JOURNAL OF ALGEBRA 196, 283᎐300Ž. 1997 ARTICLE NO. JA977071
Symmetric Composition Algebras
Alberto Elduque*
Departamento de Matematicas´´ y Computacion, Uni¨ersidad de La Rioja, CORE 26004, Logrono,˜ Spain Metadata, citation and similar papers at core.ac.uk
Provided by Elsevier - Publisher Connector Communicated by Efim Zelmano¨
Received December 18, 1996
A construction of all the Okubo algebras over fields of characteristic 3 is provided and the complete classification of the composition algebras with associa- tive norm over fields is finished. ᮊ 1997 Academic Press
1. INTRODUCTION
An algebra A over a field F is said to be a composition algebra if there exists a strictly nondegenerate quadratic formŽ. called the norm n: A ª F such that nxyŽ.snxny Ž.Ž. Ž.1 for any x, y g A. The form being strictly nondegenerate means that its polarization, defined by
nxŽ.,ysnx Žqy .ynx Ž.Ž.yny, is a nondegenerate bilinear form. Linearization ofŽ. 1 gives
nxyŽ.Ž.Ž.Ž.,xz s nyx,zx s nxny,z, Ž.2 nxyŽ.Ž.Ž.Ž.,zt q nzy,xt s nx,zny,t,
for any x, y, z, t g A.
*Supported by the Spanish DGICYTŽ. PB 94-1311-C03-03 and by the Universidad de La RiojaŽ. Convenio Caja Rioja, 95PYB06MBC and 96PYB07PBC . Present address: Departa- mento de Matematicas,´ Universidad de Zaragoza, 50009 Zaragoza, Spain. E-mail: [email protected].
283
0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. 284 ALBERTO ELDUQUE
Unital composition algebras, also called Hurwitz algebras, constitute a well-known class of algebras of dimension 1, 2, 4, or 8, and are classified by the ‘‘Generalized Hurwitz Theorem’’w ZSSS82, Chap. II; KMRT,Ž. 35.16x . Through the work of Peterssonwx P69 , Okubo w O78, O95 x , Okubo and Myungwx O-M80 , Okubo and Osborn w O-O181, O-O281 x , Faulkner wx F88 , Elduque and Myungwx E-M91, E-M93 , and Elduque and Perez´ wx E-P96 , those composition algebras in which the norm is associative, that is, it satisfies
nxyŽ.Ž.,zsnx,yz Ž.3 for any x, y, z, have been considered. Because of their nice and symmetric properties, these composition algebras of dimension G 2 with associative norm have been called sym- metric composition algebras inwx KMRT and this notation will be used here. It turns outwx M86, p. 86 that for a strictly nondegenerate quadratic form ndefined on an algebra, conditionsŽ. 1 and Ž. 3 are equivalent to the validity of
Ž.xy x s xyx Ž.snxy Ž. Ž.4 for any x, y. In particular, by taking an element 0 / x g A with nxŽ./0, Ž.4 shows that the left and right multiplications by x are bijective, whence it followsŽ seewx K53. that any symmetric composition algebra is finite dimensional, its dimension being restricted to 2, 4, or 8. The linearization ofŽ. 4 ;
Ž.xy z q Ž.zy x s xyz Ž.qzyx Ž.snx Ž,zy .,5 Ž. will be used throughout the paper. Two classes of symmetric composition algebras have played a key role: the para-Hurwitz and Okubo algebras. Given a Hurwitz algebra C, with multiplication xy, the para-Hurwitz algebra associated to C is the algebra defined on the same vector space C, but with new multiplication x и y s xy Žseewx O-M80. , where x ¬ x is the standard conjugation of C. By definition, the Okubo algebras are the forms of the pseudo-octonion algebraᎏwhich will be defined shortlyᎏthat is, those algebras which become isomorphic to the pseudo-octonion algebra after extending scalars. The original definition inwx O78 of the pseudo-octonion algebra required some restrictions on the ground field. In order to avoid this, a new definition, which extended the one given by Okubo, valid over any field was proposed inwx E-P96 , inspired in a previous construction of Petersson wxP69 . Given a Hurwitz algebra C and an automorphism of C such that 3 s 1, the Petersson algebra C is the algebra defined on C with the new SYMMETRIC COMPOSITION ALGEBRAS 285
multiplication
y1 x) y s xy
Žseew E-P96,Ž. 1.2x. . The name Petersson algebra appeared for the first time inwx KMRT, Sect. 36B . Now, in case C s CFŽ.is Zorn’s vector matrix algebra
au CFŽ.s :␣,gF,u,¨gF=F=F ½5ž/¨
with multiplication
␣ u ␣Ј uЈ ␣␣Ј q uи¨Ј ␣uЈ q Јuy¨ =¨Ј s , ž/ž/¨¨ЈЈ ž/␣Ј¨q¨Јqu=uЈЈ q u и uЈ
where u и ¨ and u = ¨ denote the usual dot and vector product in V s Ž. 3 F=F=F, let us take any g End F V with minimum polynomial X y 1 and define
au ␣ u s Ž*.1 , ž/¨ ž/¨y 
where * is the adjoint relative to the dot product. Then is an
automorphism of CFŽ.of order 3 and the Petersson algebra CF Ž. Ž.unique up to isomorphism is called the pseudo-octonion algebra and
denoted by PF8Ž.. The main results inwx E-P96 , extending those inw O-O181, O-O281x , assert that the symmetric composition algebras with nonzero idempotents are exactly the Petersson algebras, and that these are either para-Hurwitz or some Okubo algebras constructed in a specific way. Since it is knownw O-O181; O-O281; KMRT,Ž. 36.11x that given a symmetric composition algebra, either it already contains a nonzero idem- potent or one finds these idempotents after a scalar extension of degree threeŽ. see also the paragraph before Corollary 3.4 here , it follows that the symmetric composition algebras are the forms of the para-Hurwitz alge- bras and the Okubo algebras. On the other hand, any form of a para- Hurwitz algebra of dimension / 2 is again a para-Hurwitz algebraw E-P96, Lemma 3.3x , so only the forms of the two-dimensional para-Hurwitz algebras and the Okubo algebras need to be determined. Along the same lines of the definition by Okubo of the pseudo-octonion algebra, but independently, Faulknerwx F88 obtained composition algebras from separable alternative algebras of generic degree 3 over fields of 286 ALBERTO ELDUQUE characteristic /3. Following this idea, the symmetric composition alge- bras over fields of characteristic / 3 are classified inwx E-M93 as follows. Let F be a field of characteristic / 3 and assume first that F contains a primitive cube root of 1. Let A be a separable alternative algebra of degree 3 over F with generic minimum polynomial
3 2 pxŽ. s y Tx Ž.qSx Ž.yNx Ž.1, for a linear form T, a quadratic form S, and a cubic one N. Consider the Ä Ž. 4 subspace A0 s x g A: Txs0 with multiplication 2 1 2 q x y s xy y yx y TxyŽ.1. ) 3 Ž. Then A0 , ) is a composition algebra relative to S. If, on the contrary, F does not contain the primitive cube roots of 1, consider the quadratic field extension K s Fwx obtained by adjoining one of them, ,to F. Let A be a separable alternative algebra of degree 3 over K with generic minimum polynomial pxŽ. as above and equipped J 2 with an involution J of the second kind Ž s .. Consider the F-sub- Ä J Ž. 4 Ž. space A˜0 s x g A: x syx and Txs0 , with the well defined multiplication
2 1 2 q x y s xy y yx y TxyŽ.1. ) 3
Ž.˜ <˜ Then A0 , ) is a composition algebra relative to S A 0 . Moreover, we have THEOREM 1.1wx E-M93 . Let C be an algebra o¨er a field F of characteris- tic / 3 with dimF C ) 1. Ž.i If F contains the cube roots of 1 Ž.type I,then C is a composition algebra with associati¨e bilinear form if and only if there exists a separable Ž. alternati¨e algebra A of degree 3 o¨er F such that C ( A0 , ) . Two such algebras C’s are isomorphic if and only if so are their correspond- ing alternati¨e algebras. Ž.ii If F does not contain the cube roots of 1 Ž.type II and K s Fwx 2 with / 1 s , then C is a composition algebras with associati¨e bilinear form if and only if there exists a separable alternati¨e algebra A o¨er K with an Ž.˜ in¨olution of the second kind such that C ( A0 , ) . Two such algebras C’s are isomorphic if and only if so are their correspond- ing alternati¨e algebras as algebras with in¨olution. Actually, inwx E-M93 the ground field is assumed to have characteristic /2, 3, but everything is valid also in characteristic 2. The only necessary SYMMETRIC COMPOSITION ALGEBRAS 287 changes affect the argument in Section 6 of the latter reference for types IIc and Ic. For type IIc and following the notation in that paper, A s K = C JJ with K s k and C s C, the algebra D s Äx g C: x s x4is Hurwitz with the restrictions of the trace t and norm n of C, and, although the Ž. decomposition D s Fe [ D0 wxE-M93, end of p. 2496 and 6.1 inwx E-M93 are not valid for characteristic 2, the mapŽ. with x и y s xy
Ž.Ž . :D,иªA0,) 1 1 a¬ŽŽ.Ž..qtay ,qytaya may be shown to be an isomorphism valid in any characteristic /3. Similarly, for type Ic the map
Ž.Ž . :C,иªA0,) 1 1 a¬ŽŽ.Ž..qtay ,qytaya is an isomorphism. In a slightly different way, this has been very nicely developed in wxKMRT, Chap. VIII , where it has been noted too that the results in wxO-O181, and E-P96 give a proof of the known classification of the separable alternative algebras of generic degree three over fields of characteristic / 3. The Okubo algebras with idempotents are the ones that appear related to the algebra of 3 = 3 matrices over a fieldw E-M93, Proposition 7.4x . Therefore, the knowledge of the symmetric composition algebras will be complete as long as the symmetric composition algebras without nonzero idempotents over fields of characteristic 3 are classified. This is the objective of this paper, which will be structured as follows. The next section will provide a new construction of some Okubo algebras C, over fields of characteristic 3, depending on two parameters. Actually this construction provides all the Okubo algebrasŽ. Theorem 5.1 . Then some properties of the symmetric composition algebras without nonzero idempo- tents will be given in Section 3. In Section 4, it will be shown that for any Okubo algebra without nonzero idempotents, scalars and can be Ž. found so that A is isomorphic to C, characteristic s 3 . Finally, Section 5 will provide the classification of the symmetric composition algebras over fields of characteristic 3. Unless otherwise stated, the characteristic of the ground field F will be assumed from now on to be 3. Given a subset of an algebra A, the subalgebraŽ. respectively ideal or subspace generated by S will be denoted 288 ALBERTO ELDUQUE by alg²:ŽS respectively ideal ²:S or span ²:.S . If the algebra is equipped with a quadratic form n, S H will denote the orthogonal subspace to S relative to the polarization of n: S Hs Ä x g A: nxŽ.,ss0 for any s g S4.
2. A NEW CONSTRUCTION OF OKUBO ALGEBRAS
Let us consider the algebra FXwx,Y of polynomials in two variables over the ground field F and let us take 0 / , g F, and the ideal I s 33 ideal² X y , Y y :. In FXwx,Y a new multiplication is defined by
ij XYij᭛XY iЈjЈ 1 XYiqiЈjqjЈ.6Ž. sž/yiЈjЈ
Then I is still an ideal of the algebra ŽFXwx,Y,᭛.. Let a and b denote the classes of X and Y modulo I. The quotient algebra ŽFXwx,YrI,᭛.s ŽFawx,b,᭛.has dimension 9 and decomposes as the direct sum
Fawx,bsF[C,, with
² ij : C,sspan ab:0Fi, jF2, Ž.Ž.i, j / 0,0 .
For u, ¨ g C, , the product u᭛¨ decomposes accordingly as
u᭛¨ s nuŽ.,¨ qu)¨, Ž. Žij iЈjЈ. with nu,¨ gF and u)¨ g C, . Then nab,ab s0 unless both i q iЈ and j q jЈ are multiples of 3. In that case, for i q iЈ s 3r and ij iЈjЈ iiЈjjЈ rs jqjЈs3s, nabŽ ,ab.sabq qs ,sonŽ., is a nondegenerate symmetric bilinear form on C, , which is the polarization of the quadratic form given by
1 nuŽ.s2nu Ž,u .synu Ž,u ..
It is clear that the Witt index of n if 4. It will be shown later onŽ. Theorems 4.5 and 5.1 that the algebras Ž.Ž . C, ,)which will be denoted just by C ,constitute precisely the class of Okubo algebras over F. Also necessary and sufficient conditions will be given for two such algebras to be isomorphic, thus obtaining a classifica- tion of the Okubo algebras. SYMMETRIC COMPOSITION ALGEBRAS 289
Let us obtain first a suitable basis and multiplication table of C, . First, in Fawx,b the elements a and b are invertible with inverses y12a and y1 b2, respectively. Hence it makes sense to consider the ‘‘monomials’’ ij ab with i, j g ޚ. The formula, which follows fromŽ. 6 ,
ij abijab iЈjЈ Ž.1 ␦␦ 1 abiqiЈjqjЈ,7Ž. ) s y iqiЈ,0 jqjЈ ,0 ž/yiЈjЈ where i denotes the class of i modulo 3 and ␦ is the usual Kronecker delta, is valid for i, iЈ, j, jЈ g ޚ with Ž.Ž.Ži, j / 0, 0 / iЈ, jЈ .. As a basis for ij Ž. C, we take the elements xi, j syab, with y1 F i, j F 1 and i, j / Ž.0, 0 . Then
␦␦ nxŽ.i,ji,xЈ,jЈs iqiЈ,0 jqjЈ,0,8Ž. and by usingŽ. 7 , the corresponding multiplication table is easily shown in Table I. A simple inspection shows that with s 1 and s ␣, this multiplica- tion table is exactly Table 2 inwx E-P96Ž with different notation for the elements in the basis.Ž. . Therefore, the algebras C ␣ inwx E-P96 are exactly Ž.up to isomorphism the algebras C1, ␣ and, as a consequence, the class of algebras C, contains the class of Okubo algebras with nonzero idempo- tents.
PROPOSITION 2.1. For any 0 / , g F, the algebra C, is an Okubo algebra.
1 313 Proof. By extending scalars to K s FŽ r , r., it can be assumed ŽŽ.. that s s 1. But C1, 1 s C 1 is exactly the pseudo-octonion algebra.
Remark. It can be proved in an elementary way that C, is a symmet- ric composition algebra. To do this, one has to check the validity ofŽ. 5 for ij the monomials ab with 0 F i, j F 2, Ž.Ž.i, j / 0, 0 , and this is easily 2 proved by working with vectors in ކ33, where ކ is the field of three elements.
Given an algebra A, with multiplication xy, the opposite algebra Aop is the new algebra defined on A, but with the new multiplication x и y s yx. ij jЈ iЈ Interchanging the roles of a and b in C, and using that iЈjЈs ji 290 ALBERTO ELDUQUE 1, 1 1 y 1 y x x 1 y 1, 0 0 0 0 1 y 1, y x y x
y 1 y 1 x y x 1, 1 1, 0 0 0 1 1 y y x y y
y x 1 1, 1 1 1 y y y x y xx 00 0 00 0 1, 1 1 y y x
y
Ž. y 10, y 1, 11,0 y y 1 1, 0 x , 0 0 y x x x
Ž. y TABLE I 11 10, x 1 x y y 0 0 00 00 1 y 0, x y x
y 11, 11,01, y y 0, 1, 1, 1 0, 1 1, 1 1, 0 1, 1 0, 1 x 0 0 0, 1 y y y
x
y 1 10,1 y 11,1 xx y 1 1, 0 y xx 0 0 00 00 1, y y 1 y x x y
y 1, 0 1, 1, 1 1, 0 y 0 0 0 0 x 1, 0 1, 1 y xx x x x x x x x
y 11, 1 10, y y y 1,01, 1,0 1,1 0,1 0, 1 0, 1, 0 1, 1 1, y y y x x xx x x x xx x SYMMETRIC COMPOSITION ALGEBRAS 291 we arrive at:
op PROPOSITION 2.2. For any 0 / , g F, C, is isomorphic to C ,.
3. SYMMETRIC COMPOSITION ALGEBRAS WITHOUT IDEMPOTENTS
Let A be a symmetric composition algebra over F with multiplication denoted by juxtaposition and norm n. As inwxw E-P96, Sect. 5 or KMRT, 2 Ž.36.11x , let us consider the cubic form g: A ª F given by gxŽ.snx Ž,x.. Our hypotheses on the characteristic of F being 3 and on the associativ- ity of n imply that g is a semilinear map of the vector space A over F on 3 3 the vector space F over its subfield F , with gŽ.␣ x s ␣ gxŽ.for any ␣ g F and x g A. 2 2 3 LEMMA 3.1. For any x g A, gxŽ .Ž.sgx qnxŽ.. 22 2 2 2 Proof. ByŽ. 4 , for any x g A, xxqŽxxx.Žsnx,xx.and xxs 2 xx s nxxŽ.. Therefore 2 2 2 x x s gxxŽ.ynxx Ž. .9Ž. Hence,
2 222 2 2 gxŽ.snx Ž,xx .snxŽ.,gxx Ž.ynxx Ž.by Ž. 9 2 2 2 sgxnxŽ.Ž,x .ynxnx Ž.Ž,x .
22 2 3 sgxŽ.ynx Ž. nx Ž,x .sgx Ž .qnx Ž.. 3 COROLLARY 3.2. F q gŽ. A is a subfield of FŽ so it is a purely insepara- ble extension of exponent 1 of F 3.. Proof. Linearizing the assertion in Lemma 3.1, we obtain for any x, y g A that 3 2 gxgyŽ.Ž.sgxy Žqyx .y nx Ž,y .,10Ž. 3 3 so F q gAŽ.is an F -subalgebra of F of finite dimension, hence a subfield of F.
PROPOSITION 3.3. Let A be a symmetric composition algebra o¨er F. Then the following conditions are equi¨alent: Ž.i A contains some nonzero idempotent, Ž.ii there is an element x g A with g Ž. x s 1, Ž.iii F3 is contained in gŽ. A , Ž.iv there is a nonzero element x g A with g Ž. x s 0. 292 ALBERTO ELDUQUE
Proof. Assume first that there is a nonzero idempotent e g A. Then 2 2 eseesneeŽ.,sone Ž.s1 and gŽ.Žye s n ye, e .Ž.syne,e s y2neŽ.sne Ž.s1 and Ž. ii is satisfied. Obviously, assertionŽ. ii implies Ž iii . . Assume now that F 3 is contained 3 3 in gAŽ.. Then, since the field extension F q gAŽ.rF is purely insep- arable, its degree is a power of 3; but since dimF A s 2, 4, or 8, Ž 3 Ž.. Ž. Ž. dim FF33F q gA sdim gAF8 and we get that dimF3gAs1or3. Thus g is not one-to-one, soŽ. iv follows. Finally, assume that g is isotropicŽŽ.. condition iv or, equivalently, that Ž 3 Ž.. ker g / 0. Then g: A ª F is not one-to-one, so dim F 3 F q gA -9. Ž 3 Ž.. Ž. Hence dim FF33F q gA s1 or 3 and dim gAF3. If there is a nonzero element x g ker g with nxŽ./0, then byŽ. 9 the element 2 ŽŽ..y1rnx x is a nonzero idempotent. This is always the case if the dimension of A is 8, since in this case dimF ker g G 5. So assume that for 22 any x g ker g, nxŽ.s0 and take 0 / a g ker g.By9,Ž.aa s0sowe 2 2 can assume too that a s 0Ž otherwise a can be substituted by a .. Substitute now x by x q y and x y y inŽ. 9 and add to obtain
22 22 2 xyqyxqŽ.xy q yx 2 2 synxŽ.Ž.,yxyqyx y nxy Ž.ynyx Ž..
Take b g A with naŽ.,b sy1 and put x s a and y s b in the last equation to get that e s ab q ba is an idempotent, and it is nonzero byŽ. 10 . The fact that conditionŽ. iv implies condition Ž. i is valid for the symmet- ric composition algebras over any field, as shown inw KMRT,Ž. 36.11x . From the proof above, it follows that if A is a symmetric composition algebra without nonzero idempotents, then g: A ª F is a one-to-one 3 semilinear mapŽ.Ž. this will be very important in the sequel , gAlFs0, 3 3 and F q gAŽ.is a purely inseparable extension of F of exponent 1, so its degree is 3 or 9. Hence the dimension of A is either 2 or 8. This result was already known, since the symmetric composition algebras of dimension 4 are para-Hurwitz algebraswx E-P96, Lemma 3.3 , so they contain nonzero idempotents. It follows too from Proposition 3.3 that if A is any symmetric composi- tion algebra without nonzero idempotents, then there is a purely insepara- ble field extention K of F of degree 3 such that AKFs K m A does contain nonzero idempotentsŽŽ it is enough to take K s F .with a root 3 of the polynomial gaŽ.Ž.qXb s gaqXgbŽ., for linearly independent elements a and b of A.. This result, with a different proof, had been noticed inwx O-O281 . SYMMETRIC COMPOSITION ALGEBRAS 293
COROLLARY 3.4. The norm of any Okubo algebra without nonzero idem- potents represents 0. Hence its Witt index is 4. Proof. Let A be an Okubo algebra with nonzero idempotents and let n be its norm. Let KrF be a purely inseparable field extension of degree 3 such that AKFs K m A contains nonzero idempotents. Byw E-P96, Lemma 3.7x , the norm in AK represents 0 and so does n by a theorem of Springer wxS52 . Moreover, since n is also the norm of a Hurwitz algebra, and it represents 0, the Witt index has to be maximum. We close this section with a result on subalgebras generated by one element.
PROPOSITION 3.5. Let A be a symmetric composition algebra without nonzero idempotents and let x be any nonzero element of A. The alg²:xisa two-dimensional composition subalgebra of A.
2 2 Proof. It cannot be x s 0, since gxŽ.snx Ž,x./0 by Proposi- tion 3.3. By hypothesis, x and x 2 are linearly independent andŽ. 4 and Ž. 9 2 show that alg²:x s Fx q Fx , which is two-dimensional. The coordinate matrix of the restriction of nŽ., to alg ²x :in the basis Äx, x 24 is
nxŽ.,xnx Ž,x2 . ynxŽ. gx Ž. , 222s 2 ž/nxŽ.Ž.,xnx,x ž/gxŽ.ynx Ž.
3 2 2 3 with determinant ⌬ s nxŽ.ygxŽ.syŽŽgx.Ž.qnx .by Lemma 3.1. 2 3 3 Thus if ⌬ s 0, gxŽ .Ž.synx ggAŽ.lF, which is zero by Proposition 2 3.3Ž. iii . This forces x s 0, a contradiction.
4. OKUBO ALGEBRAS WITHOUT IDEMPOTENTS
In this section the symmetric composition algebras without nonzero idempotents of dimension 8 will be studied. As mentioned in the Introduc- tion, they are necessarily Okubo algebras. By Proposition 3.5, if A is any such Okubo algebra, for any 0 / x g A, alg²:x is a two-dimensional composition subalgebra of A.
PROPOSITION 4.1. Let A be an Okubo algebra without nonzero idempo- tents and norm n. Then for any nonzero isotropic element x g A there is another element 0 / y g A with nŽ. y s 0 and n Žalg ²:x , alg ²:.y s 0. Proof. Let us take any nonzero x g A with nxŽ.s0 Ž this is possible by Corollary 3.4.²: . Since alg x is a two-dimensional composition subalgebra of A, A decomposes as the direct sum A s alg²:x [ alg ²:x H. The 294 ALBERTO ELDUQUE
kernel of the left multiplication Lx by x has dimension 4, since both its kernel and its image are isotropic subspaces whose dimensions add up to 8, so there is an element z g alg²:x H with y s xz / 0. Then again alg²:y 2 sFy q Fy is a two-dimensional composition subalgebra of A, and byŽ. 2 andŽ. 3 ,
2 nxŽ.,ysnx Ž,xz .Žs nx,z .s0, 2 2 nxŽ.,ysnxŽ.Ž.Ž.,xz s nxnx,zs0, 2 nxŽ.,y snxyŽ.Ž,ysnxy,xz .Ž.Ž.s nxny,zs0,
22 2 nxŽ.,y synyxŽ.Ž.,xy q nx,y synyŽ.,xxy Ž. synxzŽ.,xxyŽ.synxnz Ž.Ž,xy .s 0, so nŽ²:²:.alg x , alg y s 0, as required. PROPOSITION 4.2. Let A be any Okubo algebra without nonzero idempo- tents and norm n, and let x and y be nonzero isotropic ¨ectors of A with nŽ²:²:.alg x , alg y s 0. Then either xy s 0 or yx s 0, but not both. More- o¨er, alg²:Ž²:²:.xy q yx : alg x [ alg y H. Proof. Because ofŽ. 10 , Proposition 3.3, and the hypothesis of the proposition,
gxyŽ.Ž.Ž.qyx s 2 gxgy/0, so at least one of xy and yx is not 0. Now byŽ.Ž. 3 , 4 , and Ž. 5
2 2 Ž.Ž.xy yx syxyxyŽ. Ž.qnxy Ž,xy .synyx Ž. qnxŽ.,yys0, and similarly Ž.Ž.yx xy s 0. Therefore Ž.Ž.Ž.Ž.xy yx q yx xy s 0. Also 22 2 nxyŽ.,yx syny Ž,x.Ž.qnx,y s0 byŽ. 2 , so by Ž 10 . again
3 2 gxygyxŽ.Ž.sgxyyxŽ. Ž.Ž.q Ž.Ž.yx xy q nxy Ž,yx .s 0, and either gxyŽ.s0 and gyx Ž.s0. Proposition 3.3 finishes the proof with the exception of the last assertion, which now follows quickly fromŽ. 2 andŽ. 3 .
33 THEOREM 4.3. Gi¨en scalars g F _ F and g F _ F Ž. , there is a unique, up to isomorphism, Okubo algebra A containing elements x, y with gxŽ.s,gy Ž.s,n Žalg ²:x , alg ²:.y s 0 and xy s 0. Moreo¨er, A does not contain nonzero idempotents and it isŽ. isomorphic to the Okubo algebra
C, . SYMMETRIC COMPOSITION ALGEBRAS 295
Proof. In case an Okubo algebra A satisfying these conditions exists, 3 33 3 then , g F q gAŽ.,soF ;FŽ.;F qgAŽ.and the degree of 3 3 the field extension F q gAŽ.rF is necessarily 9. By Proposition 3.3, A does not contain any nonzero idempotent. Besides, by repeated use ofŽ. 3 andŽ. 5
2 yx syxxyŽ.qnx Ž,yx .s0, 2 xysyŽ.yx x q nx Ž,yx .sy Ž.yx x, 2 Ž.yx s Ž.Ž.yx yx syŽ. Ž.yx x y q ny Ž,yx . x 2 2 sŽ.xyyynxŽ.,yxyqnyŽ.,xx 2 2 2 2 2 syy x qnxŽ.,yysyy x ,
22222222 Ž.xy s Ž.Ž.Ž.xy xysyŽ.xyyxqnx Ž,xyy . 222 2 2 2 2 sŽ.yxxynx Ž.,yyxqnx Žx,yy . 222 2 2 2 2 2 2 sŽ.yxxsyŽ.x x y qnxŽ.,y x 2 2 2 syŽ.gxxŽ.ynxx Ž. ysyxy Ž.byŽ. 9 . From Proposition 3.5 it follows that
2 22 0/alg²:yx s span ²yx, Ž.yx :s span ²yx, yx :, so y 2 x 2 / 0. Also
2 22 alg² xy:²sspan xy,xy :. By the last assertion in Proposition 4.2,
H alg²yx :: Ž.alg ²:x [ alg ²:y ,
22H H alg²xy ::Ž.alg ²x :[ alg ²:y s Ž.alg ²:x [ alg ²:y ,
2 and since xysyŽ.yx x,
2 H alg²xy ::Ž.alg ²:yx [ alg ²:y : alg ²:yx H . Therefore A is the orthogonal direct sum
2 Asalg²:x [ alg ²:y [ alg ²yx :[ alg ²xy:, and Äx, x 22, y, y , yx, yx 2222,xy,xy 4is a basis of A. By means ofŽ. 4 and Ž. 5 it is easy to check that all the products of elements of this basis are 296 ALBERTO ELDUQUE completely determined. For instance,
2 2 2 2 2 Ž.yxŽ. x y syyxŽ.Ž.yx syynxŽ. Ž,xy .yxyxŽ.syy ,
2 because gxŽ.sand yx s 0, or
2 2 2 Ž.Ž.x yyxŽ.syŽ. Ž.yx y x synyxx Ž. s0, in accordance with Proposition 4.2. Therefore there is at most one Okubo algebra, up to isomorphism, satisfying the hypotheses. But C, is an Okubo algebra, and it verifies
2 g Ž.yx1,0 s nŽ.yx1,0, x 1,0 s nŽ.yx1,0, y xy1,0 s
Ž. Ž. Ž ² : ² :. by 8 . In the same way g yx0, 1 s , and also n alg x1, 0 , alg x0, 1 s 0 and x1, 0) x 0, 1 s 0. This concludes the proof. Remarks.Ž. i Actually one can check in a straightforward way that, in 12 12 122 the proof above, the basis Äyx, yy x , yy, yyy , yx, Ž. yyx, y12xy,y12xy 4presents exactly the same multiplication table as in Table I. Ž.ii If the equality yx s 0 instead of xy s 0 were required in the theorem above, a unique Okubo algebra without idempotents would have been obtained and, by uniqueness, this algebra is the opposite algebra of the one obtained in the theorem. This agrees with Proposition 2.2. Given an algebra A, the symmetrized algebra Aq is the new algebra defined on A, but with the new multiplication defined by x( y s xy q yx. THEOREM 4.4. Let A and B be two Okubo algebras without nonzero idempotents, with respecti¨e norms nAB and n and associated cubic forms g A and gB. Then the following conditions are equi¨alent: Ž. Ž . Ž . i gAABsgB, Ž.ii Aqq is isomorphic to B , Ž.iii A and B are either isomorphic or anti-isomorphic.
Moreo¨er, if these conditions are satisfied, there is a unique isomorphism : AqqªB,so A is isomorphicŽ. respecti¨ely anti-isomorphic to B if and only if this unique is an isomorphismŽ. respecti¨ely an anti-isomorphism from A to B. In particular the group of automorphisms of A and of Aq is tri¨ial and no Okubo algebra without idempotents is isomorphic to its opposite algebra. SYMMETRIC COMPOSITION ALGEBRAS 297
Proof. Assume first thatŽ. i is satisfied. Since g ABand g are one-to-one Ž. by Proposition 3.3, there is a unique map : A ª B such that gxA s ŽŽ.. gB x for any x g A. This map is clearly linear and invertible. Besides, by Lemma 3.1, for any x g A
2 23 gBBŽ.Ž.xsgŽ. Ž.xqn B Ž. Ž.x
2 23 gxAAAŽ .sgx Ž.qnx Ž. Ž. Ž Ž.. and subtracting and using that gxABsgxwe obtain
2 233 gBBŽ.Ž.xy Žx .sŽ.nŽ. Ž.xynxA Ž.ggBB Ž .lFs0
22 by Proposition 3.3. Therefore Žx .Ž.s x and is an isomorphism form Aq to Bq. Conversely, let : Aqª Bq be an isomorphism. For any 0 / x g A, 222 Ž. Ž. Ž. Ž2.Ž.Ž. x(xsxx q xxs2nxx by 4 , so x ( x s 2nxA x, but Ž. Ž2.Ž.Ž.2 ŽŽ..Ž. ŽŽ.. also x ( x s x ( x s 2nBB x x . Hence n x s Ž. Ž. Ž Ž.. nxAAfor any x and is orthogonal. Therefore gxsgBxfor any Ž. Ž. Ž. Ž. xand gAABsgB. Thus conditions i and ii are equivalent. Evidently conditionŽ. iii implies Ž. ii . Finally, let us assume that condi- tionsŽ. i and Ž ii . are satisfied. By Proposition 4.1 there are nonzero Ž²: ²:. isotropic elements x, y g A with nA alg x , alg y s 0 and we can assume, by Proposition 4.2, that xy s 0Ž. otherwise interchange x and y . Ž. Ž Ž.. Let be the unique invertible linear map such that gaABsgafor any a g A. As above, is an isomorphismŽ. the only possible one from Aqqto B and it is an orthogonal map. Hence Ž.x and Ž.y are nonzero 2 isotropic vectors of B. But alg²Ž.: x s span ²Ž. x , Ž.x :²Ž.s span x , Ž 2 .: Ž ² :. ² Ž .: Ž ² :. Ž ² Ž .: x s alg x , also alg y s alg y ,sonB alg x , alg² Ž.:.y s 0. By Proposition 4.2 either Ž.Ž.x y s 0, and the unique- ness of Theorem 4.3 forces that is an isomorphism, or Ž.Ž.y x s 0 and is an anti-isomorphism. The final assertions of the theorem are now straightforward. Remark. Notice that even though the automorphism group of any Okubo algebra without nonzero idempotents A is trivial, it is not so with the derivation algebra Der A. Since Okubo algebras are flexible and Lie-admissibleŽ seewx M86.Ž. , 0 / ad A : Der A, where ad xyswxx,ys xy y yx.
If gdenotes the cubic form associated to C ,,Ž. 8 and Table I imply Ž. ij Ž. ²ij that gx i, j sy for any i, j. Hence gC , sspan F3 : y1 F Ž.Ž.: i,jF1, i, j / 0, 0 . Let us denote by G this set. The next result summarizes some of the work in this section. 298 ALBERTO ELDUQUE
THEOREM 4.5.Ž. a The Okubo algebras without nonzero idempotents are 3 exactly, up to isomorphism, the algebras C, , with g F _ F and g F _ F3Ž.. 33 3 Ž.bIf , Ј g F _ F , g F _ F Ž. , and Ј g F _ F Ž.Ј , then Ž. qq iC, is isomorphic to CЈ,Ј if and only if Gs G ЈЈ. Ž. ii If Gs G ЈЈ, then the algebras C, and CЈ,Ј are isomorphic if y1Ž.y1Ž. y1Ž.y1Ž. and only if gЈ g Ј s 0. If this is not satisfied, then gЈ g Ј s0and C, and CЈ,Ј are anti-isomorphic.
5. CLASSIFICATION OF THE SYMMETRIC COMPOSITION ALGEBRAS OVER FIELDS OF CHARACTERISTIC THREE
Assume for a while that the characteristic of the ground field is arbitrary. As mentioned in the Introduction, the symmetric composition algebras without nonzero idempotents are the forms of two-dimensional para-Hurwitz algebras and the Okubo algebras without nonzero idempo- tents. If A is a two-dimensional symmetric composition algebra without nonzero idempotents and 0 / x g A is an element with nxŽ.s1 Ž these 2 elements always exist, because nuŽ ./0 implies n ŽŽ1rnu Ž .. u..s1 , then 2 2 Asalg²:x s span ²x, x :Ž.Ž. Let s gxsnx,x.. Then the multiplica- 22 tion in A is determined byŽ. 4 and Ž. 9 , which give xx s xxsxand 2 2 2 x x s x y x , respectively. The nondegeneracy of n is equivalent to the restriction /"2. Besides, for any g F,
2222 Ž.Ž.xqxsy1xqŽ.q2x, which is / 0 since /"2. But there are no nonzero idempotents, so
2 1 2 0 / y q 2 3 1 s y y
3 and the polynomial X y 3 X y is irreducibleŽ a condition that already implies /"2.. Therefore any two-dimensional symmetric composition algebra without nonzero idempotentsŽ. that is, not being para-Hurwitz is isomorphic to the algebra, which will be denoted by A, with a basis Ä4u, ¨ and multiplication given by
2 2 u s ¨ , u¨ s ¨u s u, ¨ s u y ¨ ,1Ž.1 SYMMETRIC COMPOSITION ALGEBRAS 299
3 where g F such that X y 3 X y is irreducible. This is the content of wxE-M91, Theorem 4.3 , which is valid in any characteristic. In particular, if the characteristic is 3, the restriction on is equivalent to the condition 3 g F _ F . From now on, we return to our restriction of the characteristic being 3. 3 3 Hence the polynomial X y 3 X y becomes just X y .
THEOREM 5.1. Let A be a symmetric composition algebra with norm n and cubic form g. Then one and only one of the following possibilities occurs:
Ž. 3 1 There is a scalar g F _ F such that A is isomorphic to A. This Ž. possibility happens if and only if dim F 3gAs2. 3 Ž.2 A is a para-Hurwitz algebra and in this case gŽ. A s F . Ž.3 A is an Okubo algebra. Then there are nonzero elements , g F such that A is isomorphic to C, and either: 3 Ž.a gA Ž .sF,which occurs if and only if A is isomorphic to the pseudo-octonian algebra P8Ž. F . Ž.b g Ž A . is a subfield of F and a cubic extension of F 3. This happens if 3 and only if there is an element ␣ g F _ F such that A is isomorphic to C1, ␣ ŽŽ.(C␣inwx E-P96. . Ž. Ž . c dimF 3 gAs8. This happens if and only if A does not contain 33 any nonzero idempotent and then g F _ F and g F _ F Ž. .
3 The cases Ž.2 and Ž.Ž.Ž3 a in which g Ž A .s F. are distinguished by the commutati¨e center KŽ. A s Ä4x g A: xy s yx ᭙ y g A , which is 0 for any Okubo algebra and nonzero for para-Hurwitz algebras. Moreo¨er, two algebras A and B in case Ž.1 are isomorphic if and only if gAŽ.sg Ž. B and the same happens in cases Ž.Ž.3 a and Ž.Ž.3 b . Two para- Hurwitz algebrasŽŽ.. case 2 are isomorphic if and only if so are the correspond- ing Hurwitz algebras if and only if their norms are equi¨alent. Finally the isomorphism conditions for case 3Ž.c are shown in Theorem 4.5.
Proof. Just a few details need to be checked. If A is a para-Hurwitz algebra, then there is a multiplication ᭛ on A which makes it a Hurwitz algebra with unity element 1 and the product in A is given by xy s x᭛ y. Ä Ž. 4Ž. Ž2 . Ž . Then for x g A0s z g A: n 1, z s 0, gxsnx,x snx,x᭛xs 3 3 nxŽ.ŽŽ..Ž.,x᭛xsnx,ynx1s0, so gAsFgŽ.1sF. In the algebra Ž.Ž. Ž. Ž 2.Ž. A, it follows from 11 , 4 , and Lemma 3.1 that gusnu,u snu,¨ Ž. Ž2.Ž.2 Ž.32 Ž. sand g ¨ s gu sgu qnu s q1, so dimF 3gA s2. The condition for isomorphism among the algebras in caseŽ. 1 follows exactly as in the proof of Theorem 4.4, taking into account that these algebras are commutative. For the algebras in caseŽ. 3 it follows from 300 ALBERTO ELDUQUE
Theorem 4.4 andwx E-P96, Theorem 5.1 . Finally, if two composition alge- bras are isomorphic, their norms are equivalentwx P71 ; if moreover these are para-Hurwitz algebras this is equivalent to their Hurwitz counterparts being isomorphicŽ see alsowx O-O181, Lemma 3. .
ACKNOWLEDGMENTS
The author is very much indebted to Jose´´´ Marıa Perez, who made a thorough revision of a preliminary version of the paper and simplified several of the arguments, and to Professor Max A. Knus, who provided a draft copy ofwx KMRT . Proposition 3.3, which is crucial in this paper, was motivated byw KMRT,Ž. 36.11x .
REFERENCES wxE-M91 A. Elduque and H. C. Myung, Flexible composition algebras and Okubo algebras, Comm. Algebra 19 Ž.1991 , 1197᎐1227. wxE-M93 A. Elduque and H. C. Myung, On flexible composition algebras, Comm. Algebra 21 Ž.1993 2481᎐2505. wxE-P96 A. Elduque and J. M. Perez,´ Composition algebras with associative bilinear form, Comm. Algebra 24 Ž.1996 , 1091᎐1116. wxF88 J. R. Faulkner, Finding octonion algebras in associative algebras, Proc. Amer. Math. Soc. 104 Ž.1988 , 1027᎐1030. wxK53 I. Kaplansky, Infinite-dimensional quadratic forms admitting composition, Proc. Amer. Math. Soc. 4,Ž. 1953 , 956᎐960. wxKMRT M. A. Knus, A. S. Merkurjev, M. Rost, and J. P. Tignol, The book of involutions, to appear. wxM86 H. C. Myung, Malcev-admissible algebras, in ‘‘Progr. Math.,’’ Vol. 64, Birkhauser,¨ Boston, 1986. wxO78 S. Okubo, Pseudo-quaterion and pseudo-octonion algebras, Hadronic J. 1 Ž.1978 , 1250᎐1278. wxO95 S. Okubo, ‘‘Introduction to Octonion and Other Non-Associative Algebras in Physics,’’ Cambridge Univ. Press, Cambridge, UK, 1995. wxO-M80 S. Okubo and H. C. Myung, Some new classes of division algebras, J. Algebra 67 Ž.1980 , 479᎐490. wxO-O181 S. Okubo and J. M. Osborn, Algebras with nondegenerate associative symmetric bilinear form permitting composition, Comm. Algebra 9 Ž.1981 , 1233᎐1261. wxO-O281 S. Okubo and J. M. Osborn, Algebras with nondegenerate associative symmetric bilinear form permitting composition, II, Comm. Algebra 9 Ž.1981 , 2015᎐2073. wxP69 H. Petersson, Eine Identitat¨¨ funften Grades, der gewisse Isotope von Komposi- tions-Algebren genugen,¨ Math. Z. 109 Ž.1969 , 217᎐238. wxP71 H. Petersson, Quasi Composition Algebras, Abh. Math. Sem. Uni¨. Hamburg 35 Ž.1971 , 215᎐222. wxS52 T. A. Springer, Sur les formes quandratiques d’indice zero, C. R. Acad. Sci. Paris 234 Ž.1952 , 1517᎐1519. wxZSSS82 K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, ‘‘Rings That Are Nearly Associative,’’ Academic Press, New York, 1982.