On the axioms defining a quadratic Jordan division algebra
aa r X i v : . [ m a t h . R A ] S e p On the axioms defining a quadratic Jordandivision algebras
Matthias Gr¨uninger ∗ Universit´e catholique de LouvainInstitute pour la recherche en math´ematique et physiqueChemin du Cyclotron 2, bte. L7.01.021348 Louvain-la-Neuve, BelgiqueE-Mail:[email protected] 27, 2018
Abstract
Quadratic Jordan algebras are defined by identities that have to holdstrictly, i.e that continue to hold in every scalar extension. In this paper weshow that strictness is not required for quadratic Jordan division algebras.
Keywords : Quadratic Jordan algebras, Moufang sets
We begin with the classical definition of Jordan algebras.
Definition 1.1
Let k be a field with chark = 2 .(a) A commutative, unital k -algebra J is called a (linear) Jordan algebra if (J) a · ( ba ) = ( a · b ) · a holds for all a, b ∈ J .(b) A non-zero element a of Jordan algebra J is called invertible if there is anelement a − ∈ J with a · a − = 1 and a · a − = a .(c) A Jordan algebra is called a Jordan division algebra if every non-zeroelement is invertible. ∗ supported by the ERC (grant A be an associative k -algebra. Define a new multiplication ◦ on A by a ◦ b = 12 ( ab + ba ) . Then A + = ( J, ◦ ) is a Jordan algebra.As one can already see in this example, the constraint that chark = 2 is nec-essary. One can see that a commutative algebra over a field of characteristic 2which satisfies (J) is associative. But since Jordan algebras have been a usefultool to describe some algebraic groups that are also defined over fields of evencharacteristic, one has to alter the definition of a Jordan algebra to include thecase characteristic 2. In 1966 Kevin McCrimmon came up with new definitionfor Jordan algebras which works for a field of arbitrary characteristic (see [3]).For convenience, we first introduce some more definitions. Definition 1.2
Let k be a field of arbitrary characteristic.(a) If V, W are two vector-spaces or k , then a map Q : V → W is calledquadratic if Q ( tv ) = t Q ( v ) for all t ∈ k, v ∈ V and if there is a k -bilinearmap f : V × V → W with Q ( v + w ) = Q ( v ) + Q ( w ) + f ( v, w ) for all v, w ∈ V .(b) A quadratic algebra over k is a pair ( J, Q ) where J is a k -vectorspace and Q : J → End k ( J ) : a Q a is quadratic.(c) For a quadratic algebra ( J, Q ) and a, b ∈ J one defines the maps Q a,b , V a,b : J → J by cQ a,b = cQ a + b − cQ a − cQ b and cV a,b = bQ a,c for c ∈ J . Ofcourse, one has cV a,b = aV c,b for all a, b, c ∈ J . The map J × J → End k ( J ) : ( a, b ) V a,b is k -bilinear.(d) An element a ∈ J is called invertible if Q a is invertible. The element a − := aQ − a is called the inverse of a . We denote the se t of invertibleelements of J by J ∗ .(e) If K is an extension field of k , then one defines the quadratic algebra J K := ( K ⊗ k J, ˆ Q ) by ˆ Q P ni =1 t i ⊗ a i = P ni =1 t i Q a i + P i Let ( J, Q ) be a quadratic algebra over k and ∈ J := J \ { } .Then ( J, Q, is called a weak quadratic Jordan algebra if the following holdsfor all a, b ∈ J .(QJ1) Q = id J .(QJ2) Q a V a,b = V b,a Q a .(QJ3) Q bQ a = Q a Q b Q a . weak quadratic Jordan algebra is called a quadratic Jordan algebra if (QJ1)-(QJ3) hold strictly, i.e. if J K is a weak quadratic Jordan algebra for all extensionfields K/k . Remark 1.4 (a) If J is a weak quadratic Jordan algebra and a ∈ J is in-vertible, then we have Q a − = Q − a . Indeed, we have a = a − Q a and thuswith (QJ3) Q a = Q a − Q a = Q a Q a − Q a , hence Q − a = Q a − . Note that if a, b ∈ J are invertible, then aQ b and a − are also invertible.(b) (QJ1)-(QJ3) hold strictly iff their linearized versions holds. It is clear that(QJ1) always holds strictly if it holds, and (QJ2) and (QJ3) hold strictlyiff additionally(QJ2*) Q a V a ,b + Q a ,a V a ,b = V b,a Q a ,a + V b,a Q a .(QJ3*) Q bQ a ,bQ a ,a = Q a ,a Q b Q a + Q a Q b Q a ,a .(QJ3**) Q bQ a ,a + Q bQ a ,bQ a = Q a Q b Q a + Q a Q b Q a + Q a ,a Q b Q a ,a .hold for all a , a , b ∈ J . One sees easily that (QJ3) and (QJ3*) imply(QJ3**). If | k | ≥ , then then (QJ2*) follows from (QJ2), and if | k | ≥ , then (QJ3*) follows from (QJ3). Thus if | k | ≥ , then every weakquadratic Jordan algebra is a quadratic Jordan algebra.(c) The author doesn’t know an example of a weak quadratic Jordan algebrawhich is not a quadratic Jordan algebra.(d) Let J be a linear Jordan algebra over k . For a ∈ J define Q a : J → J by bQ a = − a · b + 2 a · ( a · b ) . Then ( J, Q, is a quadratic Jordanalgebra. If chark = 2 and J is a quadratic Jordan algebra, then we definea multiplication · on J by a · b = Q a,b . Then one can show that J is a linear Jordan algebra. Therefore for chark = 2 these two conceptscoincide (see for example [3]). Moreover, an element is invertible in thelinear Jordan algebra iff it is invertible in the quadratic Jordan algebra. Example 1.5 Let R be a unital, associative algebra over k . For a ∈ R define Q a : R → R : b bab . Then R + := ( R, Q, is a quadratic Jordan algebra. We now introduce the important concept of an isotope. Definition 1.6 Let J = ( J, Q ) be a quadratic algebra and a ∈ J ∗ . We definethe a -isotope J a = ( J, Q a ) of J by xQ ay = xQ − a Q y for all x, y ∈ J . Lemma 1.7 Let J be a quadratic algebra. For a, b, c ∈ J we have(a) Q aa = id J .(b) Q ab,c = Q − a Q b,c . c) V ab,c = V b,cQ − a . Proof. (a) and (b) are clear. For (c) let x ∈ J . Then we have xV ab,c = cQ ab,x = cQ − a Q b,x = xV b,cQ − a . Thus the claim follows. (cid:3) Proposition 1.8 Let J be a (weak) quadratic Jordan algebra and a ∈ J ∗ . Then J a = ( J, Q a , a ) is also a (weak) quadratic Jordan algebra. Proof. It is clear that (QJ1) holds. Let b, c, x ∈ J . Then we have xQ ab V ab,c = xQ − a Q b V b,cQ − a = xQ − a V cQ − a ,b Q b = bQ cQ − a ,xQ − a Q b = bQ − a Q c,x Q − a Q b = xV cbQ − a Q ab = xV ac,b Q ab . This shows (QJ2). Moreover, we have Q abQ ac = Q − a Q bQ − a Q c = Q − a Q c Q bQ − a Q c = Q − a Q c Q − a Q b Q − a Q c = Q ac Q ab Q ac . This shows (QJ3). Hence if J is a weak quadratic Jordan algebra, so is J a .Moreover, we have ( K ⊗ k J ) a = K ⊗ k J a for all extension fields K/k , thus J a is a quadratic Jordan algebra if J is. (cid:3) Definition 1.9 (a) If ( J, Q, and ( J ′ , Q ′ , ′ ) are weak quadratic Jordan al-gebras over k , then a Jordan homomorphism between J and J ′ is a homo-morphism f : J → J ′ such that f (1) = 1 ′ and f ( aQ b ) = f ( a ) Q ′ f ( b ) for all a, b ∈ J holds.(b) If ( J, Q, and ( J ′ , Q ′ , ′ ) are weak quadratic Jordan algebras over k , then J and J ′ are called isotopic iff J ′ is isomorphic to an isotope of J .(c) Let J ′ be a subspace of a weak quadratic Jordan algebra J . Then J ′ iscalled a Jordan subalgebra of J if e ∈ J and if J ′ Q a ⊆ J ′ for all a ∈ J ′ holds.(d) A quadratic Jordan algebra J is called special if there is an associative K -algebra R such that J is isomorphic to a Jordan subalgebra of R + . In this paper we are mainly interested in (weak) quadratic Jordan divisionalgebras. Definition 1.10 A (weak) quadratic Jordan algebra is called a (weak) quadraticJordan division algebra if every non-zero element in J is invertible. The theory of quadratic Jordan division algebras is connected with the theoryof Moufang sets. 4 efinition 1.11 A Moufang set consists of a set X with | X | ≥ and a family ( U x ) x ∈ X of subgroups in SymX such that the following holds:(a) For all x ∈ X the group U x fixes x and acts regularly on X \ { x } .(b) For all x, y ∈ X and all g ∈ U x we have U gy = U yg .The groups U x are called the root groups of the Moufang set. The group G † := h U x ; x ∈ X i is called the little projective group of the Moufang set. G † is a -transitive subgroup of SymX . The Moufang set is called proper if G † isnot sharply -transitive and improper else. Example 1.12 (a) Let X be a set with at least elements, G ≤ SymX bea sharply -transitive group. Then ( X, ( G x ) x ∈ X ) is an improper Moufangset with little projective group G † = G .(b) Let k be a field, X := P ( k ) and U x be the subgroup of P SL ( k ) ≤ SymX induced by the group of unipotent matrices that fix x . Then ( X, ( U x ) x ∈ X is a Moufang set with little projective group G † = P SL ( k ) . It is properiff | k | ≥ . The second construction can be generalized to weak quadratic Jordan divsionalgebras. In [2] the author showed the following. Theorem 1.13 Every weak quadratic Jordan division algebra defines a Mou-fang set M ( J ) with root groups isomorphic to ( J, +) . The algebra J is deter-mined by M ( J ) up to isotopy. De Medts and Weiss didn’t use the concept of a weak quadratic Jordan algebraand formulated their theorem for quadratic Jordan division algebras, but theirproof doesn’t make use of the strictness of (QJ1)-(QJ3), so it also holds for weakJordan division algebras.One of the big open problems concerning Moufang sets is the following conjec-ture: Conjecture 1.14 If ( X, ( U x ) x ∈ X ) is a proper Moufang set with U x abelian forall x ∈ X , then there is a field k and a quadratic Jordan division algebra J over k such that ( X, ( U x ) x ∈ X ) is isomorphic to M ( J ) . If 1.14 is true, then one has a classification of proper Moufang sets with abelianroot groups since quadratic Jordan division algebras have been classified byMcCrimmon and Zel’manov (see [4]). The proof follows from the classificationof simple quadratic Jordan algebras over an algebraically closed field, thereforeit is essential that scalar extensions are allowed. There has been progress inproving 1.14 (see [1]), but in general this conjecture is still open. However, ifthere would be a weak quadratic Jordan division algebra which is not a quada-tric Jordan algebra, then conjecture 1.14 would be false. Such an algebra couldexist over F or F . In this paper we will prove that no such algebra exists. MAIN THEOREM Every weak quadratic Jordan division algebra is a quadraticJordan algebra. 5 Some useful identities In the following let ( J, Q, 1) be a weak quadratic Jordan algebra. Lemma 2.1 yQ aQ x ,x = aQ yQ x ,x for all a, x, y ∈ J . Proof. (QJ2) implies xQ a,y Q x = yV a,x Q x = yQ x V x,a = aQ x,yQ x . Sincethe first expression is symmetric in a and y , so is the second. Hence we get aQ x,yQ x = yQ aQ x ,x . (cid:3) Lemma 2.2 For all x ∈ J we have Q x, = V x, = V ,x . Proof. By (QJ2) we have V x, = V x, Q = Q V ,x = V ,x . We have xQ ,y = yV ,x = yV x, = 1 Q x,y for all y ∈ j . Since the last expression is symmetric in x and y , so is the first. Thus we have yV ,x = xQ ,y = yQ ,x . (cid:3) Lemma 2.3 If a ∈ J ∗ and b ∈ J , then we have V x,a − = V a,xQ − a = Q − a Q x,a . Proof. We apply 2.2 for the isotope J a and have V ax,a = V aa,x = Q aa,x and therefore V x,a − = V a,xQ − a = Q − a Q a,x . (cid:3) Lemma 2.4 Q ,x Q x = Q x Q ,x for all x ∈ J . Proof. We have Q x Q ,x = Q x V x, = V ,x Q x = Q ,x Q x by (QJ2) and 2.2. (cid:3) Lemma 2.5 If x ∈ J ∗ , then Q − x V a,x = V x,a Q − x = Q a,x − for all a ∈ J . Proof. We have aQ yQ x ,x = yQ aQ x ,x = yQ x Q a,x − Q x for all a ∈ J by (QJ3)and 2.1. Replacing y by yQ − x , we get yV x,a = aQ y,x = yQ a,x − Q x . Thus thesecond equation follows. The first now follows from (QJ2). (cid:3) Lemma 2.6 If x, y ∈ J ∗ , then we have Q − x Q x + y Q − y = Q x − + y − . Proof. Using the previous lemma for x − and y , we have Q − x Q x,y Q − y = V y,x − Q − y = Q y − ,x − . Since Q − x Q x Q − y = Q − y and Q − x Q y Q − y = Q − x , we get Q − x Q x + y Q − y = Q − x ( Q x + Q y + Q x,y ) Q − y = Q − y + Q − x + Q x − ,y − = Q x − + y − . We will also make use of the following ”Hua-identity” for weak quadratic Jordandivision algebras. It was proved by De Medts and Weiss in [2] in order to showthat a quadratic Jordan division algebra defines a Moufang set. As mentionedbefore, the proof doesn’t make use of the strictness of (QJ1)-(QJ3), so it stillholds for weak quadratic Jordan division algebras. Theorem 2.7 Let J be a weak quadratic Jordan divsion algebra and a, b ∈ J ∗ with a = b − . Then we have aQ b = b − ( b − − ( b − a − ) − ) − . Definition 3.1 Let J be a weak quadratic Jordan algebra and ǫ ∈ { + , −} . Alinear map δ : J → J is called an ǫ -derivation if δ ( aQ b ) = ǫδ ( a ) Q b + aQ b,δ ( b ) holds for all a, b ∈ J . We will call the +-derivations just derivations and the − -derivations anti-derivations . Example 3.2 Let A be an associative algebra and J ⊆ A a special quadraticJordan algebra. If δ : A → A is a(n anti-)derivation of A with δ ( J ) ≤ J , then δ induces a(n anti-) derivation of J . Indeed, for a, b ∈ J we have δ ( aQ b ) = δ ( bab ) = δ ( b ) ab + ǫbδ ( ab ) = δ ( b ) ab + ǫbδ ( a ) b + ǫ baδ ( b ) = ǫδ ( a ) Q b + aQ b,δ ( b ) with ǫ = + if δ is a derivation and ǫ = − for δ an anti-derivation. Lemma 3.3 Let δ be an ǫ -derivation for ǫ = ± and a, b, c ∈ J . Then we have:(a) δ ( aQ b,c ) = ǫδ ( a ) Q b,c + aQ δ ( b ) ,c + aQ b,δ ( c ) and δ ( aV b,c ) = δ ( a ) V b,c + aV δ ( b ) ,c + ǫaV b,δ ( c ) for all a, b, c ∈ J .(b) If a ∈ J ∗ , then δ ( a − ) = − ǫδ ( a ) Q − a .(c) The identity is an anti-derivation.(d) If chark = 2 and δ a derivation, then δ (1) = 0 .(e) If chark = 2 and δ is a derivation, then Q ,δ (1) = 0 .(f ) If chark = 2 and δ is an anti-derivation, then δ ( a ) = aQ ,δ (1) . Proof. (a) The first equation follows by linearizing the defining property of an ǫ -derivation. The second equation can be obtained by the first.7b) We have δ ( a − ) = δ ( aQ a − ) = ǫδ ( a ) Q a − + aQ a − ,δ ( a − ) . Now aQ a − ,δ ( a − ) = aV a,δ ( a − ) Q − a = δ ( a − ) Q a,a Q − a = 2 δ ( a − ) by 2.5. Thus we get − δ ( a − ) = ǫδ ( a ) Q a − = ǫδ ( a ) Q − a .(c) We have id ( aQ b ) = aQ b = − id ( a ) Q b + aQ b,id ( b ) , which shows that theidentity is an anti-derivation.(d) We have δ (1) = δ (1 − ) = − δ (1) Q − = − δ (1), thus the claim follows.(e) For all a ∈ J we have δ ( a ) = δ ( aQ ) = δ ( a ) Q + aQ ,δ (1) = δ ( a ) + aQ ,δ (1) , hence the claim follows.(f) We have δ ( a ) = δ ( aQ ) = − δ ( a ) Q + aQ ,δ (1) = − δ ( a ) + aQ ,δ (1) and thus δ ( a ) = aQ ,δ (1) . (cid:3) We set D ǫ ( J ) := { δ ∈ End k ( J ); δ is an ǫ -derivation of J } and D ( J ) = D ( J ) + D − ( J ). If chark = 2, then we have D + ( J ) = D − ( J ) = D ( J ), while for chark = 2 we have D ( J ) = D + ( J ) ⊕ D − ( J ). We call the elements of D ( J ) generalized derivations . Lemma 3.4 For ǫ , ǫ ∈ { + , −} we have [ D ǫ ( J ) , D ǫ ( J )] ⊆ D ǫ ǫ ( J ) . Espe-cially D + ( J ) and D ( J ) are Lie subalgebras of End k ( J ) , and if chark = 2 , then D ( J ) is Z -graded. Proof. For i = 1 , δ i ∈ D ǫ i ( J ). For a, b ∈ J we have δ ( δ ( aQ b )) = δ ( ǫ δ ( a ) Q b + aQ b,δ ( b ) ) = ǫ ǫ δ ( δ ( a )) Q b + ǫ δ ( a ) Q b,δ ( b ) + ǫ δ ( a ) Q b,δ ( b ) + aQ δ ( b ) ,δ ( b ) + aQ δ ( δ ( b )) ,b and analogously δ ( δ ( aQ b )) = ǫ ǫ δ ( δ ( a )) Q b + ǫ δ ( a ) Q b,δ ( b ) + ǫ δ ( a ) Q b,δ ( b ) + aQ δ ( b ) ,δ ( b ) + aQ δ ( δ ( b )) ,b . Thus we get[ δ , δ ]( aQ b ) = ǫ ǫ δ ( δ ( a )) Q b − δ ( δ ( a )) Q b + aQ δ ( δ ( b )) ,b − aQ δ ( δ ( b )) ,b = ǫ ǫ [ δ , δ ]( a ) Q b + aQ [ δ ,δ ]( b ) ,b . Hence the claim follows. (cid:3) emma 3.5 If J is a quadratic Jordan algebra, then for all a ∈ J the map Q ,a is an anti-derivation of J . Proof. Q ,a is an anti-derivation iff for all b ∈ J we have Q b Q ,a = − Q ,a Q b + Q b,bQ ,a . But this is just identity (QJ3*) with a = 1 and a = a which holds by definitionin quadratic Jordan algebras. (cid:3) Remark 3.6 Let J be a linear Jordan algebra. For a, b, x ∈ J define the asso-ciator { a, x, b } := ( a · x ) · b − a · ( x · b ) . Now by 1.4(d) Q ,a corresponds to themap x a · x . Thus for all a, b ∈ J the map [ Q ,a , Q ,b ] is a derivation of J .Hence x [ Q ,a , Q ,b ] = 4( b · ( a · x ) − a · ( b · x )) = 4(( a · x ) · b − ( a · ( x · b ))) = 4 { a, x, b } . Thus the map x 7→ { a, x, b } is a derivation of J . Moreover, if δ is an anti-derivation of (the quadratic Jordan algebra) J , then δ ( a ) = 2 a · δ (1) by 3.3(f ).One can easily prove that a linear map δ : J → J is a derivation of (the quadraticJordan algebra) J iff δ ( a · b ) = δ ( a ) · b + a · δ ( b ) for all a, b ∈ J , which is theusual definition of a derivation of a linear Jordan algebra. Theorem 3.7 Let J be a weak quadratic Jordan algebra. Suppose that for all a, y ∈ J ∗ there is a generalized derivation δ with δ ( a ) = y . Then J is a quadraticJordan algebra. Proof. Let a ∈ J . We set L J ( a ) := { y ∈ J : V b,y Q a + V b,a Q a,y = Q a V y,b + Q a,y V a,b and Q bQ a ,b a,y = Q a Q b Q a,y + Q a,y Q b Q a for all b ∈ J } . Then L J ( a ) is a subspace of J . Now let b, x ∈ J . Then we have δ ( xQ a Q b Q a ) = ǫδ ( xQ a Q b ) Q a + xQ a Q b Q a,δ ( a ) = δ ( xQ a ) Q b Q a + ǫxQ a Q b,δ ( b ) Q a + xQ a Q b Q a,δ ( a ) = ǫδ ( x ) Q a Q b Q a + xQ a,δ ( a ) Q b Q a + ǫxQ a Q b,δ ( b ) Q a + xQ a Q b Q a,δ ( a ) . On the other side, δ ( xQ a Q b Q a ) = δ ( xQ bQ a ) = ǫδ ( x ) Q bQ a + xQ bQ a ,δ ( bQ a ) = ǫδ ( x ) Q a Q b Q a + xQ bQ a ,ǫδ ( b ) Q a + xQ bQ a ,bQ a,δ ( a ) = ǫδ ( x ) Q b Q a Q b + ǫxQ a Q b,δ ( b ) Q a + xQ bQ a ,bQ a,δ ( a ) . Hence we get Q a,δ ( a ) Q b Q a + Q a Q b Q a,δ ( a ) = Q bQ a ,bQ a,δ ( a ) . δ ( xV b,a Q a ) = ǫδ ( xV b,a ) Q a + xV b,a Q a,δ ( a ) = ǫδ ( x ) V b,a Q a + xV b,δ ( a ) Q a + ǫxV δ ( b ) ,a Q a + xV b,a Q a,δ ( a ) . On the other side, we have δ ( xV b,a Q a ) = δ ( xQ a , V a,b ) = δ ( xQ a ) V a,b + xQ a V δ ( a ) ,b + ǫxQ a V a,δ ( b ) = ǫδ ( x ) Q a V a,b + xQ a,δ ( a ) V a,b + xQ a V δ ( a ) ,b + ǫxQ a V a,δ ( b ) . Thus we get V b,δ ( a ) Q a + V b,a Q a,δ ( a ) = Q a V δ ( a ) ,b + Q a,δ ( a ) V a,b . This shows that δ ( a ) ∈ L J ( a ) for all δ ∈ D ǫ ( J ), ǫ = ± . Thus the claim follows. (cid:3) Theorem 4.1 Let J be a weak Jordan division algebra, ǫ ∈ { + , −} and δ ∈ End k ( J ) with δ ( a − ) = − ǫδ ( a ) Q − a for all a ∈ J ∗ . Then δ is an ǫ -derivation. Proof. Let a, b ∈ J ∗ with a = b − . Then we have by the Hua-identity aQ b = b − ( b − − ( b − a − ) − ) − . Thus we get δ ( aQ b ) = δ ( b ) − δ ( b − − ( b − a − ) − ) − ) = δ ( b ) + ǫ ( δ ( b − ) − δ (( b − a − ) − )) Q − b − − ( b − a − ) − = δ ( b ) + ǫ ( − ǫδ ( b ) Q − b + ǫδ ( b − a − ) Q − b − a − ) Q − b − − ( b − a − ) − = δ ( b ) − δ ( b ) Q − b Q − b − − ( b − a − ) − + δ ( b ) Q − b − a − Q − b − − ( b − a − ) − + ǫδ ( a ) Q − a Q − b − a − Q − b − − ( b − a − ) − . Now for x = b − and y = − ( b − a − ) − we have by 2.6 Q b − − ( b − a − ) − = Q b − Q b − ( b − a − ) Q − ( b − a − ) − = Q − b Q − a Q b − a − and with x = − ( b − a − ) − and y = b − we get Q b − − ( b − a − ) − = Q − ( b − a − ) − + b − = Q − b − a − Q − a Q − b . δ ( aQ b ) = δ ( b ) − δ ( b ) Q a Q b − a − + δ ( b ) Q a Q b + ǫδ ( a ) Q b = ǫδ ( a ) Q b + δ ( b ) Q a ( Q − a − − Q b − a − + Q b ) = ǫδ ( a ) Q b − δ ( b ) Q a Q b, − a − = ǫδ ( a ) Q b + δ ( b ) Q a Q a − ,b = ǫδ ( a ) Q b + δ ( b ) V b,a = ǫδ ( a ) Q b + aQ b,δ ( b ) . as desired.We still have to prove δ ( aQ b ) = ǫδ ( a ) + aQ b,δ ( b ) for b ∈ { , a − } . The statementis clear for b = 0, while we have by 2.5 ǫδ ( a ) Q a − + aQ a − ,δ ( a − ) = ǫδ ( a ) Q − a + aV a,δ ( a − ) Q − a = − ǫ δ ( a − ) + δ ( a − ) Q a,a Q − a = − δ ( a − ) + 2 δ ( a − ) = δ ( a − ) = δ ( aQ a − ) . (cid:3) Lemma 4.2 Let J be a weak quadratic Jordan division algebra. Then for all a ∈ J the map δ a = Q ,a is an anti-derivation of J . Proof. We have by 2.2, 2.4 and 2.5 δ a ( x − ) = x − Q ,a = x − V ,a = aQ ,x − = aQ − x V ,x = aV ,x Q − x = xQ ,a Q − x = δ a ( x ) Q − x for all x ∈ J ∗ . (cid:3) Corollary 4.3 For all a ∈ J ∗ and all b ∈ J the map Q aa,b = V aa,b = V ab,a is ananti-derivation of J a . For odd characteristic we can show that the converse of 3.5 holds. Thus 4.2implies that a weak quadratic Jordan division algebra in odd characteristic is aquadratic Jordan algebra. Theorem 4.4 Let J be a weak quadratic Jordan algebra over a field k with chark = 2 . Suppose that for all a ∈ J the map Q ,a is an anti-derivation of J . For a, b ∈ J define a · b = aQ ,b . Then ( J, + , · ) is a linear Jordan divisionalgebra. Thus J is a quadratic Jordan algebra. Proof. We have a · b = aQ ,b = aV ,b = bQ ,a = b · a by 2.2, so · iscommutative. Moreover, we have 1 · a = a · aQ , = aQ = a , so 1is the neutral element. It remains to show that a · ( b · a ) = ( a · b ) · a holdsfor all a, b ∈ J . Note that a = aQ ,a = Q a,a = 1 Q a . Since Q ,a · b is ananti-derivation, we have a · ( a · b ) = 12 1 Q a Q ,a · b = − 12 1 Q ,a · b Q a + 12 1 Q a,aQ ,a · b =11 ( a · b ) Q a + 12 aQ , a · ( a · b ) = − ( a · b ) Q a + 2 a · ( a · ( a · b )) . Moreover, we have( a · b ) · a = 14 1 Q a Q ,b Q ,a = − 14 1 Q ,b Q a Q ,a + 14 1 Q a,aQ ,b Q ,a = − bQ ,a Q a + 14 aQ , a · b Q ,a = − ( a · b ) Q a + ( a · ( a · b )) Q ,a = − ( a · b ) Q a + 2( a · ( a · b )) · a = − ( ab ) Q a + 2 a · ( a · ( a · b )) . Thus ( J, · ) is a linear Jordan algebra. (cid:3) The following proof works for a field in arbitrary characteristic. Theorem 4.5 A weak quadratic Jordan division algebra is a Jordan divisionalgebra. Proof. We have to show (QJ2*) and (QJ3*), the first only for k = F . Sincethese equalities automatically hold if one of the elements involved is zero, weonly have to show them for non-zero elements. So let a, b, c ∈ J ∗ . Since Q ca,b isan anti-derivation of J c by 4.3, we have Q ca Q cb,c = − Q cb,c Q ca + Q caQ cb,c ,a , hence Q − c Q a Q − c Q b,c = − Q − c Q b,c Q − c Q a + Q − c Q aQ − c Q b,c ,a . Multiplying Q c on the left yields Q a Q − c Q b,c + Q b,c Q − c Q a + Q aQ − c Q b,c ,a . Replacing a by aQ c and applying (QJ3) yields Q c Q a Q b,c + Q b,c Q a Q c = Q aQ b,c ,aQ c . This shows (QJ3*).We now show (QJ2*). Since Q cb,c = V cb,c = V c,b c is an anti-derivation of J c , wehave Q ca V cb,c = − V cb,c Q a + Q caV cb,c ,a and thus Q − c Q a V b,c − = − V b,c − Q − c Q a + Q − c Q aV b,c − ,a . Replacing c by c − and applying (QJ2) yields Q c Q a V b,c = − Q c V c,b Q a + Q c Q cQ b,a ,a . Multiplying Q − c on the left yields( ∗ ) Q a V b,c = − V c,b Q a + Q cQ b,a ,a . Q aa,c = V aa,c = V ac,a is an anti-derivation of J a , we have Q aa,b V aa,c = − V aa,c Q aa,b + Q aaV aa,c ,b + Q aa,bV aa,c = − V ac,a Q aa,b + Q aaV aa,c ,b + Q aa,bV aa,c . Hence we have Q − a Q a,b V a,cQ − a = − V c,a − Q − a Q a,b + Q − a Q c,b + Q − a Q a,bV a,cQ − a . Using (QJ2) and multiplying Q a on the left yields Q a,b V a,cQ − a = − V a − ,c Q a,b + 2 Q c,b + Q a,cQ − a Q a,b . Replacing c by cQ a yields Q a,b V a,c = − V a − ,cQ a Q a,b + 2 Q cQ a ,b + Q a,cQ a,b . By 2.3 we have V a − ,cQ a = V c,a . Hence we get( † ) Q a,b V a,c = − V c,a Q a,b + 2 Q cQ a ,b + Q a,cQ a,b . Adding ( ∗ ) and ( † ) yields Q a,b V a,c + Q a V b,c = − V c,b Q a − V c,a Q a,b + 2 Q cQ a ,b + 2 Q a,cQ a,b . This gives (QJ2*) for chark = 2. (cid:3) Let M = ( X, ( U x ) x ∈ X ) be a proper Moufang set with abelian root groups. M canbe written in the form M ( U, τ ) with U an (additively written) group isomorphicto a root group of M and τ a permutation of U ∪ {∞} interchanging 0 and ∞ ,where ∞ is a symbol not contained in U .By [5] M is special, so by [6], Thm. 5.2(a) U is either torsion free and uniquelydivisible or an elementary-abelian p -group for a prime p . We write charU = 0in the first case and charU = p in the second. We can view U as a k -vectorspacefor k = Q if charU = 0 and k = F p if charU = p .In order to give U the structure of a quadratic Jordan algebra, we need aquadratic map between U and End k ( U ). There is a natural candidate for thismap. Choose e ∈ U = U \ { } and set h a := µ e µ a for a ∈ U and h = 0.Let H : U → End k ( U ); a h a (see [1] for the definition of µ a ). Then ( U, H , e )satisfies (QJ1) and (QJ3) and one has h aτ = h − a and h a · s = h a · s for a ∈ U and all s ∈ k . Moreover, if ( U, H , e ) is a quadratic Jordan divsion algebra,then M ∼ = M ( U, τ ). It remains to show that (QJ2) holds and that the map( a, b ) h a,b = h a + b − h a − h b is biadditive. In [1] the authors showed thefollowing: Theorem 5.1 If charU = 2 , and if (QJ2) holds, then ( U, H , e ) is a quadraticJordan division algebra. charU ∈ { , } because (QJ1)-(QJ3) arerequired to hold strictly, which was only guaranteed if | k | ≥ 4. But our maintheorem shows that this is always the case for weak Jordan division algebras.Thus we get Corollary 5.2 charU ∈ { , } . Remark 5.3 (a) It is sufficient to prove a weaker version of axiom (QJ2)which has to hold in all isotopes of ( U, H , e ) , i.e. for all choices of e ∈ U \ { } , compare 5.6 of [1].(b) If charU = 2 , , then in order prove that ( U, H , e ) is a quadratic Jordandivision algebra, it is also sufficient to prove that the map ( a, b ) h a,b is biadditive (5.12 of [1]). In this case however the strictness is not theonly obstacle for charU ∈ { , } and therefore it is not yet clear if thestatement is also true in this case. References [1] T. De Medts, Y. Segev, Identities in Moufang sets , Trans. Amer. Math.Soc. , No. 11, 5831-5852 (2008)[2] T. De Medts, R. Weiss, Moufang sets and Jordan division algebras , Math.Ann. , No. 2, 415-433 (2006)[3] K. McCrimmon, A general theory of Jordan rings , Proc. Natl. Acad. Sci.USA , 1072-1079 (1966).[4] K. McCrimmon, E. Zel’manov, The structure of strongly prime Jordandivision algebras , Adv. Math. , 133-222 (1988)[5] Y. Segev, Proper Moufang sets with Abelian root groups are special , J. Am.Math. Soc. , No. 3, 889-908 (2009).[6] F. G. Timmesfeld, Abstract root subgroups and simple groups of Lie-type .Monographs in Mathematics.95