm ∈ T). By [P 86],
T is a torus of maximal dimension in L and Γ(L,T ) = {γ | γ ∈ Γ(L,T)},
x p x x
where
γ (t ) = γ(t)−(ξ ◦γ)(xpm)α(t) (∀t ∈ T).
x x
We say that T is obtained from T by the elementary switching corresponding
x
to x. By [P 86], there exists an invertible linear operator E = E ∈ GL(L )
x x,ξ p
such that c (T ) = E (c (T)) and L = E (L ) for all γ ∈ Γ(L,T). Moreover,
L x x L γx x γ
8
E is a lopynomial in adx. The operator E is often referred to as a generalized
x x
Winter exponential.
It is immediate from the explicit form of generalized Winter exponentials that
L(α ) = L(α) and L(α ,β ) = L(α,β) for all roots β ∈ Γ(L,T); see [P-St 99,
x x x
pp. 218–222] for more detail. Recall that the torus T is standard by our general
x
assumption. Solvable and classical roots are proper by definition. If α is Witt
or Hamiltonian, then one can always find an element x ∈ L such that the
Si6=0 iα
root α ∈ Γ(L,T ) is proper.
x x
Our main goal in this section is to show that if T is an optimal torus, then all
roots in Γ(L,T) are proper, and describe the action of optimal tori on 2-sections.
Lemma 3.1. Let K = L[α,β] and T be as in Theorem 2.2, and suppose that we
are not in case (5) of that theorem. Let u ∈ L , and assume that α 6∈ Γ (L,T)
α p
and α ∈ Γ (L,T ). Then |Γ (L(α,β),T )| > |Γ (L(α,β),T)|.
u p u p u p
Proof. We denote by u¯ the image of u in K.
(1) Since L(γ)∩I(α,β) ⊂ Q(γ) for all γ ∈ Γ(L,T), it suffices to show that under
the above assumptions we have |Γ (K,T )| > |Γ (K,T)|. If K = (0), then the
p u p
root α is solvable, hence proper. So this case is impossible.
(2) Let K be as in case (2) of Theorem 2.2. Then K = ψ(L(µ)) for some
µ ∈ F α+F β. Since all roots in (F α+F β)\F µ are solvable, hence proper,
p p p p p
it must be that K = ψ(L(α)) = ψ(L(α )), implying |Γ(K,T )| = |Γ (K,T )|.
u u p u
Since α is improper, the result follows.
(3) Let K be as in case (3) of Theorem 2.2. Then T = (T ∩ S ) ⊕(T ∩S ) by
1 2
Proposition 2.3. Hence there exist µ,µ′ ∈ Γ(L,T) such that K = K(µ)+K(µ′)
and [K ,K ] = 0 for all i,j ∈ F∗. All roots in (F α+F β)\(F µ∪F µ′) are
iµ jµ′ p p p p p
solvable. Hence α ∈ F µ ∪ F µ′. No generality will be lost by assuming that
p p
α = µ′. Since K = E (K ) for all i ∈ F∗, the preceding remark yields
iµu u¯ iµ p
K(µ ) = E (K(µ)) = c (T )⊕ K
u u¯ K(µ) u Li∈F∗ iµ
p
(as E is a polynomial in adu¯, we have that E (y) = y for all y ∈ K ). It
u¯ u¯ Si6=0 iµ
follows that µ is proper if and only if µ is. Since α = µ′ is improper and µ′ is
u u
proper, the result follows.
(4) Let K be as in case (4) of Theorem 2.2. Then K = FD⊕H(2;1)(2) and there
is an automorphism σ of the Lie algebra DerH(2;1)(2) such that σ(T) is one of
T = Fx ∂ ⊕Fx ∂ ,
0 1 1 2 2
T = F(1+x )∂ ⊕Fx ∂ ,
1 1 1 2 2
T = F(1+x )∂ ⊕F(1+x )∂ .
2 1 1 2 2
Replacing K by its isomorphic copy σ(K) ⊂ H(2;1) we may assume that σ = Id.
Theorem III.4 in[St 92]describes the1-sections of DerH(2;1)(2) relative toσ(T),
hence the 1-sections of K relative to T (the deliberations in [St 92, Sect. III] only
9
require that p > 2). It is immediate from the description in [St 92, Sect. III] that
all improper roots in Γ(K,T) are Witt (no root in Γ(K,T) is Hamiltonian by
dimension reasons).
If σ(T) = T , then [St 92, Theorem III.5] shows that all roots in Γ(K,T) are
0
proper. As α 6∈ Γ (L,T), this case is impossible.
p
Suppose T = T . Let µ be any T-root of K such that µ(x ∂ ) 6∈ {0,1}. Then
1 2 2
in the notation of [St 92, Proposition III.3] we have b 6∈ {0,1}. Hence b ≥ 2, in
which case that proposition yields H(2;1) ⊂ H(2;1) . Thus if µ(x ∂ ) 6= 0,
µ (0) 2 2
then there is a unique i ∈ F∗ with H(2;1) 6⊂ H(2;1) . Moreover, loc. cit.
0 p i0µ (0)
shows that H(2;1) ∩H(2;1) has codimension 1 in H(2;1) . ¿From this it
i0µ (0) i0µ
is easy to deduce that T normalizes a solvable subalgebra of codimension ≤ 1 in
K(µ). As a consequence, µ is a proper root in Γ(K,T).
Applying these deliberations to our improper root α we obtain α(x ∂ ) =
2 2
0. Then H(2;1)(α) is spanned by {k(1 + x )k−1x ∂ − (1 + x )k∂ | k ∈ F }.
1 2 2 1 1 p
Therefore, K(α) is isomorphic to the Witt algebra W(1;1). Since α is a proper
u
root in Γ(L,T ), the torus T = T must normalize H(2;1) . But then T is
u u u¯ (0) u
conjugate to T under the automorphism group of DerH(2;1)(2). Our remarks
0
earlier in the proof now show that all roots in Γ(K,T ) are proper.
u
Suppose T = T . Let µ be any root in Γ(K,T) and assume by symmetry that
2
µ((1+x )∂ ) 6= 0. Set
1 1
µ((1+x )∂ ) µ((1+x )∂ )−µ((1+x )∂ )
2 2 1 1 2 2
a := , b := .
µ((1+x )∂ ) µ((1+x )∂ )
1 1 1 1
Clearly, a,b ∈ F . If b = 0, then it is easy to see that H(2;1)(µ) is solvable. If
p
b 6= 0, then the 1-section H(2;1)(µ) is spanned by all k(1+x )k−1(1+x )ka+b∂ −
1 2 2
(ka + b)(1 + x )k(1 + x )ka+b−1∂ with k ∈ F . It follows from this description
1 2 1 p
∼
that H(2;1)(µ) = W(1;1) and T does not normalize the unique subalgebra of
codimension 1 in H(2;1)(µ). Since H(2;1)(µ) ⊂ H(2;1)(2), this implies that in
the present case any root in Γ(K,T) is either solvable or improper. Moreover,
the inequality |Γ (K,T)| ≤ p−1 holds.
p
Since α is a proper Witt root in Γ(K,T ), the above discussion shows that
u u
the torus T is not conjugate to T . Hence T is conjugate to either T or T . But
u 2 u 0 1
then |Γ (K,T )| > p−1 by our remarks earlier in the proof. The result follows.
p u
(5) Case (5) of Theorem 2.2 cannot occur by our general assumption.
(6) Let K be as in case (6) of Theorem 2.2. Since S is a restricted ideal of DerS,
e e
we have that t −t ∈ S for all t ∈ T. Since L(α,β) is a 2-section for T , the pair
u¯ u
e
(K,T ) appears in Proposition 2.3(6). Hence we may assume that
u
T = F(h⊗1)⊕FId⊗(1+x )∂ .
u 1 1
10