129

TRIALITY

AND

LIE

ALGEBRAS

OF TYPE

D4

by I~, dacobson (New Haven, U. S. A.)

The simple Lie algebras of types A, B, C, D, except D4, have been determined by W. Landherr [1] and [2] and by the present author [1], [2], [3] and [4]. The main result of these investigations was the establishment of a 1 - 1 correspondence between the Lie algebras and the simple associative algebras with involution (~, J). Here ~ is associative, J is an involution (anti-automorphism of period two) and simplicity means that there are no ideals invariant under J except ~ and 0. Isomorphism, homomorphism, etc. for such pairs are defined as usual for operator groups. The Lie algebras of type V 4 were first studied by the author's student, C. L. Caroll, Jr., in his University of North Carolina dissertation (1943, unpublished). In this paper Caroll introduced the notion of type D4I, D4II, D4III, and D4vI for these Lie algebras, and constructed examples of all of these types. His work was based on the half-spin representation of the split D 4 . In this paper we shall study the Lie algebras of type D4 by means of a particular realization based on Cayley algebras. This leads to a simple explicit determination of the automorphism group of the split Lie algebra D4. The Lie algebras of type D4 which are split by a Galois extension field are in 1 - 1 correspondence with certain subgroups of the automorphism group of the split D4 or equivalently with crossed homomorphisms of G into A u t ( ~ / P ) where G is the Galois group of the splitting field P and A u t ~ / P is the group of automorphisms of a split ~ over P. Besides some general results on the problem 9 - Rend.

Circ. blatem.

Palermo

- S e r i e n - T o m o XIII - A n n o 1964

130

N. JACOBSON

we shall be concerned in this paper particularly with the Lie algebras of types D4~ and D.~H (~). T h e s e are just the Lie algebras of type D4 which are special in the sense that they can be realized as algebras ~ ( ~ ,

J)

of J - s k e w elements

of an associative algebra with involution J. W e give conditions for isomorphism and obtain results on the a u t o m o r p h i s m s

of such Lie algebras. T h e s e results

are considered for the special case of 9 the real field or a p-adic field. We remark that results analogous to the ones given here can also

be

derived for the simply connected algebraic groups which c o r r e s p o n d to the Lie algebras we consider (cf. Weil [1]).

1. T h e Lie a l g e b r a of s k e w t r a n s f o r m a t i u n s

in a C a y l e y a l g e b r a

T h r o u g h o u t this paper ,,algebra ~, will mean finite dimensional (not necessarily associative) algebra over a field of characteristic -~ 2. Let C be a Cayley algebra over a field 9 of the element x E C x + x:

and

let n(x), t(x) be the norm and trace respectively

(2). Also let x be the conjugate of x so that we have

1 [n (x q- y) - - n (x) - - n (y)] ----t (x) 1, x x = x x - : n (x) 1. W e set (x, y) = ~-

__ _12(xy-[- yx) :

1

~t(xy). T h i s is a n o n - d e g e n e r a t e symmetric bilinear form

which is anistropic if C is a division algebra and o t h e r w i s e has maximal Witt index. In the latter case C is said to be split. T h e r e is only one split Cayley algebra over the given field r

For any C we have the following basic pro-

perties of (x, y):

(1)

(x a, y) = (x, y

(a x, y) = (x,

(x, y) = (x, y).

W e recall also that C is an alternative algebra, that is, if A (x, y, z) = (xy)z ~ x ( y z ) then A(x, y, z) is an alternating function of x, y, z. If a E C an for the left and right multiplications x . ~ a x , 1 shall set /?a = 2 (aL -Jr- an).

x.~xa

we write aL and

respectively. Also we

W e now consider the set ~ ( C ) of linear transformations L in C which are ) in the sense that (xL, y ) - - - - -

(x, yL). It is clear from

(1) that if a E Co the set of elements of trace 0 in C ( a = -

a) then a z , aR E ~ ( C ) .

skew relative to ( ,

(~) The Lie algebras of types D~,,, and D4~, are being studied by my student H. Allen. We have conjectured that these can all be obtained as subalgebras of the Lie algebra of derivations of an exceptional Jordan algebra annihilating a cubic subfield. (2) The characterization and properties of Cayley algebra (or the algebra of octonions) which we shall require can be found in Jacobson [5]. See also the Appendix to this paper.

TRIALITY

Hence also R~ E ~ ( r (2)

x

and if b Er (3)

If x E r

AI~ID L I E

/ kLGEBI I / k S

OF TYPE

D4

131

a E •o we can verify that = (x, 1) a -

(a, x) 1

then x fro

=

(x, b) a - - (a, x) b.

It is easy to see that every skew linear transformation in (~ (relative to ( , )) is a sum of mappings of the form x-~(x, u ) v - - ( v , x)u where u, v E r It follows from (2) and (3) that the /~a, a E (~0, generate ~((~). If r is split then ~ (r will be called the split Lie algebra D4 (t). We shall now determine the automorphisms of the Lie algebra. For this we need

Lemma 1 (Principle of local trial#y). For every skew linear transformation A in r there exist uniquely determined skew linear transformations B and C in g, such that (4)

(xy)A = (xB)y -~- x(yC)

holds for all x, y E C,. The mappings A -~ B, A -~ C are automorphisms of ~ (~,) and the three representations A -~ A, A -~ B, A -~ C are irreducible and inequivalent. Proof. Let ~ll be the subset of ~ of elements A for which elements B, C satisfying (4) exist. Then one verifies that ~1 is a subalgebra of ~. Also the alternative identity c(xy) ~ (xy)c ~ (cx)y + x(yc) s h o w s that (5)

2(xy)Rc = (XCt.)y -+- x(ycn).

Hence /?c E Za. Since the elements /?c, c E (~0, generate ~ it follows that ~ = ~I and so B, CE ~ satisfying (4) exists for any A in ~. To prove uniqueness of B, C it suffices to assume A = 0. Then we have to s h o w that if (xB)y ninu x ( y C ) = O for all x, y E C then B = 0 and C = 0 . Suppose we have this condition. Setting x = y - - - 1 we obtain 1 B = u = - - I C . Next set x = l . Then we obtain y C = uy. Similarly if we set y = 1 we obtain x B - ~ - x u . Hence we have (xu)y-~ x(uy). This implies that u = T 1 where T E l ~. Since B is skew and x B = T u it follows that T = 0 Hence B = 0 = C. Now since B and C are unique we have single-valued mappings r : A -~ B and a 2 : A ~ C. The argument at the beginning of the proof implies that these are homomorphisms. (t) In the characteristic 0 case this is the split Lie algebra D4 as defined in Jacobson [7], p. 108. In all cases, characteristic 0 or p, this is the Lie algebra /94 of classical type as defined by Mills and Seligman in [1].

132

~. JACOBSON

Since, as is well known, ~ is simple, a, and % are isomorphisms. Hence these are automorphisms. It is well known also (and is readily verified) that the enveloping associative algebra of ~ is the complete algebra of linear transformations. It follows that the representations A-~A, A ~ . B and A->-C are irreducible. Next assume that there exists a linear transformation U in C

UA ~ , = A"2U holds for all A E ~ . If we take A = 2 R c , account (5) we see that U C L = c R U . Since l ( x U ) = ( x l ) U

c in r

such that

and take into

Hence c (x U) = (x c) U, x E C ,

cECo.

we have y ( x U ) = (xy)U for all x, y E C . Then (xy)U 2 =

--- ( y ( x U ) ) U = (xU)(yU). This implies that (x U)(1 U) = (1U)(xU). If we now assume U non-singular then the last equation implies that 1 U = ~ 1 -~ 0 so if we replace U by ~ - ' U then we may assume I U = I . for x = l

gives yU-----y for all y. Then U = I

Then y ( x U ) = ( x y ) U

and we have C L = C R , c E C 0 .

This would imply that C is commutative. Hence no non-singular U satisfying

UA ~1 = A ~ U can exist and so al and % are inequivalent representations of ~. To see that A->-A and A->-A ~;, i = 1, 2, are inequivalent we choose c E Co so that n ( c ) ~ O. Then cL and cR are non-singular. On the other hand, we can choose d E C o so that (d, c ) = 0 .

Then d c q - c d = O

so d c q - c d = O .

This

shows that /~c is singular. It follows that 2Rc and cL = (2Rc) ~ are not similar. The same holds for 2R~ and cR = (2R~)~. This completes the proof. We recall that a similarity in the vector space C relative to the form n(x) is a linear transformation S in C over r

such that n(xS) = ~n(x) where cr is

a non-zero element of 9 called the multiplier of S. An equivalent condition is:

(xS, y S ) = ~ ( x ,

y). If L E ~ ; and S is a similarity then S - 1 L S E ~ . It follows

that the mapping X - ~ S - 1 X S , X E ~ , is an automorphism of the Lie algebra ~. In addition to these automorphisms we obtain also the automorphisms X - ~ S - I X ~ S , i = 1, 2, from the triality mappings ~i. We can now prove

Theorem 1. Every aulomorphism of the Lie algebra ~ (C) of skew linear transformations in C over dp relative to ( , ) has the form X - ~ S - ~ X S or X - ~ S-1X"~S, i = 1, 2, where S is a similarity in C relative to the form ( , ). Proof. This result can be deduced from a general theorem on automorphisms of a certain class of simple Lie algebras due to Steinberg [1] (~). However, we prefer to derive the result by an earlier more elementary method. We assume (~) In the characteristic 0 case the result can also be deduced from the theory of automorphisms of semi-simple Lie algebras over an algebraically closed field given in Jacobson [7], Chapter IX.

TRIALITY AND L I ~ ALGEBRAS OF TYPE

D~

133

first that the base field is algebraically closed. Then C and ~ = D4 are split. In this case it is well known that ~ contains a four dimensional Cartan subalgebra 1b with roots • ~.l+ ~.j, i < j = 1, 2, 3, 4 (Jacobson [1], [4]). If "0 is an automorphism of ~ then it is known also that ~ n acts diagonally in the space C and the only possibilities for the weights of ~ for the representation X - ~ X ~ are: (i) .+ ~.i . .(ii) . + A I A ~ = ~-()', 1 + ~'2 + ~.3 + ~',) -- Xi, (iii) + A; 1

where A~----2.(~,,+;~2+).3+).,), A ~ =

1

)'2

~-3

~'4), A~

I (),l--X2"~-X3--X4)t,

~-(~'1- ~ ' 2 - ~'3-+-)',). If ~, and "~2 are automorphisms such that the representations X - ~ X ~1, X - ~ X ~' provide the same set of weights for 1D then the argument on pp. 503-505 of Jacobson [4] shows that "~1-----"~2~. where ;~ has the form X - ~ S - ~ X S , S a similarity. The last statement in the principle of local triality shows that the three sets of automorphisms X - ~ S - 1 X S , X - ~ S - ~ X ~ X ~ X - ~ X ~ 2 S are disjoint. It is now clear that every automorphism is contained in one of these sets if q~ is algebraically closed. The general case of arbitrary ff~ can be obtained from the algebraically closed case by a standard extension field argument.

2. A n o t h e r realization of ~ and its autornorphisms We now introduce the trilinear form

l t(x(yz) ) (x, y, z) ~ 1 t ((x y) z) = -~-

(6)

on ~;. Since t(xy)----t(yx) we have (7)

(x, y, z ) = (y, z, x ) =

(z, x, y).

1

Since (x, y ) = ]t(xy),

(8)

(x, y, z ) = (xy, z).

Definition 1. An ordered triple IL,, L~, L3J of skew linear transformations L,. in ~ over @ will be called Lie related if (9)

(xL,, y, z ) + (x, y L2, z ) + (x, y, zL3) = 0

for any x, y, z 6 r It is clear from (7) that if {L,, L~, Lal is a Lie related triple then so is [L2, L3, L11 and IL3, L~, L2!. Let n denote the conjugation x - ~ x in r and if T is a linear transformation then set T = r~T~r = ~r-'T~. If T is skew then T is

134

N. JACOBSOr~

skew since = is an orthogonal transformation relative to ( , equation in (!) and (8) we have (x, y, z ) = ( x y ,

). By the last

z)=(-yx, z)=(y,x,

z). This

implies that Lie relatedness of ILl, L2, Lsl implies that of {L2, LI, Lsl.

Lemma I. A triple ILl, L2, Ls} of skew linear transformations in r is Lie related if and only if (xy)E, = (xL,+,)y + x(yL,+2)

(10)

holds for any i = 1, 2, 3 where the subscripts are taken modulo 3. Proof. Assume {L1, L2, Ls} is a Lie related triple of skew linear transformations. From (9) and (8) we obtain ((xL,)y, z) + (x(yL2), z) + (xy, zL~) ----- O. Since L'3 is skew this gives

((xL,)y + x(YL2) -- (xy)'La, z) = O. Since ( , ) is non degenerate we obtain (10) for i = 3. Since {L2, Ls, L~} and ILs, L~, L21 are Lie related the other two relations in (10) are valid. Conversely, if we assume (10) for any i we can re-trace the steps to prove the Lie relatedness

of {L,,

L3}.

We shall now consider the set r of ordered triples {x~, x2, x31, x~E r and we endow this with the vector space structure as direct sum of three copies of r We also introduce the quadratic form on r defined by n({x,, x2, x31)= = n ( x , ) + n(x~)+ n(x3). The corresponding symmetric bilinear form is given by ([xil, {Yr = (x,, y,) + (x~, y~) + (x3, Y3) and this is non-degenerate. Now suppose {L,, L~, L31 is a Lie related triple of skew linear transformations in r We use these to define the linear transformation L = {Lil by (11)

L:{xt, x2, xsl-~{xlL,, x2L2, xsLs}

in r Clearly {Lt} is skew in r We shall call such a mapping a Lie projectivity of r A direct verification shows that the set ~ (r of projectivities of r is a subalgebra of the Lie algebra of skew linear transformations in r relative to the form defined in this space. It is clear also that L = IL~, L:, Lsl-)'-L~, i = 1, 2, 3, is a homomorphism of ]D = ~)(r into ~. We now have

Lemma 2. The mapping L = {L,, L2, L3} -~ L~, i = 1, 2, 3, is an isomorphism of ~ onto ~. Proof. If L~E r then Lemma 1 and the principle of local triality show that we can imbed Li in a Lie related triple L = {L~, L2, L3/. Hence L-~L~ is

TBIALITY

AND

LIE

ALGEBRAS

OF TYPE

D4

135

sur]ective. The same argument using the uniqueness part of the principle of local triality shows that L~ = 0 implies L = 0. Hence the mapping is an isomorphism. We consider next ordered triples lTa, T 2, 7"31 where Te is a similarity in (~ relative to its norm form n.

Definition 2. An ordered triple T = called related if

(12)

(x T,, y

z

{T~, 7"2, 7"3/ of similarities in (~ is

(x, y, z),

x, y, z ~ r k~0 in @. It is clear that if {T,, T~, 7"3t is a related triple then so are ITs, 7"3, T I } and ITS, T~, L}. If p ES3 the symmetric group on l l, 2, 3} then we define T" for T = ITs, 7"2, 7"3} as {TI., T2., T3.} or IT,., T2~, T3~} according as p is even or odd. Then it is clear that if T = {T~, T2, T~} is a related triple of similarities then so is T ". Also if U - ~ - { U i} is a related triple then so is T U ~ {T1U~, T2Uz, T3U3]. Finally if " : ; ~ 0 in 9 then Iz~l is a related triple of similarities.

Lemma 3. Let T~, i = 1, 2, 3, be a similarity such that n(x T~)~-~en(x). Assume (12) holds. Then (13)

(xy) T~ = ~,-' ~,(x T,+~)(y T~+2)

holds for i = 1, 2, 3 where the subscripts are taken modulo 3. Conversely, assume (xy) 7"a = ~ (x T,) (y 7"2) holds for all x, y E if, where ~ is a fixed non-zero element of c~. Then {T1, T~, 7"3} is a related triple of similarities. The proof is almost identical with that of Lemma I and will be omitted. We recall that a similarity T with multiplier ~ : N ( x T ) = ~n(x) satisfies det T = • td and T is called proper or improper according as the sign q-- or - holds. We can now prove

Lemma 4 (Principle of triality). If {TI, T~, T~} is a related triple of similarities then every T~ is proper. Conversely, if 7"1 is a proper similarity then there exist similarities T2, T 3 which are determined up to multipliers in 9 such that {T~, T,, 7"3} is a related triple. Proof. The second statement has been proved by van der Blij and Springer [1], p. 159. Now let {Tt} be a related triple of similarities and assume T~ improper. Then g T z is proper so we have a related triple t7: T~, , }. It follows that

136

N. SACOBSO~

we have a related triple (r~,--,--}. This contradicts a known result (Jacobson [6], p. 78). We shall now use the elements of the symmetric group $3 and the related triples of similarities to define a group of linear transformations in ~;,3,.

Definition 3. Let p be a permutation of {1, 2, 3} and T = {7"1, T 2, 7"3} a related triple of similarities in (~. We define a mapping (p, T) in (~,3, by l{xl. T1, x2. Ts, x3. 7'3} if p is even (14)

Ix" Xs' x3} ~

{x,, T,, x~, r2, x-~p7'3} if p is odd.

We call (p, T) a projectivity in C '3'. If (p, T) and (q, U) are pro]ectivities where q is a permutation and U={U1, Us, U3} then (p, T)(q, U)~-(qp, TqU). It follows that the set 13(C '3') of projectivites in r is a group of linear transformations in E '3'. This group contains the subgroup of mappings of the form (p, 1) and the invariant subgroup of the mappings (1, T). Also 13(C '~') is a semi-direct product of these two groups. If L is a Lie projectivity in C '3' and (p, T) is a projectivity then (p, T)-IL(p, 7) is a Lie projectivity. Hence L-~(p, T)-IL(p, T) is an automorphism in the Lie algebra ~ ( r We now have

Theorem 2. Let ~ be a Cayley algebra over ~b, ~D(E3) the Lie algebra of Lie projectivities of E '3', 13 (E '3') the group of projectivities C '3', ~,,3, the subgroup consisting of the mappings {xi}-~ {z, xi}, % ~ 0 in r Then we have the exact sequence (15)

(3>

1-1,- ~.,3, .~ 13 ( C ) . - ~ Aut:m:(~;'s') -~ 1 -

where Aut ~ (E '3') denotes the group of automorphisms of 7~ (E '3') and the v sends (p, T) into the automorphism L-~ (p, T)-IL (p, T). Proof. We have to show that v is surjective and its kernel is ~,,3,. of the isomorphism L-~L1 surjectivity will follow if we can show mappings L~ ~ Mt where (p, T)-~L(p, T ) = {M~, )142, )1113}include a set

mapping

Because that the of generators for Aut ~((~). Taking p = 1 we obtain in this way all the automorphisms of the form Lt-~ T~-tL~ T, where 7"1 is a proper similarity. Taking p = (23), T, = 1 we obtain the automorphism L~-~rcL~rc = L, of d;. We note next that by Lemma 1 and the principle of local triality the mapping L ~ - ~ L 2 where L = {L1, Ls, L3} is the automorphism L~-~ L~. This is obtained by taking p ~---(123), T~-----1. Similarly, if we take p = (132), T~ = 1, we obtain the

T E I A L I T Y AND L I E ALCEBEA$ O F T Y P E D 4

137

automorphism L - ~ L ~ of ~. It is clear from Theorem I that the automorphisms of ~ we have indicated generate the group of automorphisms of ~ over r Hence v is surjective. Now suppose (p, T) is in the kernel of v. We can factor (p, T) -~- (p, 1)(1, T) where the first 1 is 11, 1, 1} and the second is the identity permutation. Then (p, T) ~ = 1 implies that the automorphisms L a-(p, 1)L(p, 1)-1 and L ~ ( 1 , T)-~L(1, T) are identical. If p # 1 one sees that this implies that either % or r has the form L~-~U-~L~U or LT~= U-~L~'U where U is a similarity. This contradicts Theorem 1. Hence p = 1. Then if T = ITe}, each T~ commutes with every element of ~. Since the enveloping associative algebra of ~ is the complete algebra of linear transformations, we have T~ = %1 and the proof is complete. 3. T h e p r o b l e m of D4-forms This problem concerns the classification of the Lie algebras ~L over the basefield 9 such that ~L~ for g2 the algebraic closure of 9 is the unique (split) Lie algebra D 4 (see w 1). A general discussion of this type of problem is given in the author's Lie Algebras, Chapter X. We note first that if ~L is of type D 4 in the sense that ~

= D 4 then it can be shown that ~L has a finite dimensional

Galois splitting field P/~, that is, a finite dimensional Galois extension P such that ~L ~ ~Lp is a split D4 over P (i). Then one has a 1 between the set of ~ such that ~ p ~ - ~

1 correspondence

and the isomorphisms a : s - ~ as of the

Galois group G of P / ~ into the group of automorphisms of ~ over 9 such that for each s E G, as is an s-semi-linear transformation. We shall call such an automorphism of ~L/~ an

s-semi-automorphism of i / P . The Lie algebra of type

D4 associated with a is the set ~L~ of fixed points of ~ under all as, sE G. If a and ~" are two isomorphisms of the kind we have indicated then the associated Lie algebras ~ , and ~L~, are r

if and only if there exists

an automorphism ~ of ~ over P such that a~----- ~-1%~, s E G. These results indicate that the problem of classifying the D4-forms having a given Galois splitting field is equivalent to that of classifying the isomorphisms a where a and a" are considered as equivalent if and only if there exists an automorphism [3 of ~

over P such that a~ = ~-1%~, sE G. We now select a

(~) This is well known and easy in the characteristic 0 case. For characteristic p it is proved in Barnes [1].

138

~..IACOBSON

particular isomorphism s-~ S of G such that S is an s-semi-automorphism of 5 . We use this to define an action of G on the group Aut ~L/P of automorphisms of ~ over P by setting ~ s - - S - t ~ S, ~ E Aut Y/P, s E G. Then ~s E Aut ~ / P and (~t~)s = ~ [ ~ , ~t E Aut ~L/P. In this way Aut ~L/P becomes a G-operator group. Next let ~ : a . ~ as be an isomorphism of G into Aut ~ / ~ such that as is s-semilinear. Set "~s = S-I ~ . Then ~s E Aut ~ / P and ~ t = .v-1 - S-1 a st ~ T -1 S -1 ~ ~t

= T - ~ , T T - ~ % - - - - - ~t t . Hence the mapping s-~a% is a crossed homomorphism of G into Aut ~ / P as G-operator group. Conversely, if s ~ ~s is a crossed homomorphism of G into A u t ~ / P and we set % = S~, then s ~ - % is an isomorphism of G into Aut ~L/~ such that % is s-semi-linear. It is immediate also that the crossed homomorphisms s-~'6~, s - ~

correspond to equivalent iso-

morphisms ~, ~' if and only if there exists a [3 in Aut ~ / P such that ~ =

( ~ ' ) - ~ ~,

s EG. This is an equivalence relation for crossed homomorphisms. The set H ~(G, Aut Y/P) of equivalence classes defined by this relation is called the first

cohomology set of G with coefficients in Aut ~ / P . While the consideration of H~(G, Aut 2L/P) has many important advantages, for the immediate purposes of the present paper it is more convenient to consider the problem of the D4-forms within the framework of the isomorphism *r of G into A u t ~ / ~ . We now consider the latter group. For this purpose let r the split Cayley algebra over 9 and let (~-----Cp = P |

be

(~. Then (~ is the split

Cayley algebra over P. Let ~;o, be the direct sum of three copies of ~ as in w 2 and let :~ be the Lie algebra of Lie projectivities of ~f'. Then ~3 is a split D4 and the group of automorphisms of 9

over P has been given in

Theorem 2. If s E G let S denote the s-semi-linear transformation in (~ which is the identity in r and let (1, S) denote the mapping {x~, x2, x3} -~ Ix~ S, x~ S, x3 S} in (~'~'. Then (1, S) is s-semi-linear and (xS, yS) ----(x, y)', (xS, yS, zS) ~ (x, y, z)~, x, y, z E ~. These facts imply that the mapping L-~ (1, S)-IL(1, S) is an s-semiautomorphism of ~ . It is clear also that the group of s-semi-automorphisms of :~ for all s E G is the semi-direct product of the invariant subgroup of automorphisms

of 9

L-~(1, S)-tL(I, S).

over P

with

the subgroup consisting of the mappings

TRIALITY AND LIE ALGEBRAS OF TYpE

D4

139

We can also give a description of all the s-semi-automorphisms of ~ using an extension of the considerations of the last section. We recall that an s-semilinear transformation T in ~ is an s-semi-similarity if n (x T) ~ l~n(x) ", x E r 1~ a fixed non-zero element of P. A triple {7"1, T~, 7"3} of s-semi-similarities will be called

related if (x 7"1, y 7"2, z 7"3) = v(x, y, z) ~, x, y, z E ~;, v ~ 0 in P.

If p is a permutation of {1, 2, 3} and T =

IT1, T2, 7"3} is a triple of related

s-semi-similarities then we define, as a generalization of the projectivities of the

collineation (p, T) in ~,3, by the formula (14). Then one sees easily that L-~ (p, T)-IL(p, T) is an s-semi-automorphism of ~ . Moreover, in last section, the

view of Theorem 2 and the fact that the mapping (1, S) is a collineation it is clear that the mappings

L-~(p, T)-IL(p, T) give all the s-semi-automorphisms

of ~ , s E G (~). Since (p, T) for Ti an s-semi-similarity gives an s-semi-automorphism of ~ it is clear from Th. 2 that the mapping identity if and only if p - ~ - I and T = { ' q l ,

L-~(p, T)-IL(p, T) is the

"~1, "~31}.

We recall that the semi-similarity T in ~ such that

n ( x T ) = [~n(x)s is

called proper if det T calculated for a basis of ~ which is a basis also for C/qb satisfies det T = l~4 (Jacobson [8], p. 291). It is clear that the semi-similarity S in ~ defined before ( = 1 on r

is proper. Using this fact one has an immediate

extension of the principle of triality to semi-similarities. This states that if IT1, T2, 7"3l is a related triple of s-semi-similarities then every T~ is proper. Moreover, any proper s-semi-similarity 7"1 can be imbedded in a related triple of s-semi-similarities. We now consider an isomorphism ~ : s - ~ morphisms of ~ / ~

of G into the group of auto-

such that ~ is s-semi-linear. Then ~ has the form

L

L

T,)

p~ES~ the symmetric group on {1, 2, 3} and T, = {T~(s), T2(s), T3(s)} a related triple of s-semi-similarities Ti(s). The T~(s) are determined up to multipliers in P. The mapping s~p~ is an anti-homomorphism of G into Sa. Hence where

its image is a subgroup of order 1, 2, 3 or 6. Accordingly, we call the Lie (1) If one uses the results on the eentroid given in Jacobson [7], Chapter X, it is easy to see that every automorphism of ~ over ~I, is an s-semi-automorphism for some s in G.

140

~. jACOBSON

algebra ~L~ of ~r

elements of type V4I, D4II, D4III or D4vl. This property

is given by the choice of the splitting field P. We proceed to show that it is independent of P. Since any two finite dimensional Galois extension fields of r

can be im-

bedded in the same finite dimensional extension, it suffices to show that the type is unchanged

on extension of P to a larger finite dimensional

Galois

extension A of @. We note first that if T is s-semi-linear in a vector space over P and s* is an extension of s to an automorphism of A over 9

then the

mapping Y~~iui-~ y, )~*(uiT) ~ E A, uz ~ ~J is the unique s*-semi-linear extension of T to ill'J* over h where /1~* ~ A |

If T is an s-semi-similarity then its

extension is an s*-semi-similarity relative to the extended form and if /Ib is an algebra over P and T is a semi-automorphism, then its extension is a semiautomorphism of ~1~*. Now let ~ be an isomorphism of the Galois group of P over r

into the group of automorphisms of ~ over r

such that cos is s-semi-

linear. The set of semi-linear extensions of the COs to ~ * ~

A|

~

defines an

isomorphism ~* of the Galois group of A over r into the group of automorphisms of ~ * over 9 of the type required. It is clear that the Lie algebra of ~*-fixed elements is the same as the Lie algebra of ~-fixed elements. Also if ~s has the form L -~ (Ps, T,)-I L (ps, 7",) where T, -~ {Tl (s), T2 (s), T3 (s)l then it is clear that its semi-linear extensions have the form L*-~ (p,, T,.)-IL*(p~, 7",.) where T~. ~ {Tl(s*), T; (s*), T~ (s*)} and T~(s*) is the s*-semi-linear extension of Tt(s). Evidently, this shows that the image in Sa of G and of the Galois group of A over q~ are the same. Hence the type is unchanged. It is clear that two Lie algebras of type D4~ and D4j, i ~ j , morphic. Moreover, every type D4z , i ~---I, II, lII, VI over 9 a Galois extension

are not iso-

exists if d~ has

P with Galois group which has an anti-homomorphism

s - ~ p , onto a subgroup of order i of $3. In fact, if we are given the anti-homomorphism s-~p, we let ~s be the mapping L-~(p,, S)-~L(p~, S) where S is the triple of s-semi-linear mappings in ~ which are 1 on C. Then ~ t ~ ~ t and ~

is of type D4i. Examples of this type were constructed first by Caroll

and the one we have constructed is now called the Steinberg type associated with the given anti-homomorphism s-~p,.

141

TRIALITY A~D LIE /iLGEBRAS OF TYPE D 4

4. Lie algebras of types D4! and D411 We recall that an associative algebra with involution is a pair (~, J) where ZI is associative and J is an involution in ~ . The pair (ZI, J) is of type Dz if ~,

for the algebraic closure fl of the base field ~, is the algebra of linear

transformations in a 2l-dimensional vector space over fl and the extension of J is the adjoint mapping relative to a non-degenerate symmetric bilinear form. In this case there exists a finite dimensional (3alois extension field P such that ~ p is the algebra of linear transformations in a 2/-dimensional vector space /Ib over P and the extension of J is the adjoint mapping relative to a non-degenerate symmetric bilinear form on ~

of maximal Witt index. Also we have an iso-

morphism s-~zs of the Galois group G of P over 9

into the group of auto-

morphisms of the pair (~p, J) such that ~s is s-semi-linear. The mapping zs has the form X - ~ U(s)-IXU(s) where U(s) is an s-semi-similarity relative to the form ( ,

) on ~ .

Moreover, ~

is the set of "~:fixed elements of ZIp. The

vector space l'~ can be identified with a vector space P | space over ep and ( ,

where /13 is a

) can be obtained from a form of maximal Witt index

on I'ID. Hence we can distinguish proper and improper s-semi-similarities in /1~ as we did for (~ in w 3 (cf. Jacobson [8], p. 291). Now (ZI, J) is said to be of type Dn if every U(s) is proper; otherwise, the type of (ZI, J) is Dnl. If (ZI, J) is of type D4 then the space ~(ZI, J) of J - s k e w elements of is closed under Lie commutation [a b] ~ a b - - b a and is a Lie algebra of type D 4. This is clear since ~(ZI, J)~ is the set of skew linear transformations in an 8-dimensional vector space over the algebraic closure fl relative to a nondegenerate symmetric bilinear form. We can now prove the following sharper result.

Theorem 3. A Lie algebra is of type D41 or D411 i f and only if it has the form ~ (ZI, J) where (ZI, J) is an associative algebra with involution of type D41 or D411 respectively. Proof. Suppose first that we are given a Lie algebra of type D41. Then we have the Lie algebra ~) of Lie projectivities of ~,3, where (~ : - P

|

(~ and (~

is a split Cayley algebra. Also we have an isomorphism a of the (3alois group into the group of automorphisms of ~) over ep such that at is s-semi-linear for s in the Galois group

G of P over ~.

Moreover, every as has the form

142

n. ~ACOBSON

L - ~ ( t , Ts)-~L(1, Ts). The given Lie algebra is ~,, the subset of ~ of fixed elements relative to all the %. We have T, = ITl(s), T~(s), T3(s)} where Ti(s) is an s-semi-similarity and the triple is related. The Ti(s)are determined up to scalar factors. Hence we have

Ti(s) Ti(t ) = 9,i, ,.t T,(st).

(16)

N o w we have the isomorphism L ~ {La, L2, L31-~L~ of ~ algebra of linear transformations

into the O-Lie

in (~ which are skew relative to the norm

form and commute with every Te(s), s ~G. One can see easily, as in the author's

Lie Algebras p. 310, that if ~t'" denotes the O-space of linear transformations in ~; which commute with the T~(s), then ~t" is a central simple algebra over 9 with an involution J,. of type D 4. Also the Lie algebra ~(~'~', Je) of Ji-skew elements coincides with the subset of ~l';' of linear transformations which are skew relative to the norm form. Hence the isomorphism L ~ L ;

is 1 - - 1

into

(~'~', J~). Since the dimensionalities of the two spaces are the same it is surjective so the given Lie algebra is isomorphic to the Lie algebra ~ (~'~', J~). It is clear from the definition of (~'~', J,.) that this pair is of type D41. Next assume that we are given a Lie algebra of type D41 I. Then we have the isomorphism ~ of G into the group of automorphisms of ~ over 9 such that the image of s - ~ p ,

is a subgroup of order two. By symmetry it is sufficient to

consider the case in which the subgroup is {1, (12)}. The given algebra is %~ the set of %-fixed elements of ~ . We now consider the Lie algebra isomorphism

L - ~ L 3 of %~. If p ~ - 1 we set U ( s ) = T3(s) and if p , ~ - ( 1 2 ) then we set U(s) ~ ~ T3(s). Then the L 3 are linear transformations which commute with the U, and Us is an s-semi-similarity. Also %% ~ %t implies that U(s)U(t)-~ ~., U(st). It follows as before that if ~

denotes the algebra of linear transformations

which commute with the Us then ~t has an involution J and L - ~ L 3 is an isomorphism of ~

onto ~(~t, J). Clearly (~, J) is of type D411. Conversely, let

(~, J) be an associative algebra with involution J of type D4 (I or II). Then we have a finite dimensional Galois extension P and an 8-dimensional vector space /I-I~ over P with a non-degenerate symmetric bilinear of maximal Witt index. We can identify / ~ with ~ = P | C where C is the split Cayley algebra over 9

and the form w i t h ( ,

) on ~. For each s E G

similarity U(s) in d; and the O-automorphism "q: X ~

we have an s-semi-

U(s)-IX U(s) of the algebra

TRIALITY

AND LIE AL6EBRA$

OF TYPE

143

D4

of linear transformations of ~, over P. Also %% = %t which implies that U(s)U(t)=

= p,.tU(st). The algebra ~ is the set of linear transformations in d; over P which commute with the U(s) and ~(ZA, J) is the subset of these transformations which are skew relative to ( , ). If U(s) is proper then by the principle of triality for semi-similarities, we can imbed U(s) in the related triple ITl(S), T2(s~, T3(s) = U(s)l and if U(s) is improper we can imbed rcU(s) in the related triple {Tl(s), T2(s), T3(s)= r In the first case we let % be the automorphism L-~(1, Ts)-~L(1, T s ) a n d in the second case the automorphism L-~((I, 2), Ts)-IL((1, 2), Ts) in ~3 where T s = {Tl(s), T2(s), T3(s)}. Then one checks that % % = %, and % is s-semi-linear. Also ~ (Xq, J) ~ ~L~. Hence ~ (~, J) is of type D4x

or

D4I I according as (~, J) is of type D4I

or

O4xI.

The foregoing proof shows that if ~L~ is Lie algebra of type D4x then we can associate with ~ three associative algebras with involution (~'~', J,.)of type D4~ such that ~L~ ~ ~ ( ~ ' " , J~). The trio (Z:t', J,.) is determined by the splitting field P. However, an argument like the one used to show invariance of type shows that the three algebras with involution are independent of the choice of P. We can now prove

Theorem 4. With each Lie algebra ~ of type D4x we can associate in an invariant manner three central simple associative algebras with involution ( ~ ' , J~) of type D4x such that ~ ~ ~ (~'", Jz) the Lie algebra of skew elements of "B'" relative to J~. We have Z:t'~' | Z:t'2' | Z:t'3' ~,o 1 (in the Brauer group) and two Lie algebras of type D4x are isomorphic if and only if the associated associative algebras with involution are isomorphic in some order. Proof. The first statement has already been proved. One knows also that Z:t'" is the ring of endomorphisms of ~ which commute with every endomorphism of the form ~T~(s)~,, ~ , E P (Jacobson [8], p. 290). It follows that {Y.T~(s)[3,} is simple and its division algebra part in the Wedderburn theorem is anti-isomorphic to that of Z:t". Since Z:t';' has an anti-automorphism we have ~'e' ~,o {Y, T~ (s)~,} in the Brauer group. It is clear also that the ring {2] T~(s)[3} is a homomorphic image of the crossed product (G, P, ~'~') where t~'~' the factor set ~,.t. '~' Since the crossed product is simple we have isomorphism. Hence ~ ' ' ~--, (G, P, ~'~'). We recall that {Tl(s), 7"2(s), T 3(s)} is a related triple of s-semi-similarities. Hence we have (x T, (s), y

z T3(s)) = ks(x, y, z) s

144

~. ~acoaso~

where k s ~ 0 in P. This and (16) imply that (1)

(2) ~(3) ~

P~.t ~.t ~.t Hence (G, P, ~
P, p(~)|

t

ks kt k~ 1

P, p<3')~l and consequently ~ ' " t ~ 1 , ' 2 ' ~ ' 3 ' ~ 1

The Lie algebras isomorphic to ~

have the form ~ . ,

(i).

~ = ~-'%~, ~ an auto-

morphism of ~ over P. If we take into account the form of ~ we see that the ordered triple of s-semi-similarities {T;(s), T~(s), T~(s)l associated with 0~ has either the form IU;-~T~,(s)U~, U;' T2,(s)U2, U~' T3,(s)U3I where p is an even permutation or it has the form {U;-~ T~,(s)U~, U~ ~T2,(s)U2, U~-' T3;(s)U a}, p odd. Here lUg, U2, Ua/ is a related triple of similarities

in G. Also any triple

{T~'(s), T'~(s), Td(s)l of the form indicated can be obtained for a suitable ~ ' = ~ - ~ . Let ~'~' be the algebra of linear transformations B in G such that T;(s)B = BT'~(s),

sEG. Assume first that T;(s) ~---Ui-~Tr

Uz. Then it is clear that A-~B--~ U~-~AU~

is an isomorphism of (~'~P', J~,) onto (3B'r /(~) where K~ is the involution determined in 3B''. Next assume T;(s)= U;-~Fz,(s)U~. Then A-~U;-~,4U~ is an isomorphism of (~'~", J~,) onto ()B'r /(~). Hence in either case the triple of associative algebras with involution given by ~.,~ is equivalent in some order to that given by ~ , . The converse is clear. Next let ~ be a Lie algebra of type D 4 n which we can take to be the set of fixed elements of ~ determined by the isomorphism ~ of G. If i is the index such that ip~ ~ i, s E G then we have seen that ~ is isomorphic to ffa(~'~', J ' ) where ~ " is the enveloping associative algebra of the L~ and {L~, L:, L~} E ~g~. (El '~', J~) is of type D4I I. The argument used in the foregoing proof gives the following

Theorem 5. Any Lie algebra of type D4u is isomorphic to an algebra ~ (Zt, J) where (Z~, J) is an associative algebra with involution of type D 4 I ~ . TWO Lie algebras of this kind are isomorphic if and only if the associative algebras with involution are isomorphic. 5. T h e q u a d r a t i c f o r m c a s e Let ~ be a vector space of 21 dimensions over a field 9 with a non-degenerate symmetric bilinear form ( , ). Let ~t be the algebra of linear (~) It can be seen that this result is a special case of Theorem 4 of Jacobson [8]. However, the direct proof given here is shorter than the derivation of the result based on the more general theorem.

TRIALITY

AND LIE ALGEBRAS

OF TYPE

D4

145

transformations in 1'~ over qb, j the adjoint mapping relative to ( , ). Then we have shown elsewhere that if ~ is the discriminant of the given form then (2t, ] ) is of type D~, or of type D m according as (--1)~8 is a square or a non-square in 9 (Jacobson [8], p. 295). Hence, if l ~ 4, then ~ ( ~ , J) is a Lie algebra of type D4I or D4u according as ~ is a square or non-square in ~. The following theorem gives a characterization of the Lie algebras ~ ( ~ , J) of type D4~ with ~ ~ 1 and hence of the Lie algebras of skew linear transformations in a vector space over a field relative to a non-degenerate symmetric bilinear form of square discriminant.

Theorem 6. Let ~ be a Lie algebra of type D** and let (~'~', Ji), i = 1, 2, 3, be the associated associative algebras with involution as in Th. 4. Assume ~ ' : ' , ~ ~t 'k' for distinct j and k. Then ~ " ~,~ 1 for the remaining index i. On the other hand, if ~'~'~-,~ 1 then the algebras with involution (~':', Jj) and (~,k,, Jk) are equivalent for the remaining indices j, k. Proof. If ~':' :,o ~,k,, j # k then ~':' | ~'*' ~, 1. Since ~'~' | ~'J' | ~q,k, ~,, 1 this implies ~'~' ~ 1. Conversely assume ~{';' :--, 1. For simplicity of notation we take i = 3. Let % be the automorphism L - ~ ( l , T~)-~ L(1, 7",)in ~) where

T, -~- lT,(s), T2(s), T3(s)} and this is a related triple of s-semi-similarities. If ~,a, denotes the factor set determined by the T3(s) then , 3 , 1. It follows that we may assume ~,3.;: 1, s, tE G. Then T3(st ) = T3(s ) Ts(t). Let/il3 ~ ( x E ~ l x Ts(s)= x , s E GI. Then ill3 is a (D-subspace of 12 such that P ~ 3 = 12 and Pill3 : For x E ~ V-a =

P|

we have n ( x F 3 ( s ) ) : ~,n(x) ~. Since Ta(st)=T3(s)T3(t ) this implies

I~,~-~. Hence there exists a ~,~ P

such that I~, =

Set

n*(x) ~ ~n(x).

Then n*(.~ T~(s))= n*(x)'. If we take x-----x in ill3 this gives n*(x)-~ n*(x)', s 6 G. Hence n*(x)~a) and the restriction of n* to /113 is a quadratic form on over ff~. Its extension to 12 is the multiple ~.n of the norm form n ( x ) o n 12. The algebra ~'~' is the set of linear transformations in 12 over P which commute with every 7",. These map ~

into itself and their restrictions to /113 give the

algebra of linear transformations in ~ adjoint mapping relative to ( ,

over 4). The involution .]3 in ~'~' is the

)* the symmetric bilinear form associated with n*.

We now choose an improper orthogonal linear transformation U in ~ and let U denote also its linear extension to 12=P|

over

We have T~(s)U=UTs(s).

Moreover, we can imbed 03 = z~U in a related triple of similarities {0~, 02, 03t. 10 - R e n d .

Circ, M a l e m . P a l e r m o - S e r i e II - T o m o

XUl - Anno

1964

146

~. j,co~so~

The m a p p i n g {x,, x2, x31-~ {x20,, x-~0~, x3031 which we denote as ((12), 0) defines the a u t o m o r p h i s m ~:L--~-((12), 0)-1L((12), 0) in ~ i The m a p p i n g ~ given by 0,~= i~-la~ has the form L--~(1, T~)-IL(1, Ts) where T~ = 1T;(s), T~(s), T~(s)t and 7"1'(s) = (~ 0,)-1~ 7"2(s) T: 01,

T~(s) = (n02)-' rl(s)~0~,

(17)

(s) =

03) -1 T3(s)

03.

Since ~ : 0 3 = U and T3(s)U= UT3(s) we have T ~ ( s ) = T3(s). It follows that T;(s) is a multiple of T~(s) for i = l, 2. Since ~'~' is the set of linear transformations c o m m u t i n g with the Ti(s) the first relation in (17) and the orthogonality of ~0~ implies that the m a p p i n g A - ~ ( n 0 ~ ) - ' A ( z : 0 ~ ) i s an i s o m o r p h i s m of (~'1', J1) onto (~,2,, j~). This completes the proof. T h e symmetric bilinear forms with square discriminant include the bilinear forms w h o s e quadratic forms (x, x) permit composition, or equivalently, are norm forms of Cayley algebras. W e shall call these bilinear forms Cayley bilinear forms. T h e following theorem gives two characterizations of the corresponding Lie algebras.

Theorem 7. The following three conditions on a Lie algebra ~ of type D 4 are equivalent: (1) ~L,~ is of type D4x and Zt'" ~ 1, i = 1, 2, 3, for the associated associative algebras with involution (Zt'', J~). (2) %~ is of type D4I and the three algebras with involution (~'~', J~) are equivalent. (3) ~L,~ is isomorphic to the Lie algebra of skew linear transformations relative to a Cayley bilinear form. Proof. The equivalence of (1) and (2) is clear from Th. 6. N o w a s s u m e these conditions. The argument used to establish Th. 6 s h o w s that we may assume

that T~(s) T3(t)=T~(st) and we have a quadratic form n * ( x ) = X n ( x )

such that n*(xT3(s))= n*(x) ~, s~ G. Also we have the ~ - s u b s p a c e ITJ of of elements x such that x T3(s) ~ x and the restriction of n* to ~ is a quadratic form on /115. Since (~'~', J 1 ) ~ over P such

that T~(s) and

(L~'z', J,-) there exist similarities UI and U~ in

Uj-~ T3(s)U~ differ by scalars.

Hence

we

may

a s s u m e that

(18)

7"1 (s) = U~-' T 3 (s) U,,

T~ (s) = U2-' 7"3 (s) Us.

T h e n T, (s t) ----- T~ (s) 7"1(t), T 2 (s t) = T 2 (s) T 2 (t). We have (x T~ (s), y T 2 (s), z T a (s)) = ----- k, (x, y, z) s. It follows that tc~k, = k., and there exists ~ E P such that k . = ~-~ ~.

TI~IALITY AND

LIE

ALGEBRAS

D4

OF TYPE

147

Then if we set (x, y, z)* = ~(x, y, z) we have (xTl(s), yTs(s), ~zT3(s))*=(x, y, z) *s. Since (xy, z ) : ( x ,

y, z) we have (x, y, z ) * ~ p ( x y ,

z)* for p : ~ . - ' 8 .

These

equations imply (cf. (13)) that (19)

(~ (xy)) T3(s) = p (x T,(s))(y Ts(s)).

By (18) and (19) we have for x, y E/113 that

(x u,) (y us) T3 (s) = ~ (x U, 7"1(s))(y Us rs (s)) = ~ (x T3 (s) U,)(y r~ (s) Us) = ~ (x U,) (y Us) which shows that (20)

x o y - - ~ (x U,) (y Us) E t113.

Hence (/113, o) is an algebra over ~. We have

n* (x o y) = x n (~ (x u,) (y us)) = x n (~ (x u,) (y us))

= ~.n(xU~)n(yUs) = 9 n* (x) n* (y), where "~EP. Since n*(x-y), n*(x), n*(y) E@ this implies that x E ~ and if we replace n*(x) by zn*(x) and call this n*(x) again then we have n*(xoy)=n*(x)n*(y). This implies that (1'113, o) is a Cayley algebra. Also it is clear that ~(~t. '~', J~) is isomorphic to the Lie algebra of skew linear transformations in (/il3, o). Hence we have the implication (1), (2)-~ (3). Next suppose C is a Cayley algebra over 9 and let ~ be the Lie algebra of Lie pro]ectivities in r

as in w 2. Let P be a finite

dimensional Galois extension field of 9 such that d; = CP is a split Cayley algebra. If s is in the Galois group G of P / ~ we let S be the s-semi-linear mapping in ~ which is the identity in ~; and we let =~ be the s-semi-automorphism L-~(1, S)-~L(1, S) where S is the related triple IS, S, S1. Clearly a s t = a~=e. It is easily seen that ~ is the set of extensions to d; of the elements of ~ . Hence ~ ~.o ~ . The three algebras with involution associated with ~ , are isomorphic to the enveloping associative algebras of the L~Er for L = {L~, Le, Ls}E~.

148

N. ~Aconso~

These are the complete algebras of linear transformations in (~. Hence ~l 'i'~,-,l, i ~ 1, 2, 3. Since 19 is isomorphic to the Lie algebra of skew linear transformations in the Cayley algebra ~ it follows that (3)-~(1). We can now prove

Theorem 8. Let ~ i , i ~ l, 2, be the Lie algebra of skew linear transformations in an 8-dimensional vector space ~i/~b relative to a non-degenerate symmetric bilinear form ( , )~. Then ~ i --~ ~2 if and only if ( , )~ is equivalent to a multiple o f ( , )2. The Lie algebra ~i is isomorphic to a Lie algebra ~ (7~, J) where ~ ~ 1 if and only if the discriminant o f ( , )l is a square and ( , )i is not a Cayley form. In this case (~, J) is unique in the sense of equivalence. Proof. We recall that ( , )~ is equivalent to a multiple of ( , )~ if and only if the algebra of linear transformations in ~ with the involution which is the adjoint mapping relative to ( , )~ is equivalent to the corresponding algebra given by l'De and ( , ).~. Hence the first assertion is equivalent to ~51 ~ ~ implies that the indicated algebras with involution are equivalent. Now let (~, J) be any algebra with involution which is not equivalent to the algebra of linear transformation in /ID~ with the involution given by the adjoint relative to ( , )~ and assume ~ ( L ~ , J ) ~ 5 ~ . Then Th. 4 and 5 imply that there exists a Lie algebra ~,~ of type D4~ SUCh that for the associated associative algebras with involution ( ~ ' , J,.), i ~ 1, 2, 3, we have that (~"', J~) is equivalent to the algebra of linear transformations in l'13t with the involution given by ( , )~ while (~2, J 2 ) i s equivalent to (~, J). Since the type is D41 it follows that ( , )1 has square discriminant. We cannot have ~ ~ 1 since this implies ~3 ~ ' 1 and (~t.~, J r ) ~ ' (~12, .]2) by Th. 6. This contradicts our assumption. Hence we have ~t2 ~ 1. This implies the first assertion. Also Th. 6 implies that (N~, . ] 2 ) ~ (~t~, J~). Then the second assertion is a consequence of Th. 4. 6. A u t o m o r p h i s m s

We have determined the automorphisms of the Lie algebra of skew linear transformations relative to a Cayley form in Th. 1. We shall now show that this is the only case of Lie algebras of types D4I or D4u in which we can have this complicated a group of automorphisms. We have the following result.

Theorem 9. Let ~ (~, J) be the Lie algebra of J-skew elements of the central simple associative algebra ~t with involution J. Assume ~ (2t, J) is of type D 4 and is not isomorphic to the Lie algebra of skew linear transformations relative

TRIALITY

AND LIE &LGEBRAS OF TYPE

D~

149

to a Cayley bilinear form. Then every automorphisrn of ~ (~1, J) is the restriction of an automorphism of (~l, J) unless the type is D4~ and ~ (~, J) ~ ~ (~, K) where 3B ~.~ 1 in which case the automorphisms are restrictions of the automorphisms

of

K). Proof. As usual we consider a Lie algebra ~

obtained from the isomorphism

of G into the group of semi-automorphisms of ~ . It is easily seen that the automorphisms of ~

are the restrictions to ~

of the automorphisms l] of

over P which commute with every ,s. The mapping ~ has the form L~(q, U)-IL(q, U) where q is a permutation of {1, 2, 3J and U = [U~, U~., U3/ is a related triple of similarities. Assume first that the type is D , .

Then we have the three

algebras (~';', J~) with involution associated with ~ and we may assume that one of these is the given (~, J). Suppose q = (123) then if ~ is given by (p~, T~), the condition ,~ l] ----- l] ~ implies the UT~T~+~(s)U~ and T~(s) differ by a scalar. This implies that the three algebras with involution (~t '~', J~) are equivalent. Then Th. 7 shows that ~ ( ~ , J) is isomorphic to the Lie algebra of skew linear transformations relative to a Cayley bilinear form. This contradicts our hypothesis. Thus q = (123) is ruled out and a similar argument applies to q = (132). Next assume q = (12). Then the proof of Th. 6 shows that ~'~', J " ' ) ~ (~,2,, j,~,). It follows from ~,~,|174 ~'~,1 that ~,3, ~-~1. Also one sees that (re U3)-~T3(s)(7:U3) is a multiple of T3(s). This implies that the mapping A-~(~Ua)-IA(~U3) is an automorphism of (~,3,, J3) which extends the automorphism in ~(L~ '3', J3) corresponding to the given ~. Next let q = 1. Then one sees that U;-' T~(s) U~ is a multiple of T~(s) which implies that the automorphism in ~(~'*', J~) corresponding to ~ can be extended to one in (L~'~', J,). Finally we remark that if a ~ for which q = (12) exists then the permutation q" associated with any automorphism [~" is either 1 or (12) since q' a transposition ~ (12) implies that the permutation associated with ~ [~" is a 3-cycle and this has been ruled out. Thus we see that the automorphism given in ~(~'~', Ja) by every ~" can be extended to (~'a', Ja). This proves our assertions about the D4~ algebras. Now assume ~ is of type D4n. Then we may suppose that the p~ constitute the subgroup generated by (12). Since the permutation q associated with ~ commutes with every p~ it follows that either q = 1 or q = (12). One sees easily that the automorphism given by in ~(~'~', J~) can be extended to an automorphism of (~t '~', J~). It is clear that ~(~'~', ./3)-----~(Z~, J) and this proves the result for the Lie algebras of

type D4u.

150

~. SACOBSO~ 7. Real and p-adic fields

The central simple associative algebras with involution over real and p-adic fields have been classified by the author in [3]. According to the results given in this paper the following is a complete set of representatives of the central simple algebras with involution of type D 4 o v e r the reals: The five algebras (08, J~) where 9 8 is the algebra of 8 X 8 matrices over the real field 9 and J~ is the involution X ~ S 7 1 X ' S t where ' denotes the transpose and l

(25)

S, = { - I , - - l , . . . , - I ,

I, ..., I)

i---~ 0, I, 2, 3, 4; the algebra with involution (Q4, K) where Q is the division algebra of quaternions and K is X.~ T-IX" T, T - ~ {v, v, v, v}, v ~ - - v. The algebras of the first set which are of type D4I are those for which i ~ 0 , 2, 4. The Lie algebras ~(O8, J~), i ~ 0 and 4 are isomorphic to the Lie algebras of skew linear transformations relative to the two Cayley bilinear forms over O. It follows from our results that ~ ( O 8, J2) ~-- ~(Q4, K). This is a well-known isomorphism. Hence we see that there are exactly five isomorphism classes of Lie algebras of types D4~ and D4n over O. These are represented by the algebras ~(O8, Jr), i ~ 0, I, 2, 3, 4. The automorphisms of these algebras have the X=~O-IXO where 0 ' S 0 ~ gS. We remark that it is easily seen that the group of automorphisms of (Q4, K) is connected while that of ~ ( O 8, J~) is not. It follows that there exist automorphisms of ~(Q4, K) which cannot be extended to automorphisms of (Q4, K). Now let 9 be a p-adic field, p~'2. The results of Hasse on p-adic quadratic forms imply that there are five algebras with involution of the form (08, J). These come from non=degenerate quadraHc forms. Only one of these is a Cayley form, the one of Witt index 4 and there is one other one whose discriminant is a square. According to our results the Lie algebra given by the latter form is isomorphic to a Lie algebra ~ ( ~ , J) with Z:I ~e 1. There are four algebras with involution (~, J) (Jacobson [3], p. 551). For all of these ~ is of the form Q4 where Q is the unique quaternion division algebra over 0. It follows that there are eight Lie algebras of type D4I and D4I I o v e r 0. The enumeration for p12 is also easily given. Appendix

We take this opportunity to correct two errors in Jacobson [5]. The first concerns the proof of Theorem 2 which is faulty from line 23 of p. 68 on.

TRIALITY

AND

LIE

ALGEBRAS

O1~ T Y P E

D~

151

This has been pointed out by Professor Maria Wonenburger who has also supplied the following argument to fill the gap. The argument in the text reduces the proof to the case of an automorphism for which there exists an iE r such that i n = i. Moreover, we may assume i~ 1 if C is split. We now distinguish the two cases: ~ split, C a division algebra. I. C split. We choose j E C 0 so that j_t_ i and n ( j ) ~ 0 ( N ( j ) ~ 0 in the notation of [5]). Let V be the subspace spanned by 1, i, jn, i j n, ij. We show first that there exists an element k orthogonal to V such that n ( k ) = n(j). Since the space :18 spanned by 1, i, jn, i j n is a split quaternion algebra its orthogonal complement 3B• has Witt index 2. Let ij = u~ ~ u2, u~ E 3~, u2 E ~-'-. If u 2 = 0 , V = 3B and t h e "result is clear. Hence assume u 2 ~ 0 . If n ( u ~ ) ~ O we can imbed the subspace spanned by u 2 in a non-degenerate two dimensional subspace ]9 of ~ • Then 3B q - ] 9 is non-degenerate and has Witt index three. Hence (~-q-]D) • is non-degenerate and conlains a non-zero isotropic vector. It follows that ( ~ J r - ~ ) • contains a vector k such that n ( k ) = n(j). This is orthogonal to V as required. If n(u2)~ 0 then (u2, u2i , u2j n, u2(ijn)) is an orthogonal basic for ~ • Also w = u2jn--} - u=(ij n) is a non-zero element of ( ~ --I- ~u2) • which satisfies n(w) = 0 since n(i) = - - 1. Since ( ~ -q- ~u2) • is non-degenerate there exists a k in this space such that n ( k ) = n(j). Then k is the required element orthogonal to V. Hence in all cases we have a k - a - V such that n(k) = n ( j ) . Set u = k q - j , v = k - - j . Then (u, v ) = 0 and (iu, v ) = =(ij-q-ik, k--j)=--(ik, j ) = (k, i j ) = O . If n ( u ) ~ 0 let Tt be reflection in the split quaternion algebra (1, i, u, iu) (that is, having the indicated basis). Then in

the

i ~," = i, u ~, = u, v ~ , = split

quaternion

v so k ~ ' = j

algebra

and k ~ ' n = j n. Let "~2 be the reflection

(1, i, k + j n ,

i(k-[-jn)). W e have k 1 ik, i j n

and j'~.-Lik, ij n. Hence k - - j n - a - i ( k - f f - j n) and k - - j n - t - ( 1 , i, k-q-j n, i(k--~-jn)). Then (k - - jn)~, = _ (k - - jn) and (k -1- jn)-, = k -Jr-jn. This implies that jn,, = k. Hence i ,,n', : i, k ~,n~-~-~-k. If n ( u ) : O, n ( v ) ~ 0 and we can argue in a similar fashion. Here we let "q be the reflection in the split quaternion algebra (1, i, v, iv) and we let ~2 be the reflection in the split quaternion algebra (1, i, k - - j n , i(k--jn)). Again we obtain i , , n , , = i, k , , n ~ , = k. Hence, changing notations, we may assume that "~ leaves the quaternion subalgebra ~ 6 i = ( 1 , i, k, ik) elementwise fixed. Let 1 6 ~ - with n ( 1 ) ~ O. Then ~ - = 3~il and ~i'-n C___~ - . We have l l ~ = q l where q E ~ t and n ( q ) = l . If q : l , " q : l and if q = - - l , "q is a reflection. Hence we assume q ~ ~ 1. Then u ' = : l + l ' ~ = (1-t-q)l ~ O,

152

N. ~ACOnSON

v' ~ l - - l n = (1 - - q)l # 0 and (u', v') ~ O. Either n ( u ' ) # 0 or n(v') ~ 0 and u'v'=--n(l)(1

+q)(l--q-)E~o-----~l(3r

Since ~Bi is a split quaternion

algebra we can choose s in ~t0 so that (s, u ' v ' ) = O

and s e = l .

If n ( u ' ) ~ O

we let % be the reflection in the split quaternion algebra (1, s, u', u's). We have

(v', u ' ) = O ,

v" * s = - v ' .

(v', s ) = O

Since u " * ~ u '

and

(v', u ' s ) = O

and u ' = l + l

(since (s, u ' v ' ) = O ) .

~, v ' = l - - I

Hence

~, we have l n~---l.

On the other hand, if t is any element of ~ 0 orthogonal to s then t n*~ = t *~ = - - t. It follows that ~'~3 maps 3B~ into itself and its restriction to ~

is a reflection.

Hence ~% is a reflection in a quaternion algebra containing s and "q% is a split reflection. If n ( v ' ) #

0 we let % be the reflection in the split quaternion

algebra (1, s, v', v's). Again one sees that ~% is a reflection in a quaternion algebra containing s. Hence "OTa is a split reflection. j~

II. r a division algebra. Again assume i n -----i for i - # 0 in r Choose 0 orthogonal to 1, i and let u = j + j n , v = j ~ j n . If u = 0, ~ maps the

quaternion subalgebra ~

with basis (1, i, j, i j) into itself and its restriction

to 9 is a reflection. If v = 0 ,

"q is the identity on ~ . If u # 0 ,

v#0

choose

k # 0 orthogonal to 1, i, u, v, iu, u v. Let % be the reflection in the subalgebra with basis (1, k, u, uk). Then i ~ - - - i , so "~% maps ~

u~ - - u ,

into itself and its restriction to ~

v* ~ - - - -

v. Hence j n * ~ = j

is a reflection. Hence we

may assume at the outset that ~q is either the identity or a reflection in ~ . Now let k # 0 be orthogonal to ~ and let k n = qk, q E ~ , n ( q ) = l . If q = - + - l , "q is either the identity or is a reflection. Hence we assume q = + 1. Then

n and v'----- k - - k ~. If "~ = 1 in 2~ we let s be a non-zero element of ~Bo orthogonai to u'v" and if "q is a reflection in ~ we let s be a non-zero element of ~o such that s n = - s and s is orthogonal to u'v'. Let % be the reflection in the quaternion subalgebra with basis (1, s, u', su'). Then ~% is a reflection. u'v'=--n(k)(1--q-)(1

+ q ) E ~ o = ~BCI~o if u ' = k + k

Remark Once the result on reflections has been established then we can conclude that if "~ is an automorphism then "O has a non-zero fixed point i in Co. This follows from the w e l l - k n o w n result that a proper orthogonal transformation in an odd dimensional space over a field has a non-zero fixed point. If C is a division algebra we have n ( i ) # 0 and the argument s h o w s that "q is a product of at most three reflections. It would be interesting to determine the minimum r

TR|ALITY

AND

LIE A L G E B R A ~

OF TYPE

153

D 4

such that e v e r y "~ is a p r o d u c t of r reflections in both c a s e s :

(~ split a n d r

a

division algebra. The second correction R e p l a c e the t e x t :

w e need to m a k e is the f o l l o w i n g :

,Since

i...(19)~,

of the n e x t to last line of p. 19 b y :

Since i_a_ z, w w e h a v e b y (17)~. New Haven (Connecticut), June 1964.

Bibliography F. van der Blij and T. A. Springer [1] Octaves and triality, Nieuw Archief voor Wiskunde (3), vol. VIII (1960), 158-169. C. L. Caroll Jr. [1] Normal simple Lie algebras of type D and order 28 over a field of characteristic zero, University of North Carolina dissertation, 1943. N. Jacobson [1] A class of normal simple Lie algebras of characteristic zero, Annals of Math. vol. 38 (1937), 508-517. [2] Simple Lie algebras of type A, Annals of Math., vol. 39 (1938), 181-188. [3] Simple Lie algebras over a field of characteristic zero, Duke Math. Jour. vol. 4 (1938), 534-551. [4] Classes of restricted Lie algebras of characteristic p, I, Am. J. of Math. vol. 63 (1941), 481-515. [5] Composition algebras and their automorphisms, Rend. Circolo Matem. di Palermo, ser. I! vol. VII (1958), 55-80. [6] Some groups of linear transformations defined by Jordan algebras II, Jour. f. d. reine u. Angew. Mat., vol. 42 (1960), 74-98. [7] Lie Algebras, Interscience, New York, 1962. [8] Clifford algebras for algebras with involution of type D, Jour. of Algebra, vol. 1 (1964), 288-300. W. Landherr [I] Uber einfache Liesche Ringe, Abhandl. Mat. Sere. Hamburg, vol. 11 (1935), 41-64. 12] Liesche Ringe yon Typus A, Abhandl. Mat. Sem. Hamburg, vol. 12 (1936), 200-241. W. H. Mills and 13. B. Seligman [1] Lie algebras of classical type, J. Math. Mech., vol. 6 (1957), 519-548. J. P. Serre [1] Cohomologie Galoisienne, Coll~ge de France, 1963. R. Steinberg [1] Automorphisms of classical Lie algebras, Pacific J. of Math., vol. 11 (1961). A. Well [1] Algebras with involution and the classical groups, J. Jndian Math. Soc., vol. 24 (1960), 589-623. R. T. Barnes [1] On derivation algebras and Lie algebras of prime characteristic, Yale University dissertation, 1963.