Buildings of type F4

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I0.

I0. i. of the diagram

Buildings of ty]~e F 4 .

A total ordering (resp. a numbering from I to 4) of the vertices

F4

is called natural if two consecutive vertices are joined by a

single or double stroke (resp. if the ordering determined by this numbering is natural).

Let of the vertices of a chamber of type

&

A

be a building of type

F 4 . We choose a natural ordering

diagr & , and number them from I to 4 accordingly. Let

and, for

I' C (1,2,3,4) , let

I' . By 3.12 and 6.3,

St X34

and

XI.

St X12

denote the face of

C C

o be

of

are flag complexes of projective

planes; in view of 7.10 and 7.11 applied to the polar spaces whose flag complexes are

St X 4

and

St X 1

(cf. 3.12 and 7.4), these planes are Moufang and self-dual.

We shall denote the associated alternative division rings (cf. k(~o)

and

K(~,o)

respectively.

[9]

, 7.1) by

(Notice that, because of the self-duality~ these

rings do not depend on how we number the vertices of

diagr St X54

or

diagr St Xl2

to define the projective planes in question). The polar space of rank 2 associated by 7.4 with the building diagram

St X14

for the obvious ordering of the vertices of its

(2 < 3) will be denoted by

10.2.

S(A,o) .

The main purpose of w I0 is to prove the following theorem,

which gives a complete classification of the buildings of type

THEOREM.

Let

k

be a field and

K

a division algebra over

We assume that one of the followin~ conditions is satisfied:

(i)

K = k

and

char k ? # 2

;

(ii)

K

i.s a separable quadratic extension of

(iii)

K

is a quaternion algebra over

(iv)

K

is a Cayley algebra over

k ;

k ;

F4 .

k;

k .

201

1 (v)

Let

char k = 2

nK : K ~ k

and

form

S(k,K)

K • k 4 -~k

(I)

K , and in the cases (i) and (v) the function

2

be the polar space of rank 2 associated with the quadratic

defined by

F4

and

x I, x 2, x 3, x 4 e k . Then, there exists a buildin6

and a natural orderin6

o(k,K)

up to isomorphism.

o(k,K))

K ~ K(~(k,K),

o(k,K)) . If

- and the

(~(k~K), o(k,K))

(k',K')

(A(k,K),

Conversely, this pair determines uniquely the field

k ~ k(A(k,K),

the ~air

A(k~K) of

of the vertices of its diagram, so that

S(A(k,K), o(k,K)) ~ S(k,K) . This property characterizes the pair

with

x~--~ x

(Xo, x I, x 2, x 3, x~) ~-. nK(xo)-x z x3 + x 2 x 4

fo__~r x 0 e K type

k 2.

denote the quadratic form which is, in the cases (ii) to (iv) th__~e

norm form of the al6ebra Finally, let

kCKc

k -algebra

o(k,K)

k

o(k~K)) - indeed

K , whose ring structure is given by

denotes the opposite of the ordering

is not isomorlohic to any pair

satisfying one of the conditions

(A(k~,K~),

(i) to (v), unless

o(k,K)

o(k~KI)),

(k,K)

is of

type (v), in which case we have the isomorphism

I

i

(n(k,K), 3(k,K)) ~ (n(~,~), o(K,k2))

Given an arbitrary buildin~ of the vertices of

~

of type

diagr ~ , a field

F 4 , there exists a natural ordering k

and a division algebra

fying one of the conditions (i) t2o (v), so that

o(k,~))

(~,o)

K

over

is isomorphic to

.

The proof will be given in 10.3 and 10.12.

k

o satis-

(A(k,K),

202

10.3.

Existence of

10.3.1.

(a(k,K), o(k,K))

In the cases (i) to (iv), we can repeat, mutatis mutandis ,

what has been said in 9.2. There again exists the possibility of explicit constructions, for instance along the llne of [32], VIII to XI, and an other approach, by means of the classification theory of algebraic simple groups. This time, we have to consider the following types of groups (cf. fig. 8; as before, the notations are derived from those of [96], table II, by omitting always the second index) : F~ (split groups of type

F 4 ),

2E~

@

(quasi-split groups of type

@

!

@

E 6 ),

E9

and

E~8

'. @

,

,

r

fig. 8

The theorem 2 and proposition 5 of [96] imply that, over a given field of each one of these types are classified up to central isogeny by a which, according to the type in question, is the field split group of type

F4

over

k

K

itself (there is only one k

(the

E6 ), a division quaternion

algebra, or a division Cayley algebra. We shall denote respectively by ~

k -algebra

k ), a separable quadratic extension of

splitting field of the given quasi-split group of type

k , the groups

= F4 ,

~6 ' ~

and

the ad~oint group of the considered types correspondingto a given

algebra

K . By [8], 6.13 or 6.14 (cf. also [8~], p. 184), all these groups have

203

relative root systems of type its building over

F 4 . Let

k . The vertices of

G

be one of them and let

diagr ~(k,K)

o = o(k,K)

the notations of lO. 1,

St X.. iJ

(ordering from left to right). With

is the building over

L.. 13

is obtained by removing the orbits

G , and its anisotropic kernel is the same as that of ([96], theorem 2) and therefore also quotient of L~. ij

k

of the derived group

of a Levi subgroup of a parabolic subgroup of type

The index of

G (cf. [96], 2.5.1);

be the ordering of these vertices corresponding to the ordering of

the distinguished orbits indicated on fig. 8

Lij

be

are in canonical one to one

correspondence with the distinguished orbits in the index of let

& = ~(k,K)

Lij

St X..

ij i

of

and

G j

(5.2 (iii), 5.4). from the index of

G ; this determines

up to isomorphism. Let

Lij

L~.

by its maximal normal anisotropic subgroup; replacing

be the Lij

by

has no influence on the building (5.5) and amounts to neglecting those connected

components of the index of G~/~

, one has

Lij

L!. iO = L.. iO

which contain no distinguished orbit (actually,

unless

G = ~

and

for

ij = 34 ). Let us also recall

0

that two centrally isogenous groups have the same building (5.4). Now~ a glance at fig. 8

shows that, up to central isogeny,

' LI4

orthogonal group of the quadratic form 10.2 (I), is either the algebraic group whose group of (if

G = ~

) the group

~6

is the neutral component of the LI2 '

is the group

k -rational points is

studied in 5.12. Consequently,

(&,o)

SL 5

!

and

SS(K)

L34 or

has the required

properties:

(1)

s(a,o)

~ S(k,K)

10.3.2.

,

k(~,o)

~ k

,

K(a,o)

~ K

The existence of case (v) in 10.2 is part of a more general

phenomenon related with the existence of strictly special isogenies (9.7.3), and which we now proceed to describe.

Let a field

k

G

be an adjoint split algebraic simple group of type

of characteristic

p , with

X

over

204

X = B n , Cn

or

or

Fh

X = G2

and

and

p = 2

P = 3

,

-I and let G 9 N

K

be a field such that

its normalizer,

B

kc

K C kp

,

G

slonal tmipotent group on which

T

and we call

the set of all

(resp.

U a C B . Let for

a r r

generated by

T(k9

r

)

T(k9

~" ) , N(k,K) 9 all

finally

G(k,K) = X(k,K)

and all

Ua(K )

for

relative to

U (k)

for

a

a ~ r

t ~ T

the group

and all

the group generated by

a c r

a

such that

r

(resp.

r

is denoted by r

)

Ua 9

such that

a(t) s k

(resp.

B(k,K)

the group

9

Ua(K )

T(k,K) , all

)

T . The one-dimen-

(resp.

N(k).T(k,K)

a e r

k -split torus of

T , and

operates through a root

be the group of all

(resp.

a maximal

a Borel subgroup containing

the set of all "long" (resp. "short") roots of

r

T

for Ua(k )

a e

~,,+

for

K )

, and

a c r

9 Straightforward computations, analogous to those of

[17] and using the commutation relations of the

B(k 9 =T(k,K). ~

U(k).

a~r '+

Ua's

II

given there, p. 279 show that

Ua(*)

ace ,,+

where the factors of the product in the right-hand side can be put in an arbitrary order (for this assertion, cf. [8], 3.5)9 and that BN -pair of type

X

in

G(k9

(B(k,K) ,

N(k,K))

.

It is readily verified that i

(2)

Bn(k,K) ~ C n ( K , Z ) ~ P ~ ( q )

cf. 8.2.8), where the quadratic form

q : K • k2n-~k

is defined by

q(x O, x I, ..., X2n) = x20 + x I x 2 + ... + X2n_l X2n

is a

205

Let now with the BN-pair

X = F4

and let

(B(k,K) , N(k,K))

~ = A(k,K)

be the building associated

by 3.2.6. Up to canonical isomorphism, the

diagram of this building is the Coxeter diagram underlying the Dynkin diagram of G ; let

o = o(k,K)

be the ordering of its vertices starting at the extre~mity which

represents a long simple root. Then, it is easily seen, by means of an obvious generalization of 5.2 (iii) and using (2) (for

n = 2 ), that the relations (i) of

iO.3. I hold again.

Remark. The case

10.4. building

X = G2

provides a new class of generalized hexagons.

The following theorem describes the automorphism group of the

~(k,K). When the pair

(k,K)

satisfies one of the conditions (i) to (iv)

of 10.2, it is a consequence of theorem 5.8. In case (v) one can prove it, either by a method analogous to that of 5.11 (using also 8.6

(II) applied to

S(A,o)), or

via a straightforward generalization of 5.8 ; we omit the details.

THEOREM.

Let

k,K

be as in 10.2, set

be the group of all automorphisms of

k

& = &(k,K)

and let

which can be extended to ~

Aut K k

automorphism

o_f K . The other notations are as in 10.5, 5.1 an__~d5.7.2. Then

i__f 10.2 (i) (resp. (iii); (iv)) holds, then an extension of

i_ f

F~(k)

(resp.

10.2 (ii) holds , then

(Aut ~6)(k )

~7(k) ;

Aut A = Spe ~

(itself an extension of

alently, an extension of

E~(k)

~

finally, i___f1 0 . 2 (v) holds, then Aut K k ,

~8(k) )

an__~d Aut Z ~ / S p e A ~ Z / 2 Z

E~(k)

(K,k 2)

by

Aut K k

(cf. 5.9);

and this group!is an extension o__f_f b_yy ~ / / 2 ~

) __bY Aut K k , or, equiv-

[~ e Aut K I ~(k) = k );

Spe A

is ~u extension of

F4(k,K )

o___rI accordin~ as the two Fairs of fields

1

(~,,~) and

Aut ~ = Spe a , and this group is

are isomorphic or not.

206

10.5. tion

~ .

Set

a,b e K - [0} either

k

k = K and

(b,a)C.c = c.(a,b)

and

i - (c,b)

follows from 8.13

x

.

Suppose ,

k , n

c

: L ~k

: K ~k

Let

k

Bn

.

Thus,

is a field,

,

L

c

commutes with ((c,b+l)

is central in

K

- (c,b)).(b+l) and the lemma

is commutative).

quadratic -the -

i__ss

hence

be a field of characteristic

vanishes,

K

can be written

Since

k K

k

(10.2).

from the identity

b .

and that for

extension or a quaternion

(b,a) c = (a,b)

b ~-I

a non-degenerate

the quadratic

2,

K

form representin~

k -vector space

K • k2

a vector 1 , whose ,

a nd d

form

~--~n(x0) + XlX 2

L I = [0) • k • {0}

similitudes

one has

c ---/0 and

LEMMA.

. Then,

The hypothesis

(because it is obvious when

(x0, Xl, x2)

,Set

nK

rin~ with involu-

1 e k

quadratic

~s the norm

commutes with

associated b i l i n e a r form p

separable

it readily follows

10.6. s~aee over

*--~ x~x

division

Assume that

a~b~cba = bJa~cab

In particular,

.

(c,b) that

one has

( x , y ) = xyx -1 y -1

Set

(c,b+l)

be an associative

or a (commutative)

k , and

(a,b).c = e.(a,b)

Le__~t K

_i = [x+x~l x 6 K] .

c e k

itself,

algebra over

LEMMA.

and

(relative to

p

(x 0 E K ; Xl, x 2 c k)

L 2 = [0) • {0] • k . Suppose that the group : cf. 8.2.8)

leavin6 invariant

L1

and

L2

G

of all

is transi-

i tlve on

~-I(0) - L I - L 2 . Then,

x0

~ - ~ n ( X O )2

is a

k -linear bisection

of

K

1 onto a subfield of

k2

qontainin~

k . 1

For is

k -linear (since

x 0 e K , set Bn=

0 )

V(Xo) = n(XO )2 . Clearly, and injective

(since

n

the m a p p i n g

is anisotropic).

v: K Set

-@k

=

207

{e}= v'1(1)

. By hypothesis, if

x 0 ~ K - [0} , there exists

g(e,l,l) = (x0, I, n(XO) ) . This have and

g

multiplies the form

n(Xo).n(K ) = n(K) . It follows that v(K)

LEMMA.

Let

k

be a field and

with a non-de6enerate quadratic form S

by

such that

n(XO) , and we

is a multiplicative group,

is a field, q.e.d.

10.7.

set of

n(K - {0})

p

g ~ G

q : V ~k

S

the polar space assQciated

of Witt index 2. Let

M

be a sub-

containin6 at least two non-collinear ~oints and havin6 the following

property:

(I)

if

s c S

and

collinear with line, then

Then,

x,y E M

are such that both

s , whereaw

x, y, s

(the bilinear form associated with

denotes the projective space of

Let

y

ar_~e

s c M .

M = S , o__r Gq

the codimension of

and

do not belon~ to a same

In the following proof~ the symbol P

x

pl

in

c,d r M

q ) has rank 4 .

or

I stands for

l(q)

P , is at least 5 9

be non-collinear.

and not contained in the linear span

Let

Set A

[a,c,d,Pl]

L = clA

din

be a line of of

S . Let S

a,b ~ L

containing

[a,c,d) U p l

in

LC

M . Let cln

xln of

V

to

{a]

represented by

a I . Since

I

xlN

cln

dI

S

which contains

b

to the

implies that

, it then follows that

and, using again (1), we conclude that

the line of

A

. From 8.2.7 applied to the restriction of

represented b y

q

c,d ~ M , it follows from (I) that

x ~ A - {a] . The assumption made on

dl~a

a

P ~ the

existence of such a line follows from 8.2.7 applied to the restriction of V

and

~q , equal to

V . We assume that the rank of

be distinct (and therefore non-collinear).

hyperplane of

,

q x~O

x c M . Hence

to the subspace L

Is not reduced

A C M . Let

and has a non-empty intersection with

B

be

A . For

2~

every

y c (blA

S) - B , there

line is different (bin

S) - B C M

is a line containing

from the line . Let

all but one meet

z e S - (biN

(bin

Since every point of

[y,b] , hence

y 6 M

and meeting

z e M

A , and this

(always by (I))

S) ; among the lines of

S) - B , consequently B

y

S

and we have

lies on at least two other lines,

and

going through

z ,

S - B C M .

B C M , and the proof

is complete.

10.8.

PROPOSITION.

of a field and an alsebra (v) of 10.2.

Suppose

Then, there exists

an isomo_rphism

defined by

n K = nK,

nK,

implies

the mapping

10.9. satisfies : K ~K

~

which

k

and

nK,

o @

is

one of the conditions an___w S(k',K')

extends

k'

k -linear bijection

and satisfies

K

and

K'

(i) to

are isomorphic. K -~K'

~

: K -~K'

PROPOSITION.

the relations

M(1) = 1

of the norms

nK ,

, as is well known. in the cases

0~ = 0 S(k,K)

(i), and

(ii),

of 10.2 and assume that

(iii) o_~r (v). Let the involution

x~.x = nK(x )

is isomorphic

the polar space associated

from

n K , n K,

We use the notations

one of the conditions

b e defined by

of

,

~ : K -~K'

itself turns out to be an algebra isomorphism

and the definition

fo___r x e K - {0} . Then,

to

with an alternatin~

.

and can be

The bijection

(i) to (iv), the equivalence

of the algebras

consistin5

to an isomorphism

are isomorphic

are proportional.

k -linear

be two pairs

and this is also true in the case (v), as readily follows

dual of the polar space

(i)

and

the isomorphism

the above relations

K

nK

S(k,K)

for a suitable

o ~ . In the cases

(i), (ii) and (iii),

k ~k'

(II), the fields

M(x) = ~(1)-l.@(x)

and

Actually,

forms

an___d (k',K')

ov~ r this field satisfyin~

in such a way that,

the quadratic

(k,K)

that the polar spaces

By 8.6 identified

Let

form in

k4

,

the

2O9

(ii)

the polar space associated with equivalently, with a

(iii)

c -hermitian form (or,

~ -antihermitian form) in

the polar space associated with a

K4

(~,-i) -quadratic form in

K# ,

or

1 (v)

the polar space

S(K, k 2)

respectively.

In the projective space of the right

K4

K -vector space

in the cases (i), (ii), (iii),

1 k~ x K 4

in the case (v) ,

we consider the polar space

S*

associated, in the caSes (i) and (ii), with the

-antihermitian form

in the case (iii), with the

(xl)

~-~

x~ x 3

+

(~,-I) -quadratic form

x~ x 4

+ k

(cf.

8.2.1;

here

K

_l : k

)

and in the case (v), with the quadratic form 1

(Xo, x i) ~-* x~+ xlx 3 + x2x 4

where the index

i

always runs from i to 4 and

(xo ~ k 2)

x i e K . To prove our proposition

210

it clearly suffices to produce a correspondence (a "duality") that, if

x e S(k,K)

y e S* , then

then

[y c S*I (x,y) e D} is a line of

Ix e S(k,K) I (x,y) e D]

lations show that the relation index

i

varies from

0

D

is a line of

D C S(k,K) • S* S*

such

and, conversely, if

S(k,K) . Straightforward calcu-

defined as follows has these properties (here, the

to 4 , the index

j

varies from

0

to 4 in the case (v)

and from 1 to 4 in all other cases, and we use the obvious "homogeneous coordinates" to represent the points of

S(k,K)

and

S* ):

xlx[+xox ~

x~x~=0

O

x2x ~ + ~0x~

((xi),(x~)) ~ D ~:~

- xlx ~ : 0

x3x~-x0x ~

x2x~ = 0

x 4 x ~ - XoX ~ .I _ x3x ~ = 0

The proposition is proved.

I0. i0. that both

S

PROPOSITION.

and its dual

S(k~K) , where

k

S*

Let

S

be a polar space of rank 2, and assume

are embeddable. Then

is a field and

K

~

S

o_~r S*

is isomorphic to some

k -algebra satisfyin~ one of the conditions

(i), (ii), (iii) or (v) of 10.2.

The lines of the points of S*

S

and

S*

S

consist of at least three points, therefore

S* belong to at least three lines;

in other words,

S

and

are thick.

Let

(p, ~, ~)

snd

respectively (cf. 8.7), where vector space with x~

and

~(S)

V (resp. and

y, ... of

S* S*

V' )

with

(P', ~', $')

P (resp. P' )

be dominant embeddings of

S*

and

is the projective space of a right

over a division ring

~'(S*) . The lines of

will be denoted by

S

k (resp. S

x*, y~, . . . .

k' ). We identify

corresponding to points

S

211

From the thickness

(i)

if

~ (resp.

then

~' )

dim P

of

S

and

S* , it follows that

is represented by a symmetric bilinear form,

(resp.

dim P' )

is

> 3 9

To begin with, let us suppose that bilinear form. Let

x, y e S*

~' , there exists of

S*

z e S*

be non-eollinear

such that

which is co]linear

(in

x* , y*

span pairwlse, through

and

z*

and the mapping

x*

the polarity

meeting

S* . From the assumption made on

are collinear in

x*

x

and

and

y*

~x*,y*: y~ -~x*

~x.,y.(S) =

(~) si

with the plane spanned by ~

in

is not represented by a symmetric

y

also meets

s

In other

P

which they

is the inter-

s e y*

z* . This projection being linear,

is represented by a bilinear form. Furthermore,

an easy computation

shows that the above situation cannot occur if this form is antisymmetric, char k = 2 . This discussion and that ~'

~

f r o m n o w on, that

is represented by a symmetric bilinear form; indeedj

k

unless

is commutative

if it were not so,

would be represented by such a form and it would suffice to exchange

Since the embedding q : V -~ k

(P, ~, ~)

is dominant,

S

e i e S* (i = l, 2, 3, 4)

P' . The set

of

[el, e2, e 3, e4]

S*

either

contained in M--S

on

[el, e3}

, that

M

of all points of

is P' : [el, e2, e3, e4] , or

m u s t then be degenerate

and

char k = 2

[e2, e4)

S

~

~

denote

which correspond to lines of 10.7; hence,

has rank 4.

has rank 4. Because of (i),

(cf. 8.2.4).

(frame)

are not

[el, e2, es, e4]

clearly fulfils the hypothesis

We first consider the case where

S* .

~'

and

S* , and the six other pairs are), and let

their linear span in

and

be the four vertices of a quadrangle

S* , ordered in a natural cyclic way (i.e.

collinear in

S

is associated to a quadratic form

(cf. 8.7). We now drop the above restriction

Let in

allows us to assume,

z .

is just the "projection

of a point

and

z~ .

subspace of

(cf. 7.2.3)

A x*

P . Then, every point

is also collinear with

are contained in a 5-dimensional

z* " : the image

section of

x~ y, z

S* ) with both

This means that every line of. S words,

~'

Since the embedding

212

(P', ~', ~')

is dominant,

ratic form in

S~

P' (cf. 8.7).

of all automorphisms