The normal structure of the unipotent subgroup of a Chevalley group of type E6; E7; E8 Текст научной статьи по специальности «Математика»

УДК 512.542.5

The Normal Structure of the Unipotent Subgroup of a Chevalley Group of Type E6, E7, E8

Galina S.Suleimanova*

Institute of Mathematics, Siberian Federal University, av. Svobodny 79, Krasnoyarsk, 660041,

Russia

Viktor V.Yakoby

Institute of Natural Sciences and Mathematics, Khakas State University, av. Lenin 90, Abakan, 660041,

Russia

Received 10.02.2008, received in revised form 20.03.2008, accepted 05.04.2008 The normal structure of the unipotent subgroup of a Chevalley group of Lie type E6, E7, E8 over an arbitrary field is found.

Keywords: normal structure, unipotent subgroup, Chevalley group, associated Lie ring, ideal.

Introduction

In any Chevalley group over a field K, associated with the root system $, the unipotent subgroup U$(K) is generated by the root subgroups corresponding to the positive roots. The group U$(K) of Lie type An-1 is isomorphic to the unitriangle group UT(n, K); its normal subgroups are described in [1] on the basis of the correspondence with the ideals of the associated Lie ring. The approach from [1] was applied to investigate the normal structure of the unipotent subgroups of some certain types for the case K = 2K in [2] —[5]. However, some particular features of the descriptions have shown an inadequacy of the method.

A new approach was developed and applied in [6] for the classical types. In the present work this approach made it possible to investigate the normal structure of the groups U$(K) for the exceptional types E6, E7, E8.

Let $(K) be a Chevalley group over a field K, associated with the root system $. For the case of 2K = K the normal structure of UEm(K) was studied by L.A.Martynova [3]. We revise the cases when the known for the type An correspondence of the normal subgroups of UEm (K) and the ideals of the associated Lie ring is realized. In this paper the normal structure of its unipotent subgroup U$(K) = (Xr | r G $+} for the type $ = Em (m = 6, 7, 8) over a field of characteristic 2 is investigated.

* e-mail: [email protected] © Siberian Federal University. All rights reserved

1. The Representation of the Unipotent Subgroups

The unipotent subgroup U$(K) is generated by the root subgroups xr (K) = Xr, corresponding to the roots r e $+. Each element A of U$(K) is uniquely represented by the product of the root elements xr (tr), r e $+, disposed corresponding to the fixed ordering of roots [7, 5.3.3], [8, Lemma 18]. We'll use the representation n of the group U$(K), which was found in [9]. Choose the subalgebra N$(K) with the base er (r e $+) in the Chevalley algebra of type $ over K with the base er (r e $),... (cf. [7, § 4.4]) and let

n(A) = ^ trer, a o p = n(n-i(a)n-i(p)) (a,p e N$(K)).

The adjoint multiplication o is a group operation on N$(K) and the mapping n : U$(K) ^ (N$(K), o) is a group isomorphism. Instead of o in the product we'll usually write +, when the cofactors don't depend on the choice of n.

Further we'll use the concepts of a corner and a frame from [6].

Let {r}+ for r e $ is a set of all s e $+ with non-negative coefficients in the linear expression of s — r through the base n($). Let

T(r) = (Xs | s e {r}+), Q(L) = (Xs | s e Ur£L{r}+ \ L), L C $+.

Definition 1. If H C T(ri)T(r2) .. . T(rm) and the inclusion is not fulfilled for any replacement of T (rj) by Q(ri), then call {ri,r2, ••• ,rm} = L(H) the set of corners for H. Call the frame for H the set F(H) such that

F(H) = H mod n Q(s), F(H) C ^ Xs. (1)

seC(H) seC(H)

Call r, s from $ connected in H, if s-projection of each element from H is equal to the product of its r-projection and a fixed scalar = 0; and call them p-connected for p e $+, if also r + p, s + p e $.

This terminology for the U$(K) will be also used for N$(K). An element er of the Chevalley base we denote for brevity by r, first of all, in the notations Ker = Kr of the root subgroups. As in [10, Tables V - VII], the root system of type Em (m = 6, 7, 8) with the base

1 8

ai = £i + £8 — 2 e®, a2 = £2 + £i, a.j = £,• — £j_i (1 < j < m) 2 i=i

we choose in 8-dimentional Euclidean space with the orthonormalized base {£i,... ,£8}. Consider the next conditions for the root r in H C NEm(K) and the fundamental root

p:

(A) F([H,XP]) + Q(r + p) C H,

(B) there exists a corner s in H, p-connected with r, and there exist fundamental roots pj and roots rj = r + pi + p2 + • • • + pj, sj = s + pi + p2 + • • • + pj with pi = p, 1 ^ j ^ t, 1 < t ^ m — 3, such that (r, s)-projection and (rj, sj)-projections of H for j < t — 1 generate in

K-module (K, K) the submodule K(a, b), (rt-i, st-i)-projection is equal to K(a, b) or in H there is pt-connected with rt-i corner = st-i,

t

Q(r2,...,rt) + F([H,XPt])+ ^K(aerj + beSj) C H,

j=2

and also if there exists a fundamental root q = p2 such, that in [H, Xp] the corner r + p is not q-connected, that T(r + p + q) C H. Moreover, either

(Bi) F([H,XP])+ T(r + s + p) C H, or

(B2) |Hr | = 2, there exists a corner u = s, p-connected with r, the set {r, s,u} coincides with one of the sets of form

{«2, kiai + «3, «5 + k2«6 + k3«7 + k^g}, kj = 0, 1,

and K{aer+p + bes+p + aber+s+p + ce„+p | a G H*, b G H*, c G H^} C H.

Theorem 1. The subgroup H of the adjoint group NEm(K), over a field of characteristic 2, is normal if and only if for each its corner r and each fundamental root p with the root r + p one of the conditions (A), (B) is satisfied.

As the theorem shows, the normal subgroups are not the ideals of the Lie ring N$(K) if and only if they don't contain at least one frame F([H, Xp]) (and such p = «4 is unique). Earlier L.A.Martynova [3] has proved that the class of all normal subgroups of the adjoint group NEm(K) coincides with the class of all ideals of the associated Lie ring for the case 2K = K.

2. Proof of the Main Theorem

We now need the following lemmas.

Lemma 1. Let H C N$(K), p G and [H, Xp] = 0. Then the corners in [H, Xp] have the form p + Sj, where Sj G ) {r}+, 1 ^ i ^ k, and 1 ^ k ^ 3. When k = 3, then

$ = Dn or Em, and {p, si, S2, S3} is a base of the system of type D4.

Proof. It is obvious that |L(H)| < rank of $ and

[H,Xp] C (T(s + p) | s GUr££(H) {r}+, S + p G $+),

so L([H, Xp]) = {p + si,p + s2, • • • ,p + sk} and the sets {p + sj}+ are pairwise not incidental. The least in $ subsystem of roots, which contains L([H, Xp]) and all roots p, sj, have the connected Coxeter graph. When its rank k + 1 > 3, then from the known classification of the root systems, the subsystem has type D4 and $ is of type Dn or Em. □

As we can observe from the Definition 1, elements of H in the Lemma 1 give the frame F(H), if in their canonical decompositions we throw out all cofactors aes with s G L(H). The addition and the multiplication in H coincide modulo ^ Q(r). Hence from the

reC(H)

Chevalley commutator formula we see that for the subgroup H of the additive or adjoint group N$(K) the frame in [H, Xp] is a K-module. So we have

Lemma 2. If H is a subgroup of the additive or adjoint group N$(K), then under the conditions of lemma 3 the frame in [H, Xp] is a K-submodule in N$(K) and equals to the frame of the Lie product of H and Xp in subalgebra N$(K).

The next lemma is established by direct calculations.

Lemma 3. Let $ be a system roots of Lie type Em. Let $+ contain fundamental roots p, q and not incidental roots r, s with r + p, s + p, r + q G $+. Then s + q G $+.

It is clear that r-projection Hr of corner r in H does not depend on the root ordering. It is also clear, that r + p is a corner in [H, Xp], and we have

Lemma 4. If H < U = UEm(K), s G Ur££(H) {r}+ \ L(H), then s is a corner of a subset in [U, H].

Lemma 5. Let H < N$(K), $ = Em, L(H) = {r}. Then H = Q(r) + Hrer.

Proof. Let h($) be the Coxeter number of the system $ and ht(r) be the height of r. The derived group [H, Xp] for p G n($) with r + p G $+ by lemma 5 has a unique corner r + p. The induction on h($) — ht(r) gives the inclusion T(r + p) C H. □

Lemma 6. Let A, B C K, ^ : B ^ K. The set A{(x, xM) |x G B} additively generates (K, K), if either A = K and there exists two K-linear independent elements in {(x, xM) | x G B}, or B = K, xM = cxe,c G K* and 0 is an automorphism of K, not identical on A(A O K *)-1.

Proof. The case with A = K is obvious. For all elements s G A n K*, t G As-1 in the case B = K and xM = cxe we obtain the equalities for x G K:

[s(xt, (xt)^) — st(x, x^)] = (0, csx0(te — t)), csKM(te — t) = K(te — t).

When there exists t = te, we obtain the conclusion of the lemma. □

Consider the following conditions for the corner r in H C NEm(K) and the fundamental root p:

(C) there exist a corner s, p-connected r, and there exist fundamental roots pj and roots rj = r + pi + p2 + • • • + pj, sj = s + pi + p2 + • • • + pj with pi = p, 1 ^ j ^ t, 1 ^ t ^ m — 3, such

that (r, s)-projection and (rj, sj)-projections in H for j < t generate in K-module (K, K)

t

the submodule K(a, b), and Q(r1,..., rt) + ^ K(aer. + 6es.) C H.

j=i j j

Lemma 7. Let a subgroup H < N$(K), $ = Em, have exactly two corners. Then for each its corner r and each fundamental root p with r + p one of the conditions (A) and (C) is satisfied.

Proof. Under the conditions of the theorem r +p is a corner in [H, Xp]. When the corner is unique, the normal closure of the derived group [H, Xp] by Lemma 5 contains Q(r + p), and hence also contains F([H, Xp]). The same inclusions are obtained by Lemmas 3-5, if L([H, Xp]) = {r + p, s + p} and corners in [H, Xp] are not connected, in particular, when s G L(H).

Further assume Q{r + p} % H. Then the corners in [H, Xp] are connected and there exist fundamental roots pj and the roots rj = r+pi+p2 + • • •+pj-, sj = s+pi +p2+• • •+pj with pi = p, 1 ^ j ^ t, 1 ^ t ^ m — 3, where t is the maximal index, such that Xrt % H. The inclusion Q(ri,..., rt) % H we obtain by Lemmas 3 and 5. Let (r, s)-projection in H generates in K-module (K, K) the submodule K(a, b). Using the relations H D [[... [[H, Xpi ], Xp2] ... ], Xpt], H D [[... [[H,XPj+1 ],Xp2 ] ... ],Xpt ] (1 ^ j < t) and Lemma 6, we obtain that (rj, sj )-projections in H for j < t generate in K-module (K, K) the submodule K(a, b), since otherwise Xrt % H, against the choice of t. □

Proof of the theorem. It's sufficient to consider the case when the derived group [H, Xp] has three connected corners, the other cases by analogy with the proof of Lemma 7 give (A) or (B) with case (B1).

Assume that there exists a corner s in H, p-connected with r, and that there exist fundamental roots pj and the roots rj = r + pi + p2 + • • • + pj, sj = s + pi + p2 + • • • + pj with pi = p, 1 ^ j ^ t, 1 ^ t ^ m — 3, where t is the maximal index, such that Xrt % H, and let u = s is a corner, p-connected with r.

Since for t = 1, it is clear that the case (A) is satisfied, we may further assume t > 1. Then the frame F ([H, XPj ]) for any j > 1 is situated in H .If j < t — 1, then (rj, sj )-projections in H for j < t generate in K-module (K, K) the submodule K(a, b), otherwise T(rt) C H.

If F([H, Xp]) ^ H, then the subgroup T(r+p+s) is not situated in H. Directly calculating all roots which in addition with p = «i is again a root, we note, that among them there are no three pairwise not incidental roots, such that at least two of them have the height ^ 4 (otherwise H D T(r + p + s)).

Let $ be a system roots of Lie type E8. Consider the case p = a2. In the set ($+ + a2) + p for p = «4 all roots are pairwise incidental, and for this case [[H, Xa2],Xp] has the unique corner, hence H again contains the subgroup T(r + p + s). The set (($+ + «2) + «4) + «3 contains the pair of not incidental roots «i + «2 + 2«3 + 2«4 + «5 + «6 + «7 + «8 and «i + 2«2 + 3«3 + 4«4 + 3«5 + 2«6 + «7, but the height of both input roots > 4. The set (($+ + «2) + «4) + «5 contains the pairs of not incidental roots from the set {«i + «2 + «3 + 2«4 + 2«5 + «6, «2 + «3 + 2«4 + 2«5 + «6 + «7, «i + «2 + «3 + 2«4 + 2«5 + «6 + «7, «2 + «3 + 2«4 + 2«5 + «6 + «7 + «8}, but again all input roots have the height > 4.

If we consider the other cases by analogy (the sets of form ($+ p) + q for all fundamental roots p, q were calculated using a computer program in Turbo Pascal)), we obtain triples of corners of form {«2, ki«i + «3, «5 + &2«6 + ^«7 + k4«8}, k = 0,1.

The following equality is obtained modulo the sum of the subgroup Q(r2,...,rt) +

t

Q(s2,..., st) + K(aer. + bes.) and the subgroups of form T(r + p + q) and T(s + p + q)

j=2 j j

[Xp, H] = K{er+p + caes+p + caer+s+p + dae„+p | a G H*, c, d G K*}.

Hence using the condition T(r + s + p) ^ H we have |Hr| = 2, and (B), case (B2). □

The work is supported by the Russian Fund of Fundamental Researches (grant 06-01-00824a).

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