Fine gradings on Kantor systems of Hurwitz type
aa r X i v : . [ m a t h . R A ] A ug FINE GRADINGS ON KANTOR SYSTEMSOF HURWITZ TYPE
DIEGO ARANDA-ORNA AND ALEJANDRA S. C ´ORDOVA-MART´INEZ
Abstract.
We give a classification up to equivalence of the fine group gradingsby abelian groups on the Kantor pairs and triple systems associated to Hurwitzalgebras (i.e., unital composition algebras), under the assumption that the basefield is algebraically closed of characteristic different from 2. The universalgroups and associated Weyl groups are computed. We also determine, in thecase of Kantor pairs, the induced (fine) gradings on the associated Lie algebrasgiven by the Kantor construction. Definitions and preliminaries
Throughout this paper, we will always assume that the base field F is alge-braically closed of characteristic different from 2, unless otherwise stated.This paper is structured as follows:In this section we recall some basic definitions and results related to gradings,structurable algebras, Kantor systems (i.e., Kantor pairs and Kantor triple sys-tems), Peirce decompositions, and Hurwitz algebras; a few original results will beproven here too.In Section 2, some general results related to automorphisms and gradings onKantor systems are proven.Section 3 is aimed to study the automorphisms and orbits of Kantor systems ofHurwitz type. Unexpectedly, an exceptional case occurs for 2-dimensional Kantorpairs of Hurwitz type if char F = 3. This case, in Section 6, is shown to be relatedto the Lie algebra a , which is also exceptional if char F = 3.A classification of the fine gradings up to equivalence, for Kantor systems ofHurwitz type, is given in Section 4. The Weyl groups of these fine gradings arecomputed in Section 5, and the induced gradings on Lie algebras via the Kantorconstruction are described in Section 6.1.1. Gradings on algebras.
Now we will recall the basic definitions of gradingson algebras.Let A be an F -algebra (not necessarily associative) and G a group. A G -grading on A is a vector space decompositionΓ : A = M g ∈ G A g such that A g A h ⊆ A gh for all g, h ∈ G . Given a G -grading on A , we will alsosay that A is a G -graded algebra . The nonzero elements x ∈ A g are said to be The second author is supported by grant MTM2017-83506-C2-1-P (AEI/FEDER, UE). Andacknowledges support by grant S60 20R (Gobierno de Arag´on, Grupo de investigaci´on Investi-gaci´on en Educaci´on Matem´atica). homogeneous of degree g , and we have a degree map deg Γ ( x ) = g (also denotedby deg if there is no ambiguity with other gradings). The subspace A g is called homogeneous component of degree g . The set Supp Γ := { g ∈ G | A g = 0 } is calledthe support of the grading.Recall that an involution is an F -linear antiautomorphism of order 2. For thecase of an algebra with involution ( A , − ), we will also require the homogeneouscomponents to be invariant by the involution, that is, A g = A g for each g ∈ G .Let Γ : A = L g ∈ G A g and Γ ′ : A = L h ∈ H A ′ h be two gradings on A . Then wewill say that Γ is a refinement of Γ ′ , or that Γ ′ is a coarsening of Γ, if for any g ∈ G there exists h ∈ H such that A g ⊆ A ′ h ; if the inclusion is strict for some g ∈ G ,then we will say that we have a proper refinement or coarsening. A grading is fine if it has no proper refinements.If Γ is a grading on a finite-dimensional algebra A , a sequence of natural numbers( n , n , . . . ) is called the type of Γ if n i is the number of homogeneous componentsof dimension i , for i ∈ N .If Γ : A = L g ∈ G A g and Γ ′ : A = L h ∈ H A ′ h are gradings on A , we will say thatΓ and Γ ′ are compatible if A = L g ∈ G,h ∈ H A ( g,h ) where A ( g,h ) := A g ∩ A ′ h . Notethat compatibility implies that the subspaces A ( g,h ) define a G × H -grading on A .A subspace V of A is said to be graded for Γ if V = L g ∈ G ( V ∩ A g ).The choice of a grading group G for a grading Γ is not unique, so it is convenientto consider gradings by their universal group. Recall that a group G is said to bea universal group of a grading Γ on A if it satisfies the following universal property:for any other realization of Γ as a G -grading, there exists a unique homomorphism G → G that restricts to the identity on Supp Γ (with the natural identification ofboth supports). It is well-known that universal groups always exist and are uniqueup to isomorphism. We will denote the universal group of Γ by U (Γ). Note that U (Γ) is the group generated by Supp Γ with the defining relations g g = g for all g , g , g ∈ Supp Γ such that 0 = A g A g ⊆ A g .If Γ can be realized as an abelian group grading, then its universal abelian group can be defined similarly, by adding the restriction of being abelian. In this paperwe will only consider gradings by abelian groups, so that additive notation willbe used for their products, and by universal group we will mean universal abeliangroup.Let Γ be a G -grading on an algebra A and Γ ′ an H -grading on an algebra B .Then Γ and Γ ′ are said to be equivalent if there exist an isomorphism of algebras ϕ : A → B and a bijection α : Supp Γ → Supp Γ ′ such that ϕ ( A g ) = B α ( g ) for all g ∈ Supp Γ.A degree map related to the universal property that defines U (Γ) will be calleda universal degree map of the U (Γ)-grading Γ. Note that Γ may be realizable as a U (Γ)-grading with different degree maps, and some may not satisfy the universalproperty that defines U (Γ). Since U (Γ) is unique up to isomorphism, it followsthat the universal degree is unique up to equivalence of gradings; in other words,deg and deg ′ are universal degrees of Γ as a U (Γ)-grading if and only if there is ϕ ∈ Aut( U (Γ)) such that deg ′ = ϕ ◦ deg. When we say that a G -grading Γ, withdegree map deg, is given by its universal group , we usually mean that G ∼ = U (Γ)and deg is equivalent to the universal degree of Γ. INE GRADINGS ON KANTOR SYSTEMS OF HURWITZ TYPE 3 If V is a vector space and G an abelian group, a decomposition Γ : V = L g ∈ G V g will be called a G -grading on V . Let Γ ′ : W = L g ∈ G W g be a G -grading on a vectorspace W and g ∈ G , then a linear map f : V → W is said to be homogeneous ofdegree g if f ( V h ) ⊆ W g + h for each h ∈ G . In that case, ker f and im f are gradedsubspaces for Γ and Γ ′ , respectively.Given a G -grading Γ on A , the automorphism group of Γ, Aut(Γ), is the group ofself-equivalences of Γ. The stabilizer of
Γ, Stab(Γ), is the group of G -automorphismsof Γ, i.e., the group of automorphisms of A that fix the homogeneous components.The diagonal group of Γ, Diag(Γ), is the subgroup of Stab(Γ) consisting of theautomorphisms that act by multiplication by a nonzero scalar on each homogeneouscomponent. The
Weyl group of
Γ is the quotient group W (Γ) = Aut(Γ) / Stab(Γ),which can be regarded as a subgroup of Sym(Supp Γ) and of Aut( U (Γ)).In this paper, fine gradings will be classified up to equivalence and by theiruniversal abelian groups. Recall that for finite-dimensional algebras, the non-finegradings can be obtained by computing the coarsenings of the fine gradings, andthese are induced by quotients of the universal groups.The definitions of affine group schemes and their relation with gradings can beconsulted in [EK13, Appendix A].1.2. Kantor pairs and Kantor triple systems.
Let ( A , − ) be an F -algebra withinvolution. Then A = H ( A , − ) ⊕ S ( A , − ), where: H ( A , − ) = { a ∈ A | ¯ a = a } and S ( A , − ) = { a ∈ A | ¯ a = − a } . The subspaces H ( A , − ) and S ( A , − ) are called, respectively, the hermitian (or sym-metric ) subspace and the skew-symmetric subspace. Definition 1.1.
A unital F -algebra with involution ( A , − ) is said to be structurable if(1.1) [ V x,y , V z,w ] = V V x,y z,w − V z,V y,x w for all x, y, z ∈ A , where V x,y ( z ) = { x, y, z } := ( x ¯ y ) z + ( z ¯ y ) x − ( z ¯ x ) y . The U -operator is defined by U x,z ( y ) := { x, y, z } and U x := U x,x .Note that, in the literature, fields of characteristic 3 are usually excluded in thedefinition of structurable algebra; this is due to problems arising in characteristic 3,including some results of general theory that may not hold in that case. However,here we will include that case.Recall that the associative center of A , denoted by Z ( A ), is the set of elements z ∈ A satisfying the equalities xz = zx and ( z, x, y ) = ( x, z, y ) = ( x, y, z ) = 0 forall x, y ∈ A ), where ( x, y, z ) := ( xy ) z − x ( yz ) denotes the associator. The center of ( A , − ) is defined by Z ( A , − ) = Z ( A ) ∩ H ( A , − ), and a structurable algebra A issaid to be central if Z ( A , − ) = F Theorem 1.2 (Allison, Smirnov) . If char F = 2 , , , then any central simple struc-turable F -algebra belongs to one of the of the following six (non-disjoint) classes: (1) central simple associative algebras with involution, (2) central simple Jordan algebras (with identity involution), (3) structurable algebras constructed from a non-degenerate Hermitian formover a central simple associative algebra with involution, D. ARANDA-ORNA AND A.S. C ´ORDOVA-MART´INEZ (4) forms of the tensor product of two Hurwitz algebras, (5) simple structurable algebras of skew-dimension 1 (forms of structurable ma-trix algebras), (6) an exceptional 35-dimensional case (Kantor-Smirnov algebra), which canbe constructed from an octonion algebra. (cid:3)
This classification was given by Allison in the case of characteristic 0 ([Al78]),where case (6) was overlooked. Smirnov completed the classification and gave thegeneralization for the case with char F = 2 , , Definition 1.3. A Kantor pair (or generalized Jordan pair of second order [F94,AF99]) is a pair of vector spaces V = ( V + , V − ) and a pair of trilinear products V σ × V − σ × V σ → V σ , denoted by { x, y, z } σ , satisfying the identities:[ V σx,y , V σz,w ] = V σV σx,y z,w − V σz,V − σy,x w , (1.2) K σK σx,y z,w = K σx,y V − σz,w + V σw,z K σx,y , (1.3)where V σx,y z = U σx,z ( y ) := { x, y, z } σ , U σx := U σx,x and K σx,y z = K σ ( x, y ) z := { x, z, y } σ − { y, z, x } σ . The map V σx,y is sometimes denoted by D σx,y or D σ ( x, y ),because ( V + x,y , − V − y,x ) is a derivation of the Kantor pair. The superscript σ willalways take the values + and − , and may be omitted when there is no ambiguity. Definition 1.4. A Kantor triple system (or generalized Jordan triple system ofsecond order [K72, K73]) is a vector space T with a trilinear product T × T × T → T ,denoted by { x, y, z } , which satisfies:[ V x,y , V z,w ] = V V x,y z,w − V z,V y,x w , (1.4) K K x,y z,w = K x,y V z,w + V w,z K x,y , (1.5)where V x,y z = U x,z ( y ) := { x, y, z } , U x := U x,x and K x,y z := { x, z, y } − { y, z, x } .Given a structurable algebra A , we can define its associated Kantor triple systemas the vector space A endowed with the triple product { x, y, z } of A . Similarly,with two copies of a Kantor triple system T and two copies of its triple productwe can define the associated Kantor pair V = ( T , T ). In particular, a structurablealgebra A with its triple product defines a Kantor pair ( A , A ). Note that Jordanpairs (respectively, Jordan triple systems) are particular cases of Kantor pairs (re-spectively, Kantor triple systems); they are exactly those where K x,y = 0 for all x, y .We will now recall (see [AF99, § K ( V ) := K ( V ) − ⊕ K ( V ) − ⊕ K ( V ) ⊕ K ( V ) ⊕ K ( V ) , INE GRADINGS ON KANTOR SYSTEMS OF HURWITZ TYPE 5 where K ( V ) − = (cid:18) K ( V − , V − )0 0 (cid:19) , K ( V ) − = (cid:18) V − (cid:19) , K ( V ) = span (cid:26)(cid:18) D ( x − , x + ) 00 − D ( x + , x − ) (cid:19) | x σ ∈ V σ (cid:27) , K ( V ) = (cid:18) V + (cid:19) , K ( V ) = (cid:18) K ( V + , V + ) 0 (cid:19) . Then, the vector space S ( V ) := K ( V ) − ⊕ K ( V ) ⊕ K ( V ) = span (cid:26)(cid:18) D ( x − , x + ) K ( y − , z − ) K ( y + , z + ) − D ( x + , x − ) (cid:19) | x σ , y σ , z σ ∈ V σ (cid:27) is a subalgebra of the Lie algebraEnd (cid:18) V − V + (cid:19) = (cid:18) End( V − ) Hom ( V + , V − )Hom ( V − , V + ) End( V + ) (cid:19) , with the commutator product. Now define an anti-commutative product on K ( V )by means of[ A, B ] = AB − BA, [ A, (cid:18) x − x + (cid:19) ] = A (cid:18) x − x + (cid:19) , [ (cid:18) x − x + (cid:19) , (cid:18) y − y + (cid:19) ] = (cid:18) D ( x − , y + ) − D ( y − , x + ) K ( x − , y − ) K ( x + , y + ) − D ( y + , x − ) + D ( x + , y − ) (cid:19) where x σ , y σ ∈ V σ and A, B ∈ K ( V ) i for i = − , ,
2. Then K ( V ) becomes a Liealgebra, called the Kantor Lie algebra of V . The 5-grading is a Z -grading which iscalled the standard grading on K ( V ), but we will also refer to it as the main grading on K ( V ). The subspaces K ( V ) and K ( V ) − are usually identified with V + and V − ,respectively. The Kantor construction of a structurable algebra or Kantor triplesystem is defined as the Kantor construction of the associated Kantor pair.Conversely, it is well-known (see [AFS17, § L = L i ∈ Z L i produces a Kantor pair V = ( L − , L ) with tripleproducts defined by { x σ , y − σ , z σ } := [[ x σ , y − σ ] , z σ ] . Let A be a structurable algebra and V = ( V + , V − ) = ( A , A ) the associatedKantor pair. Recall that ν ( x − , x + ) := ( D x − ,x + , − D x + ,x − ) is a derivation called inner derivation associated to ( x − , x + ) ∈ V − × V + . The inner structure algebra of A is the Lie algebra innstr ( A ) := span { ν ( x, y ) | x, y ∈ A } . Let L x denote the leftmultiplication by x ∈ A and write S = S ( A ). Then, the map S → L S , s L s , isa linear monomorphism, thus we can identify S with L S . Also, note that the map A × A → S given by ψ ( x, y ) := x ¯ y − y ¯ x is an epimorphism (because ψ ( s,
1) = 2 s for s ∈ S ). By [AF84, (1.3)], we have the identity L ψ ( x,y ) = U x,y − U y,x = K ( x, y ) forall x, y ∈ A . Consequently, we can identify the subspaces K ( V ) and K ( V ) − with L S , and also with S . This allows to write the main grading on K ( A ) in a well-knownsecond form, as follows:(1.7) K ( A ) = S − ⊕ A − ⊕ innstr ( A ) ⊕ A + ⊕ S + . D. ARANDA-ORNA AND A.S. C ´ORDOVA-MART´INEZ
This construction can be used to induce gradings on the Kantor Lie algebra fromgradings on a structurable algebra, Kantor pair, or Kantor triple system. That isone of the aims of this work, where the particular case related to Hurwitz algebrasis studied.Finally, note that for a Kantor pair V , each automorphism ϕ ∈ Aut( V ) has anatural extension e ϕ ∈ Aut( K ( V )) whose action on S ( V ) is given by(1.8) A = (cid:18) A A A A (cid:19) (cid:18) ϕ − A ( ϕ − ) − ϕ − A ( ϕ + ) − ϕ + A ( ϕ − ) − ϕ + A ( ϕ + ) − (cid:19) for each A ∈ S ( V ), and we have that Aut ( V ) ≤ Aut ( K ( V )).1.3. Gradings on Kantor pairs and Kantor systems.Definition 1.5.
Let G be an abelian group. Given two decompositions of vec-tor spaces Γ σ = L g ∈ G V σg , we will say Γ = (Γ + , Γ − ) is a G -grading on V if { V σg , V − σh , V σk } ⊆ V σg + h + k for any g, h, k ∈ G and σ ∈ { + , −} . The vector space V + g ⊕ V − g is the homogeneous component of degree g . If 0 = x ∈ V σg we say x is homogeneous of degree g and we write deg( x ) = g .The rest of definitions related to gradings (universal group, equivalence andisomorphism of gradings, etc) are analogous to the algebra case.Let Γ be a G -grading on a Kantor pair V with degree deg. Fix g ∈ G . Forany homogeneous elements x + ∈ V + and y − ∈ V − , set deg g ( x + ) := deg( x + ) + g ,deg g ( y − ) := deg( y − ) − g . This defines a new G -grading, which will be denotedby Γ [ g ] and called the g -shift of Γ. Note that, although Γ and Γ [ g ] may fail to beequivalent (because the shift may collapse or split a homogeneous subspace of V + with another of V − ), the intersection of their homogeneous components with V σ coincide for each σ . It is clear that (Γ [ g ] ) [ h ] = Γ [ g + h ] . Similarly, if Γ is a G -gradingon a Kantor triple system T and g ∈ G has order 1 or 2, we can define the g -shift Γ [ g ] with the new degree deg g ( x ) := deg( x ) + g .A G -grading on K ( V ) is called Kantor-compatible if K ( V ) − and K ( V ) are G -graded spaces. Note that in this case K ( V ) = span[ K ( V ) , K ( V ) − ], K ( V ) =span[ K ( V ) , K ( V ) ] and K ( V ) − = span[ K ( V ) − , K ( V ) − ] are graded too. In otherwords, a grading on K ( V ) is Kantor-compatible if and only if it is compatible withthe Z -grading associated to the Kantor construction. A G -grading Γ on V canbe extended to a Kantor-compatible G -grading E G (Γ) : K ( V ) = L g ∈ G K ( V ) g bysetting K ( V ) g = L i = − K ( V ) ig where K ( V ) ig := K ( V ) i ∩ K ( V ) g . Note that we have: K ( V ) − g = span (cid:26)(cid:18) K ( V − g , V − g )0 0 (cid:19) | g + g = g (cid:27) , K ( V ) − g = (cid:18) V − g (cid:19) , K ( V ) g = span (cid:26)(cid:18) D ( x − , x + ) 00 − D ( x + , x − ) (cid:19) | x σ ∈ V σ , deg( x + ) + deg( x − ) = g (cid:27) , K ( V ) g = (cid:18) V + g (cid:19) , K ( V ) g = span (cid:26)(cid:18) K ( V + g , V + g ) 0 (cid:19) | g + g = g (cid:27) . Conversely, any Kantor-compatible G -grading e Γ on K ( V ) restricts to a G -grading R G ( e Γ) on V , because { x σ , y − σ , z σ } = [[ x σ , y − σ ] , z σ ] for x σ , z σ ∈ V σ , y − σ ∈ V − σ and σ ∈ { + , −} . (We identify x − with (cid:18) x − (cid:19) and x + with (cid:18) x + (cid:19) for all x σ ∈ V σ .) INE GRADINGS ON KANTOR SYSTEMS OF HURWITZ TYPE 7
Denote by Grad G ( V ) the set of G -gradings on V , and by KGrad G ( K ( V )) theset of Kantor-compatible G -gradings on K ( V ). We will call E G : Grad G ( V ) → KGrad G ( K ( V )) the extension map and R G : KGrad G ( K ( V )) → Grad G ( V ) the re-striction map .The following result is an extension of the Jordan case in [Ara17, Th. 2.13]. Proposition 1.6.
Let V be a Kantor pair with associated Lie algebra K ( V ) andlet G be an abelian group. Then, the maps E G and R G are inverses of each other.Coarsenings are preserved by the correspondence, that is, given a G i -grading Γ i on V with extended G i -grading e Γ i = E G (Γ i ) on K ( V ) for i = 1 , , and a homomorphism α : G → G , then Γ = α Γ if and only if e Γ = α e Γ . Let Γ be a G -grading on V .If G = U (Γ) , then G = U ( E G (Γ)) . Moreover, Γ is fine and G = U (Γ) if and onlyif E G (Γ) is fine and G = U ( E G (Γ)) .Proof. By construction E G and R G are inverses of each other.Assume that Γ = α Γ for some homomorphism α : G → G . Since V ± g ⊆ V ± α ( g ) for any g ∈ G , we have K ( V ) ± g ⊆ K ( V ) ± α ( g ) and K ( V ) i + jg = X g + g = g [ K ( V ) ig , K ( V ) jg ] ⊆ X g + g = g [ K ( V ) iα ( g ) , K ( V ) jα ( g ) ] ⊆ K ( V ) i + jα ( g ) for i, j ∈ { , − } . Then K ( V ) g ⊆ K ( V ) α ( g ) for any g ∈ G . Hence e Γ refines e Γ and e Γ = α e Γ . Conversely, if e Γ = α e Γ , by restriction we obtain Γ = α Γ . This provesthat the coarsenings are preserved.Consider e Γ = E G (Γ) with G = U (Γ). Note that Supp Γ generates U (Γ) and U ( e Γ). Since the U ( e Γ)-grading e Γ restricts to Γ as a U ( e Γ)-grading, there is a uniquehomomorphism G = U (Γ) → U ( e Γ) which restricts to the identity in Supp (Γ).Conversely, Γ extends to e Γ as a G -grading, so there is a unique homomorphism U ( e Γ) → G which restricts to the identity map in Supp ( e Γ) = Supp (Γ). Thereforethe compositions G → U ( e Γ) → G and U ( e Γ) → G → U ( e Γ) are the identity map,and G = U ( e Γ).Suppose again that e Γ = E G (Γ). Note that Γ is a fine G -grading on V with G = U (Γ) if and only if Supp Γ generates G and Γ satisfies the following property:if Γ = α Γ for some G -grading Γ on V , where G is generated by Supp Γ and α : G → G is an epimorphism, then α is an isomorphism. The same is true for Kantor-compatible gradings. Since the coarsenings are preserved in the correspondence,so does this property (also note that Supp Γ generates G if and only if Supp e Γgenerates G ). Then we get that Γ is fine and G = U (Γ) if and only if E G (Γ) isfine in the class of Kantor-compatible gradings and G = U ( E G (Γ)). Moreover, if e Γis fine in the class of Kantor-compatible gradings, then the supports of K ( V ) i for i = − , − , , , e Γ is also fine in the class of all abeliangroup gradings on K ( V ). Therefore Γ is fine and G = U (Γ) if and only if E G (Γ) isfine and G = U ( E G (Γ)). (cid:3) Note that if Γ is a grading on a Kantor pair V given by its universal group G = U (Γ), since automorphisms extend from V to K ( V ), we have that W (Γ) ≤ W ( E G (Γ)). D. ARANDA-ORNA AND A.S. C ´ORDOVA-MART´INEZ
Gradings on Hurwitz algebras.
Recall that for a Hurwitz algebra C , theequation(1.9) x − n ( x, x + n ( x )1 = 0is satisfied by any x ∈ C , where n denotes both the norm of C and its polar form.The norm is nondegenerate and multiplicative ( n ( xy ) = n ( x ) n ( y ) for any x, y ∈ C ),and the trace of x ∈ C is defined by t ( x ) := n ( x, C has a homogeneous basis B CD ( C ) := { x g } g ∈ Z associatedto a (fine) Z -grading, with degree map deg( x g ) := g , and such that the product isgiven by(1.10) x g x h := σ ( g, h ) x g + h with σ ( g, h ) = σ g,h := ( − ψ ( g,h ) ,ψ ( g, h ) := h g g + g h g + g g h + X i ≤ j g i h j , for any g = ( g , g , g ), h = ( h , h , h ) ∈ Z . The multiplication constants of thisbasis satisfy the property σ g,h + k = σ g,h σ g,k for any g, h, k ∈ Z ; in other words, σ g ∈ b Z for all g ∈ Z , where we denote σ g ( h ) := σ g,h for h ∈ Z . Moreover, x = 1, B CD ( C ) is an orthonormal basis relative to the norm, and the involution isgiven by ¯ x g = σ g,g x g for any g ∈ Z . We will refer to B CD ( C ) as a Cayley-Dicksonbasis of C .Furthermore, for each subgroup Z m ∼ = H ≤ Z with m ∈ { , , } , we have that C := span { x h } h ∈ H is an H -graded Hurwitz algebra of dimension 2 m , and the basis B CD ( C ) of C inherits the good properties from the basis B CD ( C ); we will say that B CD ( C ) is a Cayley-Dickson basis of C .Let { a i } i =1 be the canonical basis of Z . Then, if we consider Z with the ordergiven by (0 , a , a , a + a , a , a + a , a + a , a + a + a ) , the multiplication constants for the Cayley-Dickson basis on C are given by(1.11) ( σ g,h ) g,h ∈ Z = − − − − − − − −
11 1 − − − − − − − − − − − − − − − − − − − − , where the highlighted submatrices above correspond to the constants for the casesdim C = 1 , , B Z ( C ) := { e , e , u , u , u , v , v , v } of the split Cayley algebra C where the multiplication is given as in Figure 1 ([EK13, § Cartan basis of C .If C is a Hurwitz algebra with dim C = 1 , ,
4, respectively, then we define the
Cartan basis B Z ( C ) of C as { e } , { e , e } or { e , e , u , v } , respectively, with thesame products as in Figure 1. INE GRADINGS ON KANTOR SYSTEMS OF HURWITZ TYPE 9 e e u u u v v v e e u u u e e v v v u u v − v − e u u − v v − e u u v − v − e v v − e u − u v v − e − u u v v − e u − u Figure 1.
Multiplication for the Cartan basisRecall from [EK13, § Z -grading on C , where the Cartanbasis is homogeneous and the degree map is given by(1.12) deg( e ) = (0 ,
0) = deg( e ) , deg( u ) = (1 ,
0) = − deg( v ) , deg( u ) = (0 ,
1) = − deg( v ) , deg( v ) = (1 ,
1) = − deg( u ) . If dim C = 4, we have a fine Z -grading on C where the Cartan basis is homogeneouswith degree map given by deg( e i ) = 0 for i = 1 , u ) = 1 = − deg( v ).For the cases where dim C = 1 ,
2, the only grading on C where the Cartan basis ishomogeneous is the trivial grading (see [EK13, Remark 4.16]).It is straightforward to see that we can obtain a Cayley-Dickson basis from aCartan basis via the expressions:(1.13) x = e + e , x a = − i ( e − e ) ,x a = u + v , x a + a = − i ( u − v ) ,x a = u + v , x a + a = i ( u − v ) ,x a + a = − ( u + v ) , x a + a + a = − i ( u − v ) , where i is a square root of − F , or equivalently:(1.14) e = ( x + i x a ) / , e = ( x − i x a ) / ,u = ( x a + i x a + a ) / , v = ( x a − i x a + a ) / ,u = ( x a − i x a + a ) / , v = ( x a + i x a + a ) / ,u = − ( x a + a − i x a + a + a ) / , v = − ( x a + a + i x a + a + a ) / . Notation 1.7.
In order to study gradings on Kantor pairs and triple systems infurther sections, it may be convenient to introduce the following notation, which isbetter suited to describe the new symmetries appearing.Let i be a square root of − F . Consider the parity operator | a | := ord( a ) − ∈{ , } for a ∈ Z ; also note that(1.15) i | a | + | b | = σ a,b i | a + b | and ( − | a | = σ a,a for a, b ∈ Z , where σ a,b denotes the corresponding multiplication constant for aCayley-Dickson basis of the 2-dimensional Hurwitz algebra. Let C be a Hurwitz algebra with dim C = 2 m > B CD ( C ) = { x g } g ∈ Z m . Using the same basis { a i } mi =1 associated to (1.11), we will denote x ag := x ( g,a ) for ( g, a ) ∈ Z m − × Z ≡ Z m , where the group Z is generated by a , and onthe other hand, Z m − is generated by { a , a } if m = 3 and by { a } if m = 2. It isalso straightforward to see that(1.16) σ a,k σ k,a = ( − | a || k | = σ | k | a,a = σ | a | k,k for each ( k, a ) ∈ Z m − × Z . We will denote(1.17) v αg := √ X a ∈ Z α ( a ) σ a,g i | a | x ag , for each g ∈ Z m − and α ∈ b Z . We can recover the original Cayley-Dickson basisvia the expression:(1.18) x ag = √ X α ∈ b Z α ( a ) σ a,g ( − i ) | a | v αg , for each ( g, a ) ∈ Z m − × Z . Indeed, √ X α ∈ b Z α ( a ) σ a,g ( − i ) | a | v αg = 12 X α ∈ b Z α ( a ) σ a,g ( − i ) | a | X b ∈ Z α ( b ) σ b,g i | b | x bg = X b ∈ Z σ a,g σ b,g ( − i ) | a | i | b | X α ∈ b Z α ( a + b ) x bg = X b ∈ Z σ a,g σ b,g ( − i ) | a | i | b | δ a,b x bg = ( − | a | i | a | + | a | x ag = σ a,a i | a + a | x ag = x ag , where we have used (1.15).Also, it is easy to see from (1.13) that the basis { v αg | g ∈ Z m − , α ∈ b Z } is ahomogeneous basis for the Cartan grading because when m = 3 we have that:(1.19) v = e / √ , v ω = e / √ , v a = u / √ , v ωa = v / √ ,v a = u / √ , v ωa = v / √ , v a + a = − u / √ , v ωa + a = − v / √ , where b Z = h ω i .Furthermore, we claim that(1.20) x hg v αk = ( αω | k | σ k )( h ) σ ( g,h ) , ( k,h ) i | h | v αω | g | g + k ∈ F v αω | g | g + k INE GRADINGS ON KANTOR SYSTEMS OF HURWITZ TYPE 11 for each g, k ∈ Z m − , h ∈ Z , α ∈ b Z , where b Z = h ω i . Indeed, we can first checkthat: x hg v αk = x hg √ X a ∈ Z α ( a ) σ a,k i | a | x ak = √ X a ∈ Z α ( a ) σ a,k σ ( g,h ) , ( k,a ) i | a | x h + ag + k = √ X a ∈ Z α ( a + h ) σ a + h,k σ ( g,h ) , ( k,a + h ) i | a + h | x ag + k = α ( h ) σ ( g,h ) , ( k,h ) i | h | √ X a ∈ Z (cid:0) ασ ( g,h ) σ h (cid:1) ( a ) ·· h σ k,a + h ( − | a + h || k | ih σ a,g + k σ g + k,a ( − | a || g + k | i i | a | x ag + k = α ( h ) σ ( g,h ) , ( k,h ) σ k,h i | h | √ X a ∈ Z (cid:0) ασ ( g,h ) σ g + k σ h σ k (cid:1) ( a ) ·· σ a,g + k ω | g + k | ( a ) ω | k | ( a + h ) i | a | x ag + k = ( αω | k | σ k )( h ) σ ( g,h ) , ( k,h ) i | h | v ασ ( g,h ) σ g + k σ h σ k ω | g + k | + | k | g + k where we have used (1.15), (1.16) and the property ( − | a | = ω ( a ) for a ∈ Z . Ifwe denote ψ := ασ ( g,h ) σ g + k σ h σ k ω | g + k | + | k | , we need to prove that ψ = αω | g | . Thesecond column of σ in (1.11) shows that σ ( g,h ) ( a ) = ( σ g σ h )( a ) for a ∈ Z = h a i ,so that ψ = ασ g σ g + k σ k ω | g + k | + | k | . It is clear that (1.20) follows from the equalitiesabove in the case g = 0, so we can assume from now on that g = 0. If k ∈ { , g } ,then σ g σ g + k σ k = and ω | g + k | + | k | = ω = ω | g | ; on the other hand, if k / ∈ { , g } , wehave that ω | g + k | + | k | = ω = and ψ = ασ a σ a σ a + a = αω ; in all cases (1.20)holds, which proves the claim. Remark . In further sections we will see that, on the Kantor pairs of Hurwitztype, the pairs ( v αg , v ωαg ) are homogeneous idempotents for a grading that is relatedto the Cartan grading on the associated Kantor-Lie algebra. Furthermore, we willuse the fact that (1.20) describes certain automorphisms of the associated Kantortriple systems that permute the subspaces F v αg .1.5. Kantor systems of Hurwitz type.
It may be convenient to introduce thefollowing terminology:
Definition 1.9.
Kantor pairs and triple systems associated to a Hurwitz algebra C will be referred to as Hurwitz pairs and
Hurwitz triples , and denoted by V C and T C , respectively; the term Hurwitz (Kantor) system will be used to refer to anyof them. For each possible case for dim C = 1 , , ,
8, the corresponding Hurwitzsystems will be called Hurwitz systems of types unarion , binarion , quaternion and octonion , respectively. Remark . A straightforward calculation shows that the U -operator for a Kantorsystem associated to a Hurwitz algebra C is given by(1.21) U x ( y ) := { x, y, x } = 2 n ( x, y ) x − n ( x ) y for any x, y ∈ C . Proposition 1.11.
Let C be a Hurwitz algebra of dimension greater than . Then,the main grading on the Kantor-Lie algebra K ( C ) := K ( V C ) produces a Jordan pair given by (cid:0) K ( C ) , K ( C ) − (cid:1) that is isomorphic to a simple Jordan pair of type IV k with k = dim S ( C ) .Proof. Denote V = (cid:0) K ( C ) , K ( C ) − (cid:1) and S = S ( C ). Let W k = ( F k , F k ) denote thesimple Jordan pair of type IV k (see [L75, Chapter 4]); recall that the triple productof W k is determined by Q x ( y ) := { x, y, x } = q ( x, y ) x − q ( x ) y , where q is thequadratic form associated to the standard scalar product of F k . By restriction ofthe main grading on K ( C ), it is clear that V generates a Kantor-Lie algebra whosemain grading is a 3-grading, which forces V to be a Jordan pair.By definition of the Kantor construction, we have that V σ = { ι σ ( L x ) | x ∈ S } for σ = ± , where ι σ : V σ → K ( C ) σ ⊆ K ( C ) are the inclusions given by(1.22) ι + ( f ) := (cid:18) f (cid:19) , ι − ( f ) := (cid:18) f (cid:19) . A straightforward computation shows that { ι σ ( L x ) , ι − σ ( L y ) , ι σ ( L z ) } = (cid:2) [ ι σ ( L x ) , ι − σ ( L y )] , ι σ ( L z ) (cid:3) = ι σ ( L x L y L z + L z L y L x ) . By the left Moufang identity ([ZSSS82, Chapter 2]) we know that L x L y L x = L xyx .Since ¯ x = − x , from the identities ¯ ab + ¯ ba = n ( a, b ), ( ab ) b = ab and ¯ aa = n ( a ), weget that xyx = − (¯ xy ) x = − n ( x, y ) x + y (¯ xx ) = n ( x ) y − n ( x, y ) x . Thus12 { ι σ ( L x ) , ι − σ ( L y ) , ι σ ( L x ) } = ι σ ( L x L y L x ) = ι σ ( L xyx ) = ι σ ( L n ( x ) y − n ( x,y ) x ) . Let i be a square root of − F . Then the map W σk ≡ S −→ V σ , x i ι σ ( L x )defines an isomorphism, where the quadratic form on S is just n (cid:12)(cid:12) S . (cid:3) Peirce and root space decompositions.
In this section, we will give a re-sult that, under certain restrictions, gives a relation between a Peirce decompositionon a Kantor pair and the root system of the associated Kantor-Lie algebra.
Definition 1.12.
An element e = ( e + , e − ) ∈ V = ( V + , V − ) is called an idempotent of the Kantor pair V if { e σ , e − σ , e σ } = e σ for each σ ∈ { + , −} . Define the linearoperators L σ ( e ) , R σ ( e ) ∈ End( V σ ) as L σ ( e ) x σ := { e σ , e − σ , x σ } and R σ ( e ) x σ := { x σ , e − σ , e σ } for x σ ∈ V σ and σ ∈ { + , −} .Similarly, an element e of a Kantor triple system T is called a tripotent if { e, e, e } = e , and we define the operators L ( e ) , R ( e ) ∈ End( T ) as L ( e ) x := { e, e, x } and R ( e ) x := { x, e, e } for x ∈ T .It is well-known [KK03] that, if char F = 2 , ,
5, a tripotent e of a Kantor triplesystem T produces a Peirce decomposition given by(1.23) T = T , ⊕ T , ⊕ T , ⊕ T , ⊕ T − , ⊕ T , ⊕ T , ⊕ T , where(1.24) T λ,µ = T λ,µ ( e ) := { x ∈ T | L ( e ) x = λx, R ( e ) x = µx } . For the Peirce subspace T λ,µ , the scalars λ and µ are called its associated Peirceconstants . Peirce decompositions are important in theory of Jordan pairs and Jor-dan triple systems (see for instance [MC03]), and have also been studied in the caseof structurable algebras, but it is unknown to the authors if these have appeared
INE GRADINGS ON KANTOR SYSTEMS OF HURWITZ TYPE 13 in the literature in the context non-Jordan Kantor pairs. Note that in the case ofKantor pairs, the Peirce subspaces associated to an idempotent can be defined by:(1.25) V σλ,µ = V σλ,µ ( e ) := { x ∈ V σ | L σ ( e ) x = λx, R σ ( e ) x = µx } . Note that it is unknown if an idempotent of a Kantor pair will always produce aPeirce decomposition under our general assumptions on char F . Definition 1.13.
Let L be a semisimple Lie algebra over an algebraically closedfield F of characteristic 0, with Killing form κ . Let H be a fixed Cartan subalgebraof L . We have the well-known root space decomposition : L = H ⊕ M α ∈ Φ L α , where L α = { x ∈ L | [ h, x ] = α ( h ) x for all h ∈ H } . The elements in Φ ⊆ H ∗ \ { } are called roots and Φ = Φ + ∪ Φ − where Φ + (resp. Φ − ) is the set of positive (resp.negative) roots. Let ∆ be a system of simple roots, which generates Φ. (See [EK13, § { α , ..., α n } for n ∈ N . We can consider a map ω : ∆ → Z . Since ∆generates Φ, we get a linear operator ω : Φ → Z . By defining L i := ⊕ ω ( α )= i L α for α ∈ Φ we get a Z -grading on L , L = ⊕ i L i . It is clear that this decomposition is agrading since [ L α , L β ] ⊆ L α + β for all α, β ∈ Φ.In particular, we can choose an ω : Φ → Z such that ω (∆) ⊆ { , } and thatfor all α ∈ Φ + we get ω ( α ) ≤
2, analogously for α ∈ Φ − ; then we get a 5-grading.Recall from the Kantor construction that V = ( L , L − ) defines a Kantor pair.For any α ∈ H ∗ , let t α ∈ H be such that α ( h ) = κ ( t α , h ) for all h ∈ H . Sincethe restriction of κ to H is nondegenerate, it induces a nondegenerate symmetricbilinear form ( | ) : H ∗ × H ∗ → F given by ( α | β ) = κ ( t α , t β ). Remark . Consider L , H , H ∗ and Φ as above. We claim that for each α ∈ Φ, the Kantor pair defined by ( L α , L − α ) is isomorphic to V F (the 1-dimensionalsimple Kantor pair), or equivalently, the Kantor pair ( L α , L − α ) is spanned by anidempotent e = ( e α , e − α ).Indeed, it is well-known that each pair of roots { α, − α } generates a Lie alge-bra isomorphic to sl ( F ); hence there exist x ± α ∈ L ± α such that [ x α , x − α ] = h α ,[ h α , x α ] = 2 x α and [ h α , x − α ] = − x − α (see [EK13, §
3] and references therein).Therefore, it is straightforward to see that ( x α , x − α ) is an idempotent of ( L α , L − α ).(Note that this also follows from the fact that sl ( F ) is the Kantor-Lie algebra as-sociated to the 1-dimensional Kantor pair ( F , F ).)More in general, given an idempotent e = ( e α , e − α ) of ( L α , L − α ), it is easy tosee that any idempotent of ( L α , L − α ) has the form ( λe α , λ − e − α ) for λ ∈ F × . Anyof these idempotents will be called an idempotent associated to the root α . Notethat given an idempotent e of ( L α , L − α ), the Peirce operators L σ ( e ) and R σ ( e ) donot depend on the choice of the idempotent, and this remains true for any Kantorpair containing ( L α , L − α ).Next result shows a relation between roots and left Peirce constants on Kantorpairs. Proposition 1.15.
Let L be a semisimple Lie algebra over an algebraically closedfield F of characteristic . Let H be a Cartan subalgebra of L and Φ a set of roots. Let V be a nonzero Kantor pair associated to a Z -grading compatible with theCartan grading on L , that is, V = ( L , L − ) for some Z -grading that is a coarseningof the Cartan grading. Let e α := ( e α , e − α ) be an idempotent of V associated tosome α ∈ Φ . Then, for any β ∈ Φ such that L ± β ⊆ V ± , the idempotent e α actsmultiplicatively on L ± β with left Peirce constant λ = ( α | β )( α | α ) = k β kk α k cos( α, β ) .Proof. Since e α is an idempotent and [ x, y ] = κ ( x, y ) t α for x ∈ L α and y ∈ L − α ([H78, § e α = { e α , e − α , e α } = [[ e α , e − α ] , e α ] = κ ( e α , e − α )[ t α , e α ] = κ ( e α , e − α ) α ( t α ) e α = κ ( e α , e − α )( α | α ) e α . Then 1 = κ ( e α , e − α )( α | α ).Let β ∈ Φ and x β ∈ L β such that x β ∈ V σ for some σ = ± . Then { e α , e − α , x β } = [[ e α , e − α ] , x β ] = κ ( e α , e − α )[ t α , x β ] = β ( t α )( α | α ) x β = ( α | β )( α | α ) x β , with ( α | β )( α | α ) = k α k k β k cos( α, β ) k α k = k β kk α k cos( α, β ). Similarly, for x − β ∈ L − β , wehave that { e − α , e α , x − β } = ( − α | − β )( − α | − α ) x − β = ( α | β )( α | α ) x − β . (cid:3) Generalities on Kantor systems
On gradings and automorphisms.
We now prove some general results onKantor systems. Some of these are generalizations of analogous results for Jordansystems given in [Ara17, § Proposition 2.1.
Let A be a structurable F -algebra with unity and denote by T and V the associated Kantor triple system and Kantor pair, respectively. Then theautomorphism group scheme Aut ( A , − ) is the stabilizer of in Aut ( T ) .In addition, suppose that either char F = 3 , or that A is the algebra generatedby H ( A , − ) ∪ S ( A , − ) . Then Aut ( A , − ) is the stabilizer of (1 + , − ) in Aut ( V ) .Proof. Let R denote a commutative associative unital F -algebra and consider the R -algebra A R = A ⊗ R with the extended R -linear involution. It is clear thatAut R ( A R , − ) ≤ Aut R ( T R ) ≤ Aut R ( V R ). Denote H = H ( A R , − ) and S = S ( A R , − ).First, we claim that the involution and product of A R are determined by 1 σ ( σ = ± ) and the triple product. On the one hand, we can recover the involutionusing that ¯ x = 2 x − { x, , } , which determines the subspaces H and S . Let h ∈ H and z ∈ A R ; then, hz = { h, , z } and zh = h ¯ z . Now, take s, t ∈ S ; we have st = −{ t, s, } . Therefore, the product is recovered too, which proves the claim.Let ϕ = ( ϕ + , ϕ − ) ∈ Aut R ( V R ) with ϕ σ (1 σ ) = 1 σ for σ = ± . Notice that theinvolution of A R commutes with ϕ σ because ϕ σ (¯ x ) = ϕ σ (2 x − { x, , } ) = 2 ϕ σ ( x ) − { ϕ σ ( x ) , , } = ϕ σ ( x ) , and therefore the subspaces H and S are ϕ σ -invariant. Note that U has eigenspaces H and S with associated eigenvalues 1 and −
3, respectively. Take h ∈ H ; then ϕ σ ( h ) ∈ H and ϕ + ( h ) = ϕ + ( U ( h )) = U ( ϕ − ( h )) = ϕ − ( h ), from where it followsthat the maps ϕ + and ϕ − coincide in H .Fix s ∈ S . In the case that char F = 3 we have that ϕ + ( s ) = ϕ + ( − U ( s )) = − U ( ϕ − ( s )) = ϕ − ( s ), thus ϕ + and ϕ − coincide in S too, and therefore ϕ + = ϕ − . INE GRADINGS ON KANTOR SYSTEMS OF HURWITZ TYPE 15
Now, consider the case where A is the F -algebra generated by H ( A , − ) ∪ S ( A , − ) .Then A R is the R -algebra generated by H ∪ S . Since we have that { t, s, } = − st for each s, t ∈ S , it follows that ϕ σ ( st ) = ϕ σ ( −{ t, s, } ) = −{ ϕ σ ( t ) , ϕ − σ ( s ) , } = ϕ − σ ( s ) ϕ σ ( t )and ϕ σ ( st ) = ϕ σ ( ts ) = ϕ σ ( ts ) = ϕ − σ ( t ) ϕ σ ( s ) = ϕ σ ( s ) ϕ − σ ( t ) . In consequence, ϕ + ( st ) = ϕ + ( s ) ϕ − ( t ) = ϕ − ( st ), that is, ϕ + and ϕ − coincide in S .Let x , x , x ∈ A R be such that ϕ + ( x i ) = ϕ − ( x i ) and set z = { x , x , x } . Then ϕ + ( z ) = ϕ − ( z ). Indeed, ϕ + ( z ) = ϕ + ( { x , x , x } ) = { ϕ + ( x ) , ϕ − ( x ) , ϕ + ( x ) } = { ϕ − ( x ) , ϕ + ( x ) , ϕ − ( x ) } = ϕ − ( { x , x , x } ) = ϕ − ( z ). Since the product of A R can be recovered from the triple product and the unity, an inductive argumentshows that ϕ + and ϕ − coincide in the algebra generated by H ∪ S , which is A R .Thus we get again ϕ + = ϕ − .We need to prove that ϕ := ϕ + ∈ Aut R ( A R , − ). If h ∈ H and z ∈ A R , wehave ϕ ( h ) ∈ H and hz = { h, , z } , and so ϕ ( hz ) = ϕ ( { h, , z } ) = { ϕ ( h ) , , ϕ ( z ) } = ϕ ( h ) ϕ ( z ). If s, t ∈ S , we have that st = −{ t, s, } and again it follows easily that ϕ ( st ) = ϕ ( s ) ϕ ( t ). Therefore ϕ ∈ Aut R ( A R , − ). This proves the second statementof the result. The first statement can be proven with similar arguments. (cid:3) The following example shows that the conditions in Proposition 2.1 cannot bedropped:
Example 2.2.
Let F be a field of characteristic 3 and consider the 2-dimensionalsplit Hurwitz algebra K = F × F with involution ( x, y ) ( y, x ). (Note that thesubalgebra generated by H ( K, − ) ∪ S ( K, − ) is F
1, which does not equal K .) Theelement s = (1 , − ∈ K is skew-symmetric and satisfies s = 1. Let V = ( K, K )denote the Kantor pair associated to K and fix λ ∈ F × . We claim that the maps(2.1) ϕ σ : K −→ K, , s λ σ s, for σ ∈ { + , −} , define an automorphism ϕ of V . Indeed, this is straightforwardbecause the triple product of K is determined by { , , } = −{ s, s, } = 1 , { , , s } = −{ s, s, s } = s, { , s, } = { s, , } = { , s, s } = { s, , s } = 0 . However ϕ + = ϕ − and ϕ σ / ∈ Aut( K ) unless λ ∈ {± } . Proposition 2.3.
Let Γ be a fine grading on a Kantor pair V and let G be itsuniversal group. Then, there is a group homomorphism π : G → Z such that π ( g ) = σ if V σg = 0 for some σ ∈ { + , −} . In particular, Supp Γ + and Supp Γ − aredisjoint.Proof. This follows from the same arguments of the proof for the particular case ofJordan pairs, given in [Ara17, Prop. 2.2]. (cid:3)
Proposition 2.4.
Let A be a structurable F -algebra with unity , and G an abeliangroup. In case that char F = 3 , assume also that Z ( A ) ∩ S ( A , − ) = 0 and that A is generated as an algebra by H ( A , − ) ∪ S ( A , − ) . Consider the associated Kantorpair V = ( A , A ) . If Γ is a G -grading on V such that + (or − ) is homogeneous, then the restrictionof the shift Γ [ g ] to A = V + , with g = − deg(1 + ) , defines a G -grading Γ A on A .Moreover, if G = U (Γ) then the universal group U (Γ A ) is isomorphic to the subgroupof U (Γ) generated by Supp Γ [ g ] ; if in addition Γ is fine we also have that U (Γ) ∼ = U (Γ A ) × Z , and the universal degree of Γ is equivalent to deg Γ ( x σ ) := (deg A ( x ) , σ where deg A is the universal degree of Γ A .Proof. Denote H := H ( A , − ) and S := S ( A , − ). Let Γ be a G -grading on V where1 + is homogeneous. We claim that 1 − is homogeneous.First, consider the case where char F = 3. Then U +1 ( h − ) = h + and U +1 ( s − ) = − s + for h ∈ H , s ∈ S . Since the homogeneous map U +1 is invertible, there existsa homogeneous element y − ∈ A − such that U +1 ( y − ) = 1 + , and since this is onlypossible for y − = 1 − , we get that 1 − is homogeneous, which proves the claim inthis case.Consider now the case with char F = 3. Note that V ,s ( x ) = [ x, s ] for any x ∈ A , s ∈ S ; since Z ( A ) ∩ S ( A , − ) = 0, it follows that V ,s = 0 if and onlyif s = 0; in other words, the linear map S → End( V + ), s V ,s is injective.The subspaces im U +1 = H + and ker U +1 = S − are graded. Since 1 + ∈ im U +1 ,there exists a homogeneous element y − such that U +1 ( y − ) = 1 + , and we have thatdeg(1 + )+deg( y − ) = 0. It is clear that y − − − ∈ ker U +1 = S − , so that y − = 1 − + s − for some s ∈ S . If V ,s = 0, then s = 0 and 1 − is homogeneous. Assume now that V ,s = 0. Then 0 = V ,s = V ,y − V , = V ,y − id is a homogeneous map of degree0. Since S − is graded, we can write s = P i s i where s − i are homogeneous and havedifferent degrees in S − . Consequently, we have that V ,s = P i V ,s i with V ,s i = 0for each i , and since V ,s has degree 0, it follows that s = s i for some i , that is, s is homogeneous in V − . Moreover, deg( s − ) = − deg(1 + ) = deg( y − ), and 1 − ishomogeneous.We have proven that 1 − is homogeneous in any case. If deg g is the degree map ofthe grading Γ [ g ] on V , we have that deg g (1 + ) = 0 = deg g (1 − ) because U +1 (1 − ) = 1 + .By Proposition 2.1, the stabilizer of (1 + , − ) in Aut ( V ) is Aut ( A , − ). Note thatthe “automorphisms” in Aut ( V ) producing the grading must be in the stabilizerof (1 + , − ), which is Aut ( A , − ). Therefore, using the correspondence betweengradings and morphisms of affine group schemes, it follows that the grading Γ [ g ] on V is produced by a morphism Hom Alg F ( F G, − ) → Aut ( A , − ), which also defines agrading Γ A on A . It also follows that the homogeneous components of Γ [ g ] coincidein V + and V − .Assume now that G = U (Γ) and call H = h Supp Γ [ g ] i . Note that Γ [ g ] canbe regarded as a U (Γ)-grading and also as an H -grading; similarly Γ A can beregarded as a U (Γ A )-grading and as an H -grading. By the universal property ofthe universal group, the H -grading Γ A is induced from the U (Γ A )-grading Γ A byan homomorphism ϕ : U (Γ A ) → H that restricts to the identity in the support.On the other hand, the U (Γ A )-grading Γ A induces a U (Γ A )-grading (Γ A , Γ A ) on V that is a coarsening of Γ, and therefore (Γ A , Γ A ) is induced from Γ by someepimorphism ϕ : U (Γ) → U (Γ A ). Let ϕ : H → U (Γ A ) be the restriction of ϕ to H . Note that g ∈ ker ϕ , which implies that the U (Γ A )-grading (Γ A , Γ A ) is inducedfrom the H -grading Γ [ g ] by ϕ , and also that ϕ is an epimorphism which is theidentity in the support. Since each epimorphism ϕ i is the identity in the support,both compositions ϕ ϕ and ϕ ϕ must be the identity and therefore U (Γ A ) ∼ = H . INE GRADINGS ON KANTOR SYSTEMS OF HURWITZ TYPE 17
Suppose now that Γ is fine and denote by Γ H the grading Γ [ g ] regarded as an H -grading. Note that U (Γ) = h Supp Γ [ g ] , g i = h Supp Γ H , g i = h H, g i . Consider H as a subgroup of H × Z ∼ = H × h g i , where the element g has infinite order. The H -grading Γ H can be regarded as an H ×h g i -grading, and the shift (Γ H ) [ g ] definesanother H × h g i -grading where deg(1 + ) = g . Since the H × h g i -grading (Γ H ) [ g ] is a coarsening of the U (Γ)-grading Γ (because Γ is fine and by Proposition 2.3),by the universal property there is an epimorphism U (Γ) = h H, g i → H × h g i thatsends − g g and fixes the elements of H . Consequently, H ∩ h g i = 0, h g i ∼ = Z ,and we can conclude that U (Γ) = h H, g i ∼ = H × Z ∼ = U (Γ A ) × Z . The statementabout the universal degree follows too. (cid:3) Proposition 2.5.
Let A be a structurable F -algebra with unity , and T its associ-ated Kantor triple system. If Γ is a G -grading on T such that is homogeneous, thenthe shift Γ [ g ] with g = deg(1) induces a G -grading Γ A on A . Moreover, if G = U (Γ) then deg(1) has order and U (Γ A ) is isomorphic to the subgroup of U (Γ) generatedby Supp Γ [ g ] ; and if in addition Γ is fine we also have that U (Γ) ∼ = U (Γ A ) × Z ,and the universal degree of Γ is equivalent to deg Γ ( x ) := (deg A ( x ) , ¯1) where deg A is the universal degree of Γ A .Proof. Let Γ be a G -grading on T such that 1 is homogeneous. Then Γ V := (Γ , Γ) isa G -grading on the Kantor pair V := V A such that 1 + and 1 − are homogeneous. ByProposition 2.1, and using the same argument from the proof of Proposition 2.4, itfollows that the restriction of the shift Γ [ g ] V to A = V + , with g = − deg(1 + ), definesa G -grading Γ A on A . Since U (1) = 1, we get that 2 g = 0, so that Γ [ g ] defines agrading on T , and also Γ [ g ] V = (Γ [ g ] , Γ [ g ] ). Thus Γ [ g ] defines the same G -grading Γ A on A , which proves the first claim.Assume now that we also have G = U (Γ). We have shown above that 2 deg(1) =0. The Z -grading Γ Z given by T = T ¯1 is a coarsening of Γ, which implies bythe universal property that ϕ : G → Z is a group epimorphism and Γ Z is inducedby ϕ from Γ. Consequently, deg(1) has order 2. Set H = h Supp Γ [ g ] i . The restof the proof follows with the same arguments from the last part of the proof ofProposition 2.4, but using T instead of V A . (cid:3) Automorphisms and orbits
In this section, the automorphism groups and their orbits are studied for Kantorsystems of Hurwitz type.
Notation 3.1.
Let V C and T C be the Kantor pair and Kantor triple system,respectively, associated to a Hurwitz algebra C . For any λ ∈ F × , consider the pairof maps c λ := ( c + λ , c − λ ) defined by(3.1) c + λ ( x ) := λx, c − λ ( y ) := λ − y, for any x ∈ V + C , y ∈ V − C . It is easy to see that c λ is an automorphism of V C . Notethat the 1-torus h c λ | λ ∈ F × i produces the Z -grading that is associated to theKantor construction. Also, it is clear that c λ is an automorphism of T C if and onlyif λ = ± L a and R a , respectively, the left and right multiplications by a ∈ C . We will also consider the triple system T ′ C := C with the triple productgiven by { x, y, z } ′ := ( x ¯ y ) z . Remark . Let V be a finite-dimensional vector space and q : V → F a nondegen-erate quadratic form, and denote by O ′ ( V, q ) the reduced orthogonal group. Recallfrom [J89, 4.8]) that O ′ ( V, q ) E O + ( V, q ) and O + ( V, q ) / O ′ ( V, q ) ∼ = F × / ( F × ) , where( F × ) is the multiplicative group of squares of F × . Since we assume that the basefield is algebraically closed, we have that O ′ ( V, q ) = O + ( V, q ).On the other hand, recall from [Eld00, Section 1] that if C is a Cayley algebrawith norm n , then O ′ ( C , n ) = h L a | a ∈ C , n ( a ) = 1 i = h R a | a ∈ C , n ( a ) = 1 i . Lemma 3.3.
Let C be a Hurwitz algebra and x ∈ C a traceless element with n ( x ) = 1 . Then L x ∈ Aut( T C ) ∩ Aut( T ′ C ) .Proof. Recall from [Eld96, Section 3] that if C is a Cayley algebra, we have that(3.2) (( xa )( xb ))( xc ) = n ( x ) x (( a ¯ b ) c )for all x, a, b, c ∈ C where x is traceless. Since any Hurwitz algebra is contained ina Cayley algebra, (3.2) holds for any Hurwitz algebra, and consequently it is easyto see that L x behaves well with the triple products of T C and T ′ C . (Note that L − x = L ¯ x = − L x .) (cid:3) Theorem 3.4.
Let C be a Hurwitz algebra, T C its associated Kantor triple system,and T ′ C the triple system defined in 3.1. Then Aut ( T C ) = Aut ( T ′ C ) ≤ O ( C, n ) .Proof. The result follows in the case that dim C = 1 or 2, because the triple prod-ucts of T C and T ′ C coincide. Assume from now on that dim C = 4 or 8. Denote T = T C and T ′ = T ′ C . For any unital associative commutative algebra R , we willconsider T R := T ⊗ R and T ′ R := T ′ ⊗ R , with their R -linear triple products, and the R -algebra C R := C ⊗ R . It is clear that Aut R ( T ′ R ) ≤ Aut R ( T R ). Let f ∈ Aut R ( T R );we need to prove that f ∈ Aut R ( T ′ R ). Set a = f (1).Note that { x, x, x } = n ( x ) x for any x ∈ T R . Hence, f ( { x, x, x } ) = n ( x ) f ( x )and f ( { x, x, x } ) = { f ( x ) , f ( x ) , f ( x ) } = n ( f ( x )) f ( x ), so that n ( f ( x )) = n ( x ) and f ∈ O ( C R , n ). In particular, n ( a ) = 1.If we had a = 1, by Proposition 2.1 we would have that f ∈ Aut R ( C R ) ≤ Aut R ( T ′ R ) and the result holds. Assume from now on that a = 1. If t ( a ) = 0, thenby Lemma 3.3 we have that L − a ∈ Aut R ( T R ) ∩ Aut R ( T ′ R ); but we also have that( L − a ◦ f )(1) = 1, so that by Proposition 2.1 we deduce that L − a ◦ f ∈ Aut R ( C R ) ≤ Aut R ( T ′ R ) and f ∈ Aut R ( T ′ R ). Finally, consider the case where t ( a ) = 0. Sincedim C = 4 or 8, there exists b ∈ C with n ( b ) = 1, t ( b ) = 0 and such that L b ( a ) = ba is traceless (it suffices to consider a 2-dimensional Hurwitz subalgebra A containing1 and a , and then take b as an element with n ( b ) = 1 in the subspace orthogonalto A ), and it is clear that L b ∈ Aut R ( T R ) ∩ Aut R ( T ′ R ); therefore, if we replace f by L b ◦ f , we can assume again that t ( a ) = 0 and the result holds by the previouscase. (cid:3) Corollary 3.5.
Let C be a Hurwitz algebra with norm n .1) If dim C = 1 , then Aut( T C ) = O ( C, n ) ∼ = Z .2) If dim C = 2 , then Aut( T C ) = O ( C, n ) ∼ = F × ⋊ Z .3) If dim C = 4 , then Aut( T C ) ∼ = O + ( C, n ) .4) If dim C = 8 , then Aut( T C ) ∼ = Spin( S , n ) , where S is the skew-symmetric sub-space of C . INE GRADINGS ON KANTOR SYSTEMS OF HURWITZ TYPE 19
Proof.
Cases 1), 2) and 3) are consequence of [Eld96, Corollary 6], where they wereproven for the triple system T ′ C . Note that in case 2), the torus F × corresponds tothe automorphisms acting on the Cartan basis via(3.3) f λ ( e ) = λe , f λ ( e ) = λ − e . for λ ∈ F × , and the group Z corresponds to the swapping automorphism e ↔ e .Finally, recall from [Eld96, Theorem 10] that we have an isomorphism Aut( T ′ C ) ∼ =Spin( W, q ), where W is the set of traceless elements of C with the quadratic form q = − n . It is clear that W = S . Finally, note that if i is a square root of − F , then the map S → S , s i s extends to an isomorphism Spin( S , q ) → Spin( S , n ). (cid:3) Corollary 3.6.
Let C be a Cayley algebra. Then, Aut( T C ) = h L a | a ∈ C , n ( a ) = 1 , t ( a ) = 0 i . Proof.
Set G = h L a | a ∈ C , n ( a ) = 1 , t ( a ) = 0 i . By Lemma 3.3, we have that G ≤ Aut( T C ). It is well-known that Aut( C , − ) = Aut( C ). By [Eld96, Corollary 8] itfollows that Aut( C ) ≤ G . Let ϕ ∈ Aut( T C ) and set z = ϕ (1). Then, by the Cayley-Dickson doubling process, it follows that there exists s ∈ C such that n ( s ) = 1, t ( s ) = 0 and t ( sz ) = 0. Then L s ∈ Aut( T C ). Since Aut( T C ) ≤ O ( C , n ), we havethat n ( sz ) = n ( L s ϕ (1)) = 1, so that L sz ∈ Aut( T C ). Besides, L sz L s ϕ (1) = 1. ByProposition 2.1, we get that L sz L s ϕ ∈ Aut( C ) ≤ G , which implies that ϕ ∈ G .This proves that Aut( T C ) ≤ G , and the result follows. (cid:3) Corollary 3.7.
Let C be a Hurwitz algebra. Then, gradings and equivalence classesof fine gradings are the same on T C and on T ′ C .Remark . It is well known that if char F = 3, conjugate inverses of elements ofsimple structurable algebras are unique if they exist. Assume now that char F = 3and let C be a Hurwitz F -algebra.Consider first the case where C is the 2-dimensional Hurwitz algebra K . Then,for any skew element s , we have that V ,s = 0, but also V , = id, so that V , s = id.Therefore, the conjugate inverse of 1 is not unique in this case.Now consider the case dim C >
2. Let y be a conjugate inverse of 1, that is V ,y = id. Set y = λ s with λ ∈ F and s ∈ S ( C, − ). Then 1 = V ,y (1) = V ,λ (1) + V ,s (1) = λ λ
1, from where it follows that λ = 1. Thus id = V ,y = V , + V ,s = id + V ,s and V ,s = 0; but we also have that V ,s ( x ) = [ x, s ] foreach x ∈ C . Hence, [ x, s ] = 0 for all x ∈ C , that is, s ∈ Z ( C ) ∩ S ( C, − ) = 0, so that s = 0. This proves that 1 is the only conjugate inverse of 1 in this case.We will now classify the orbits of Kantor pairs and triple systems associated toa Hurwitz algebra. Notation 3.9.
Let C be a Hurwitz algebra and λ ∈ F × . Let n denote the (qua-dratic) norm of C as a Hurwitz algebra. (Note that in the 1-dimensional case, thenorm as structurable algebra is linear, and so does not coincide with n .) • If dim
C >
1, where we assume that either dim C = 2 or char F = 3, we define: O := { } , O := { = x ∈ C | n ( x ) = 0 } , O := { x ∈ C | n ( x ) = 0 } , O ( λ ) := { x ∈ C | n ( x ) = λ } . • In the case dim C = 1, we define: O := { } , O := { x ∈ C | x = 0 } , O ( λ ) := { x ∈ C | n ( x ) = λ } . In any case, we will say that an element x of C has rank i if x ∈ O i . Proposition 3.10.
Let V C be the Kantor pair associated to a Hurwitz algebra C ,and consider the orbits of the action of Aut( V C ) on V + C (or V − C ). Then:1) If dim C = 1 , then the orbits are O and O .2) If dim C > , with either dim C = 2 or char F = 3 , then the orbits are O , O and O .Furthermore, conjugate inverses are unique and only exist for elements in the orbitof nonzero norm.Proof. We will only consider the subspace V + C = C (for V − C the proof is analogous).It is clear that O is always an orbit. It is easy to see that Aut( V F ) = h c λ | λ ∈ F × i ,from where the classification follows in the case dim C = 1. Consider from now onthe case dim C > F = 3, for e ina Cartan basis of C , it is easy to see that im U e = F e , im U = C , which havedifferent dimensions, hence the claim follows. Now consider the case char F =3. Since V , = id and V ,s = 0 for each skew element s ∈ S , it is clear that { , C, ·} = F id, and this subspace consists of invertible maps and the zero map.On the other hand, the map { e , e , ·} is neither zero nor invertible. Consequently,1 and e cannot be in the same orbit, and again, it follows that there are at least2 nontrivial orbits.Assume now that dim C = 2. Then, for each λ ∈ F × , we have an automorphism f λ of T C as in (3.3). Recall that c λ is an automorphism of V C . Also, since C is commutative, the involution defines a swapping automorphism e ↔ e on C .These automorphisms of V C show that any nonzero element of C is either in theorbit of e or in the orbit of 1. The fact that there are at least 2 nontrivial orbitsproves the result in this case. We can assume from now on that dim C > = x ∈ C such that n ( x ) = 0. We claim that x is in the orbit of theelement e for some Cartan basis of C .Assume that λ := n ( x, = 0. By applying the automorphism c λ − , we canassume n ( x,
1) = 1. Then e := x and e := e = 1 − e are isotropic orthogonalidempotents, and we can use the construction given in [Eld98] (or in [EK13, Chapter4]) to complete them to a Cartan basis with multiplication table as in Figure 1.This proves the claim in this case.Now suppose n ( x,
1) = 0. Then x = 0 because n ( x ) = 0 = t ( x ). Since the normis nondegenerate, there exists y ∈ C such that n ( x, y ) = 1. Thus n (1 , xy ) = 1, andthe elements e := xy and e := e = 1 − e are isotropic idempotents. Again,we can complete { e , e } to a Cartan basis. Notice that n ( x, e ) = n ( x, xy ) = n ( x , y ) = 0, because x = 0, and n ( x, e ) = n ( x, − e ) = n ( x, − n ( x, e ) = 0.Hence x ∈ ( F e + F e ) ⊥ . Furthermore, xe = x ( xy ) = ( xx ) y = n ( x ) y = 0 and0 = xe = e x = e ( − x ), so xe = 0 = e x . We have xe = x (1 − e ) = x andsimilarly e x = x , so that xe = x = e x . We have that x ∈ U := e Ce , and( F e + F e ) ⊥ = U ⊕ V where V := e Ce . One more time, using the constructionin [Eld98], we can assume that x = u in a Cartan basis. With the relation in(1.19), equation (1.20) shows that there is an automorphism of the form L z , for INE GRADINGS ON KANTOR SYSTEMS OF HURWITZ TYPE 21 some traceless element z ∈ C of norm 1, such that L z ( x ) ∈ F e . Then, we can take λ ∈ F × such that c λ L z ( x ) = e . We have proven the claim for the orbit of rank 1in all cases.Fix x ∈ O , that is, n ( x ) = 0. We now claim that x and 1 belong to the sameorbit.Take λ ∈ F × such that λ n ( x ) = 1; then n ( c + λ ( x )) = 1 and without loss ofgenerality we can assume that n ( x ) = 1. If n ( x,
1) = 0, then L ¯ x ∈ Aut( V C ) and L ¯ x ( x ) = ¯ xx = n ( x )1 = 1, which proves the claim in this case. From now on,assume that λ := n ( x, = 0. By the Cayley-Dickson doubling process appliedto the algebra generated by x , there exists a traceless element y ∈ ( F F x ) ⊥ of norm 1. Then, the element z = L y ( x ) = yx has norm 1, and is also tracelessbecause n ( z,
1) = n ( yx,
1) = n ( x, ¯ y ) = − n ( x, y ) = 0. Consequently, L y and L ¯ z areautomorphisms and we have that L ¯ z L y ( x ) = 1, which proves the claim.Since there are at least 2 nontrivial orbits in the case dim C >
2, the classificationfollows for the remaining cases. The last statement follows from Remark 3.8 in thecase char F = 3, and was proven in [AF92] for the case char F = 3. (cid:3) Proposition 3.11.
Let T C be the Kantor triple system associated to a Hurwitzalgebra C , and consider the orbits of the action of Aut( T C ) on C . Then:1) If dim C = 1 , the orbits are O and O ( λ ) , with λ ∈ F × .2) If dim C > , the orbits are O , O and O ( λ ) , with λ ∈ F × .Proof. The result in the case dim C = 1 follows easily from Corollary 3.5. The casewith dim C = 2 is easy to check using a Cartan basis { e , e } and the automorphismgroup in Corollary 3.5 (which is generated by the automorphisms of the form f λ asin (3.3), and the swapping automorphism e ↔ e ). We can assume from now onthat dim C >
2. Note that O is obviously an orbit in all cases.Fix 0 = x ∈ C such that n ( x ) = 0. We need to prove that x is in the orbit ofthe element e in a Cartan basis of C .First, consider the case with λ := n ( x, = 0. Then e := λ − x is an isotropicidempotent. As in Proposition 3.10, we can extend e to a Cartan basis, provingthe claim in this case.Now consider the case where n ( x,
1) = 0. With the same arguments used in theproof of Proposition 3.10, we deduce that x = u and that there is an automorphism f ∈ Aut( T C ) such that f ( x ) ∈ F e , where e and u belong to some Cartan basisof C . From the 2-torus producing the Cartan grading on C , it follows that for each α ∈ F × there is some g α ∈ Aut( C ) ≤ Aut( T C ) such that g α ( u ) = αu . Therefore, f ( g α ( x )) = αf ( x ) ∈ F e for each α ∈ F × , which forces x and e to be in the sameorbit. We have proven the claim for all elements of rank 1.Consider the last case, that is, x ∈ O . Then µ := n ( x ) = 0, and since F isalgebraically closed, we can take λ ∈ F × such that n ( x ) = λ . Then we can write x = λy with n ( y ) = 1. With the same arguments used in the proof of Proposi-tion 3.10, it follows that there is f ∈ Aut( T C ) such that f ( y ) = 1. Consequently, f ( x ) = λ
1, so that x is in the orbit of λ
1. It follows that the elements of O ( µ ) arein the same orbit.Finally, by Theorem 3.4, we have that Aut( T C ) ≤ O ( C, n ), which forces thesets O and O ( λ ), for λ ∈ F × , to be contained in different orbits, and the resultfollows. (cid:3) Now we will deal with the automorphisms and orbits of the most exceptionalcase in our list:
Theorem 3.12.
Let K be a -dimensional Hurwitz F -algebra, with char F = 3 ,and V K its associated Kantor pair. Then there is a group isomorphism Ψ : Aut( V K ) → GL( K ) ∼ = GL ( F ) , ( ϕ + , ϕ − ) ϕ + , where ϕ − is the dual inverse of ϕ + relative to the bilinear trace. Consequently,there is only one nontrivial Aut( V K ) -orbit on K + (or K − ).Proof. It is clear that Ψ is a homomorphism. Let ϕ ∈ Aut( V K ). By (1.21) andsince char F = 3, we have that2 n ( x, y ) ϕ + ( x ) = ϕ + (cid:0) U + x ( y ) (cid:1) = U + ϕ + ( x ) (cid:0) ϕ − ( y ) (cid:1) = 2 n (cid:0) ϕ + ( x ) , ϕ − ( y ) (cid:1) ϕ + ( x )for all x ∈ V + K , y ∈ V − K , which implies that n ( x, y ) = n (cid:0) ϕ + ( x ) , ϕ − ( y ) (cid:1) for all x ∈ V + K , y ∈ V − K . Hence ϕ − and ϕ + are dual inverses relative to the trace, andthey determine each other. It follows that Ψ is injective.Since we assume that F is algebraically closed, we have that there exists a Cartanbasis { e , e } of K , K = F e ⊕ F e ∼ = F × F , and the involution is given by e ↔ e .It is straightforward to see that the triple product of K on the Cartan basis is givenby:(3.4) { e , e , e } = 2 e , { e , e , e } = − e , { e , e , e } = 2 e , { e , e , e } = − e , { e i , e i , e i } = { e , e , e } = { e , e , e } = 0 . Let τ ∈ Aut( K, − ) ≤ Aut( V K ) be the automorphism τ : e ↔ e that swaps theidempotents of the Cartan basis (i.e., the involution). Fix α, β ∈ F × and considerthe maps T + α,β : e +1 αe +1 , e +2 βe +2 ,T − α,β : e − β − e − , e − α − e − . By (3.4), it is clear that T α,β := ( T + α,β , T − α,β ) is an automorphism of V K . Finally,fix λ ∈ F and define A λ := ( A + λ , A − λ ) where(3.5) A σλ : e e , e e + σλe , for σ = ± . Then, using (3.4), it is straightforward to prove that A λ ∈ Aut( V K ).For instance, we have that { A σλ ( e ) , A − σλ ( e ) , A σλ ( e ) } = { e , e − σλe , e } = { e , e , e } = 2 e = A σλ (2 e ) = A σλ ( { e , e , e } ) , { A σλ ( e ) , A − σλ ( e ) , A σλ ( e ) } = { e + σλe , e − σλe , e + σλe } = { σλe , e , e + σλe } + { e , − σλe , e + σλe } = − σλe + 2 λ e − σλe + λ e = 0= A σλ (0) = A σλ ( { e , e , e } ) , and the other cases are proven similarly. The maps of the form τ , T + α,β and A + λ act on the vector space V + K as elementary matrices, and these generate a groupisomorphic to GL ( F ); we can conclude that Ψ is surjective, and therefore an iso-morphism. (cid:3) INE GRADINGS ON KANTOR SYSTEMS OF HURWITZ TYPE 23
Theorem 3.13.
Let C be a Hurwitz algebra and V C and T C the associated Kantorpair and triple system, respectively. Assume either that dim C = 2 or that char F =3 . Then, we have that Aut( V C ) = h c λ , Aut( T C ) | λ ∈ F × i ∼ = ( F × × Aut( T C )) /C with C := h ( − , − id) i ∼ = Z . Consequently, each ϕ ∈ Aut( V C ) satisfies the prop-erty t ( ϕ + ( x + ) , ϕ − ( y − )) = t ( x + , y − ) for any x + ∈ V + , y − ∈ V − ; equivalently, ( ϕ + ) − = [ ( ϕ − ) , where b denotes thedual relative to the bilinear trace.Proof. First, consider the case with dim
C >
2. Fix f = ( f + , f − ) ∈ Aut( V C ).Recall that if a ∈ C is traceless of norm 1, then L a ∈ Aut( T C ) ⊆ Aut( V C ). Noticethat if x ∈ V + C and y ∈ V − C are conjugate inverses, then g + ( x ) and g − ( y ) areconjugate inverses for any automorphism g of V C . Since 1 + is invertible, f + (1 + )is invertible too, so that λ := n ( f + (1)) = 0 (see Proposition 3.10). Without lossof generality, we can replace f with c / √ λ f and assume that n ( f + (1)) = 1. Sincedim C >
2, the Cayley-Dickson doubling process shows that there is some tracelesselement a ∈ C of norm 1 such that z := L a f + (1) is traceless, and we also have n ( z ) = 1. Then, L ¯ z L a f + (1) = 1. Again, without loss of generality we can replace f with L ¯ z L a f and assume that f + (1) = 1. Since f − (1) must be a conjugate inverseof 1 + (which is unique by Remark 3.8), we also have f − (1) = 1. By Proposition 2.1,it follows that f := f + = f − ∈ Aut( C ). Since the automorphisms c λ commutewith any automorphism, it follows that Aut( V C ) = h c λ , Aut( T C ) | λ ∈ F × i and wehave an epimorphism F × × Aut( T C ) −→ Aut( V C ) , ( λ, f ) c λ f. (3.6)The kernel of the isomorphism in (3.6) is C , which finishes the proof in this case.Now, consider the case dim C = 2. Fix f = ( f + , f − ) ∈ Aut( V C ). Again,composing f with some automorphism of type c λ we can assume that n ( f + (1)) = 1.Then, there exists λ ∈ F × such that f + (1) = λ − e + λe , where { e , e } is aCartan basis of C . By (3.3), there is an automorphism f λ ∈ Aut( T C ) such that f λ f (1 + ) = 1 + . Hence, replacing f with f λ f , we can assume that f + (1) = 1. Thesame arguments used in the case above prove the result in this case (here we requirechar F = 3 in order to apply Proposition 2.1).Finally, for the case with dim C = 1, the result is trivial because we have thatAut( V F ) = h c λ | λ ∈ F × i and Aut( T F ) = h c λ | λ = ± ∈ F × i .The last statement of the result follows from Theorem 3.4, and because theautomorphisms c λ behave well with the trace form. (cid:3) Definition 3.14.
Recall that the general orthogonal group associated to an algebra A with a norm form n is the group of similarities of the norm, that is,GO( A , n ) := { f ∈ GL( A ) | ∃ λ ∈ F × such that n ( f ( x )) = λn ( x ) ∀ x ∈ A } . Next result classifies the automorphism groups for Kantor pairs of Hurwitz type.
Corollary 3.15.
Let C be a Hurwitz algebra with either dim C = 2 or char F = 3 .1) If dim C = 1 , then Aut( V C ) = h c λ | λ ∈ F × i ∼ = F × .
2) If dim C = 2 , then Aut( V C ) = GO ( C, n ) ∼ = ( F × ) ⋊ Z .2) If dim C = 4 , then Aut( V C ) ∼ = ( F × × O + ( C, n )) /C with C ∼ = Z .3) If dim C = 8 , then Aut( V C ) ∼ = Γ + ( S , n ) , where S is the skew-symmetric subspaceof C and Γ + ( S , n ) denotes the associated even Clifford group.Proof. Consequence of Theorem 3.13 and Corollary 3.5. (cid:3)
Definition 3.16.
Let A be a graded algebra. Recall that a bilinear form b : A × A → F is said to be homogeneous if we have g + h = 0 whenever b ( A g , A h ) = 0.(Note that this definition is consistent with the fact that the only grading up toequivalence on F is the trivial grading: F := F .) The definition is analogous for abilinear form on a graded Kantor triple system. On the other hand, given a gradedKantor pair V , a bilinear form b : V + × V − → F will be called homogeneous if wehave g + h = 0 whenever b ( V + g , V − h ) = 0.The following Lemma can be regarded as an extension from the case of Hurwitzalgebras (see [EK13, proof of Proposition 4.10]) to the case of Kantor pairs andtriple systems of Hurwitz type. Lemma 3.17.
Let C be a Hurwitz algebra and consider its associated Kantor pair V C and triple system T C . Assume also that either dim C = 2 or char F = 3 .Then, for any grading on V C (resp. T C ), the bilinear (resp. linear) trace of C ishomogeneous on V C (resp. T C ).Proof. Recall that any grading Γ on T C induces a grading (Γ , Γ) on V C . Also, thelinear trace can be recovered from the bilinear trace via t ( x ) = t ( x, V := V C . Fix a G -grading Γ on V and let x ∈ V + g , y ∈ V − h be such that t ( x, y ) = 0. We need to prove thatdeg + ( x ) + deg − ( y ) = 0. Without loss of generality, we can replace y by 2 t ( x, y ) − y and assume that t ( x, y ) = 2. We will denote H := H ( C, − ) and S := S ( C, − ).First consider the case that x is in the orbit of rank 2. Note that for anyautomorphism f , by Theorem 3.13 we have that t ( f + ( x ) , f − ( y )) = t ( x, y ); hence,up to automorphism (by Proposition 3.10), we can assume that x = 1 + . It followsthat y = 1 + s with s ∈ S . In case char F = 3, we have that U +1 ( s ) = 0, so that U +1 ( y ) = 1 and taking degrees we get deg + ( x ) + deg − ( y ) = 0. Now consider thecase char F = 3; thus U is invertible. Since U +1 is invertible and homogeneous with U +1 (1 − ) = 1 + , it follows that 1 − is homogeneous and deg(1 + ) + deg(1 − ) = 0. Onthe other hand, z − := U − U +1 ( y ) = U − − (1 − s ) = 1 + 9 s ∈ V − h . Since y = 1 + s ,1 and 1 + 9 s are linearly dependent and homogeneous in V − , they must have thesame degree. Thus deg − ( y ) = deg − (1) = − deg + ( x ).Finally, consider the case where x has rank 1. This time, we can assume withoutloss of generality that x = e +1 where e denotes the corresponding idempotent in aCartan basis of C . From t ( x, y ) = 1 we deduce that y = e + z with z ∈ ( F e ) ⊥ =span { e , u i , v i | i = 1 , , } . But we also have that ker U + x = span { e − , u − i , v − i | i =1 , , } . We conclude that z ∈ ker U + e , so that U + x ( y ) = U + e ( e + z ) = 2 e +1 = 2 x and again deg + ( x ) + deg − ( y ) = 0. (cid:3) Classification of fine gradings on Hurwitz Kantor systems
In this section, we will first describe some important Peirce decompositions onHurwitz pairs. Secondly, some examples of fine gradings by their universal groups
INE GRADINGS ON KANTOR SYSTEMS OF HURWITZ TYPE 25 on Kantor pairs and triple systems of Hurwitz type will be given. And finally, wewill prove that any fine grading on a Kantor pair or triple system of Hurwitz typeis equivalent to exactly one of those examples.4.1.
Peirce decompositions on Hurwitz pairs.Definition 4.1.
Given a Hurwitz algebra C with dim C = 2 m >
1, with two copiesof the basis in (1.17) we get a basis of V C that will be denoted B Z ( V C ) and calleda Cartan basis of V C . Note that B Z ( V C ) consists of the idempotents of V C givenby e αg := ( v αg , v ωαg ) for g ∈ Z m − , α ∈ b Z = h ω i . Proposition 4.2.
Let C be a Hurwitz algebra with dim C = 2 m > . Then, theidempotents e αg of the Cartan basis of V C produce a simultaneous Peirce decompo-sition that is given by: V σ , ( e αg ) = F ( e αg ) σ , V σ − , ( e αg ) = F ( e ωαg ) σ , V σ , ( e αg ) = M g = h ∈ Z m − F ( e αh ) σ , V σ , ( e αg ) = M g = h ∈ Z m − F ( e ωαh ) σ , for σ = ± .Proof. Consider first the case with m = 3, i.e., C = C . The character ω ∈ b Z (as in Notation 1.7), identified with its natural extension ω × ∈ b Z × c Z ∼ = c Z ,produces an automorphism f ω of the Z -grading on C that is given on the Cayley-Dickson basis via f ω ( x g ) = ω ( g ) x g . It is easy to see that f ω permutes the subspaces F e αg ↔ F e αωg for each g ∈ Z and α ∈ b Z . (Note that the automorphism f ω of C appears in [EK13, Proof of Th. 4.17] with a different guise.) Also, recall fromLemma 3.3 that for each element x g of the Cayley-Dickson grading, L x g definesan automorphism, which is given by (1.20). By the symmetries given by theseautomorphisms, it is clear that it suffices to prove the result for the case e = e ,which is easy to calculate using (1.19) and the products of the Cartan basis B Z ( C )of C . For the cases m = 1 ,
2, the result follows by restriction to the correspondingsubspaces. (cid:3)
Remark . The left Peirce decompositions in Proposition 4.2 are produced by D -operators, which are derivations, so that we have gradings associated to them.Namely, we have a grading by G = Z determined by V σσλ := V σλ,µ ( e αg ), and there-fore, by composition with the group isomorphism Z → Z ,
1, we get a Z -grading with homogeneous subspaces given by(4.1) V σσ λ := V σλ,µ ( e αg ) . Also, note that the grading in (4.1) can be properly refined with a shift on thedegree, or just combining it with the Z -grading given by V σσ = V σ .4.2. Examples of fine gradings on Hurwitz Kantor systems.Example 4.4.
For the 1-dimensional Hurwitz algebra C = F , it is clear that up toequivalence we have only one fine grading on V C ; its universal group is isomorphicto Z , and its universal degree is equivalent to deg(1 + ) = − deg(1 − ) = 1. Similarly,it is easy to see that up to equivalence there is only one fine grading on T C , whoseuniversal group is Z , and its associated universal degree is equivalent to deg(1) = ¯1. Example 4.5.
Let C be a Hurwitz algebra of dimension 2 m for some m ∈ { , , } ,where we also require to exclude the case with char F = 3 and dim C = 2 (the grad-ing obtained in that case with the construction below is not given by its universalgroup, and is actually equivalent to a grading given in Example 4.7). Set G = Z m and take a Cayley-Dickson basis B CD ( C ) = { x g } g ∈ G of C . There is a gradingΓ CD ( V C ) by the group Z × G = Z × Z m on the Kantor pair V C , where B CD ( C ) isa homogeneous basis in V σC for σ = ± , and that is given by(4.2) deg( x σg ) := ( σ , g )for σ = ± . The grading Γ CD ( V C ) will be called the Cayley-Dickson grading on V C . Example 4.6.
Let C , G and B CD ( C ) be as in Example 4.5. Then, we have agrading Γ CD ( T C ) by the group Z × G = Z m +12 on the Kantor triple system T C ,with homogeneous basis B CD ( C ), and determined by(4.3) deg( x g ) := (¯1 , g ) . We will refer to Γ CD ( T C ) as the Cayley-Dickson grading on T C . Example 4.7.
Now we will define a grading on the Cayley pair V C by the group Z using two copies of a Cartan basis of C . Define the following map:(4.4) deg( e +1 ) = − deg( e − ) := (1 , , , , deg( e +2 ) = − deg( e − ) := (0 , , , , deg( u +1 ) = − deg( v − ) := (0 , , , , deg( u +2 ) = − deg( v − ) := (0 , , , , deg( u +3 ) = − deg( v − ) := (1 , , − , − , deg( v +1 ) = − deg( u − ) := (1 , , − , , deg( v +2 ) = − deg( u − ) := (1 , , , − , deg( v +3 ) = − deg( u − ) := (0 , − , , . Note that the second coordinate of deg coincides with the Z -grading in (4.1) as-sociated to the idempotent e a + a = − √ ( u , v ) of V C . On the other hand, thelast two coordinates of deg coincide with the Cartan grading on C given in (1.12),which extends to a grading on V C . Furthermore, the sum of all coordinates of degis the Z -grading given by V σσ := V σ , and since a linear combination of compatiblegradings defines a grading, it follows that the first coordinate of deg is also a grad-ing on V C . Consequently, deg defines a grading, which will be referred to as the Cartan grading on V C . Analogous Cartan gradings are defined for Hurwitz pairs ofdimensions 2 and 4 by restriction of the same degree map to the bases { e , e } and { e , e , u , v } , where the grading groups are Z and Z , respectively. It is obviousthat the associated Cartan basis of V C is homogeneous. Given a Hurwitz algebra C with dim C >
1, the Cartan grading on V C will be denoted by Γ Z ( V C ). Example 4.8.
Consider the Cartan grading on V C . If we force the relationdeg( x + ) = deg( x − ) for x in the Cartan basis of C , we obtain a grading by Z on V C which also restricts to a grading on T C . This grading will be called the INE GRADINGS ON KANTOR SYSTEMS OF HURWITZ TYPE 27
Cartan grading on T C and denoted by Γ Z ( T C ), and its degree map is given by:(4.5) deg( e ) = − deg( e ) := (1 , , , deg( u ) = − deg( v ) := (0 , , , deg( u ) = − deg( v ) := (0 , , , deg( u ) = − deg( v ) := ( − , − , − . For a Hurwitz algebra C of dimension 2 or 4, we similarly define its Cartan gradingΓ Z ( T C ) by restriction of deg to the corresponding basis { e , e } or { e , e , u , v } ;in these two cases, the grading groups are Z and Z , respectively. Proposition 4.9.
The gradings given in Examples 4.4, 4.5, 4.6, 4.7 and 4.8 arefine and given by their universal groups and their universal degrees.Proof.
It is clear that all these gradings are fine because their homogeneous com-ponents are 1-dimensional. For the 1-dimensional cases, the result is trivial. Forthe Cayley-Dickson gradings, the result follows from Propositions 2.4 and 2.5 (notethat the Kantor pair case with dim C = 2 and char F = 3 is excluded, hence therequirements to apply Proposition 2.4 hold).Now consider the Cartan grading on the Cayley pair and let G and deg denoteits universal group and degree, respectively. By Lemma 3.17, the bilinear trace ishomogeneous, from where it follows that deg is determined by its restriction to V + C (or V − C ), and also that the following relations hold: deg( e σ ) + deg( e − σ ) = 0 anddeg( u σi ) + deg( v − σi ) = 0 for i ∈ { , , } and σ = ± . Denote a := deg( e +1 ), b :=deg( e +2 ), c := deg( u +1 ), d := deg( u +2 ). From the relations 0 = { e +1 , u − , e +2 } ∈ F u +1 ,0 = { e +1 , u − , e +2 } ∈ F u +2 , 0 = { e +1 , u − , u +2 } ∈ F v +3 and 0 = { e , u , e } ∈ F u weget that deg( v +1 ) = a + b − c , deg( v +2 ) = a + b − d , deg( v +3 ) = − b + c + d anddeg( u +3 ) = a + 2 b − c − d . Since G is generated by its support, it is also generatedby { a, b, c, d } . By the universal property, the map f : G → Z sending { a, b, c, d } tothe canonical basis of Z is a group epimorphism inducing the Cartan grading; thisalso implies that f is actually a group isomorphism and that the Cartan gradingis given by its universal degree. (Note that this proof avoids computing the tripleproduct for all the 8 possible cases to check all relations between generators.) Forthe Cartan gradings on Hurwitz pairs of dimensions 2 and 4, the result follows fromthe same arguments.Finally, consider the Cartan grading on the Cayley triple system T C ; let H anddeg denote its universal group and degree. Note that H is defined exactly bythe same relations as the universal group G of the Cartan grading on V C and theadditional relation given by deg( x + ) = deg( x − ). It is easy to see that the lastrelation is equivalent to a + b = 0, from where it follows that H ∼ = Z and that theCartan grading on T C is given by its universal degree. Again, the same argumentshold for Hurwitz triple systems of dimensions 2 and 4. (cid:3) Classification of fine gradings on Hurwitz Kantor pairs.
Remark . Recall that in the case where dim C = 1 we have only one fine gradingon V C (see Example 4.4). So it remains to deal with the classification in the caseswith dim C > Theorem 4.11.
Let Γ be a fine grading on V C and dim C ≥ . Then: If char F = 3 and dim C = 2 , then Γ is, up to equivalence, the Cartangrading. If either char F = 3 or dim C = 2 , then Γ is, up to equivalence, the Cayley-Dickson grading or the Cartan grading.Proof. Let Γ be a fine G -grading on V C . We will deal first with the case 1), soassume that char F = 3 and dim C = 2. By Theorem 3.12, the group Aut( V C ) ∼ =GL ( F ) acts transitively on the set of bases of V + C , so that by applying an auto-morphism we can assume that a Cartan basis { e , e } of C is homogeneous in V + C .Then, the subspaces ker U + e i = F e − i , for i = 1 ,
2, are graded, so that the Cartan ba-sis is also homogeneous in V − C . Consequently, since Γ is fine, it must be equivalentto the Cartan grading.From now on, consider the case with either char F = 3 or dim C = 2. Assumethat dim C = 8; since the cases with dim C = 2 , x + ∈ V + C ofrank 2 (for the case x − ∈ V − C , the proof is analogous). By applying an automor-phism we can assume that x + = 1 + . Take g = − deg(1 + ). By Proposition 2.4,the shifted grading Γ [ g ] restricts to a grading Γ C on C = V + C . Notice that (Γ [ g ] ) + and (Γ [ g ] ) − have the same homogeneous components, although the degrees are notnecessarily the same. Since Γ is fine, Γ C is fine on C . Then Γ C cannot be equivalentto the Cartan grading on C because its extension admits a proper refinement on V C (namely, the Cartan grading on V C ). Thus Γ C must be equivalent to the Cayley-Dickson grading on C , and consequently, Γ is equivalent to the Cayley-Dicksongrading on V C .Now consider the case where there are no homogeneous elements of rank 2,that is, all homogeneous elements have rank 1 (which forces them to be isotropic).Since n is nondegenerate, we can take two homogeneous elements x, y ∈ V + C suchthat n ( x, y ) = 0. Since n ( x, y ) = n ( x + y ) − n ( x ) − n ( y ) = n ( x + y ), we alsohave that n ( x + y ) = 0. Up to automorphism (see Proposition 3.10), we canassume that x + y = 1 + . Then 2 = t (1 + ) = t ( x ) + t ( y ), and so we have either t ( x ) = 0 or t ( y ) = 0. Without loss of generality, consider the case t ( x ) = 0.Since x is isotropic, we have x = λx for λ = t ( x ) = 0 and e := λ − x is anisotropic idempotent. Using the same arguments as in [EK13, Chapter 4], we cancomplete e to a Cartan basis where e := 1 − e is an isotropic idempotent. Since y = 1 − x = ( e + e ) − λe = (1 − λ ) e + e is isotropic it follows that λ = 1.Therefore x = e and y = e are homogeneous in V + C .Notice that U e +1 ,e +2 and U e + i , for i = 1 ,
2, are homogeneous maps; hence we havethat Ker( U e +1 ,e +2 ) = F e ⊕ F e and Ker( U e + i ) = F e i ⊕ U ⊕ V are graded subspacesin V − for i = 1 ,
2, where U := span { u i | i = 1 , , } and V := span { v i | i = 1 , , } .Intersecting these subspaces we get that U ⊕ V , F e and F e are graded subspacesin V − . Analogously, U ⊕ V is graded in V + . Now we will prove that U and V are graded subspaces in V σ for σ = ± . Let deg( e + i ) = g i for i = 1 , e − ) = − g anddeg( e − ) = − g . Take 0 = u + v ∈ ( U ⊕ V ) − g for some g ∈ G , with u ∈ U , v ∈ V . Then { e +1 , u + v, e +2 } = − ( u + 2 v ) ∈ ( U ⊕ V ) + g + g + g and { e − , u + 2 v, e − } = − ( u + 4 v ) ∈ ( U ⊕ V ) − g . It follows that v and u are homogeneous of degree g in V − C .Therefore U and V are graded in V + , and similarly in V − . If 0 = u ∈ U − g for some g ∈ G , then { e +1 , u, e +2 } = − u ∈ U + g + g + g , and it follows that the homogeneouscomponents of U + and U − coincide. Similarly, the homogeneous components of V + and V − coincide. INE GRADINGS ON KANTOR SYSTEMS OF HURWITZ TYPE 29
Following again [EK13, Chapter 4], we will now construct a homogeneous Cartanbasis. Denote U := e Ce and V := e Ce . Take a homogeneous basis { u i } i =1 of U + such that n ( u u , u ) = 1 (this is possible because n ( U , U ) = 0 and n ( U ) = 0).Note that { u i } i =1 are homogeneous in U − too. Then { v = u u , v = u u , v = u u } is the dual basis in V . Since { e σ , u − σi , u σj } are homogeneous for i, j = 1 , , v i is homogeneous in V σC for i = 1 , , σ = ± . We have obtaineda homogeneous Cartan basis of V C . Since Γ is fine, it must be equivalent to theCartan grading on V C . (cid:3) Classification of fine gradings on Hurwitz Kantor triple systems.
Remark . Recall from Corollary 3.7 that the classification of fine gradingscoincide on T C and T ′ C . It turns out that fine gradings on the triple system T ′ C ,where C is a Cayley algebra, have also been classified in an independent work[DETpr], where this triple system has been used to classify gradings on a 3-foldcross product denoted by ( C , X C ). Remark . We already know that in the case where dim C = 1 there is only onefine grading on V C (see Example 4.4). We will now deal with the remaining cases. Theorem 4.14.
Let Γ be a fine grading on T C and dim C ≥ . Then Γ is, up toequivalence, the Cayley-Dickson grading or the Cartan grading.Proof. Let Γ be a fine G -grading on V C . Only the case dim C = 8 will be considered(the cases with dim C = 2 , x in some orbit O ( λ ). Since F is algebraically closed, we can scale x and assume that n ( x ) = 1.Furthermore, up to automorphism of T C , we can assume that x = 1 is homogeneous.By Proposition 2.5, for g = deg(1), Γ [ g ] restricts to a grading Γ C on C , which isfine because Γ is fine. With exactly the same argument used in the first part ofthe proof of Theorem 4.11, it follows that Γ is equivalent to the Cayley-Dicksongrading on T C .Now consider the case where all homogeneous elements are in the orbit of rank1. Since n is nondegenerate we can take two homogeneous elements x, y ∈ T C such that n ( x + y ) = 0. Following the proof of Theorem 4.11, we deduce that x and y are isotropic idempotents in C , which will be denoted by e and e . ByTheorem 3.4, we can use the product of T ′ C instead of the one in T C , because theclassification of fine gradings (and the orbits under their automorphisms groups)coincide on both triple systems. Denote U := e Ce and V := e Ce . Since e and e are homogeneous, { e , C, e } ′ = U and { e , C, e } ′ = V are graded subspaces.Following the proof of Theorem 4.11, we can take a homogeneous basis { u i } i =1 of U such that n ( u u , u ) = 1, and construct its dual basis { v i } i =1 in V , so that weobtain a homogeneous Cartan basis of T C . Since Γ is fine, it must be equivalent tothe Cartan grading on T C . (cid:3) Weyl groups of fine gradings on Hurwitz Kantor systems
In this section we compute the Weyl grups of the fine gradings on Kantor pairsand triple systems of Hurwitz type.
Remark . Consider the Kantor pair V F and triple system T F associated to the1-dimensional Hurwitz algebra F . Let Γ be the only fine grading on V F or on T F . In both cases, it is clear that W (Γ) is the trivial group. Thus, it remains to deal withthe Kantor pairs and triple systems associated to Hurwitz algebras of dimensiongreater than 1. Theorem 5.2.
Let C be a Hurwitz algebra of dimension m for some m ∈ { , , } ,and Γ CD the Cayley-Dickson Z × Z m -grading (resp., Z m +12 -grading) on V C (resp.,on T C ). Then we have: W (Γ CD ) ∼ = (cid:26)(cid:18) a A (cid:19) | a ∈ Z m , A ∈ GL m ( Z ) (cid:27) . GL m +1 ( Z ) . Proof.
Denote the Cayley-Dickson grading on V C , resp. on T C , by Γ V , resp. byΓ T . Also denote W := (cid:26)(cid:18) a A (cid:19) | a ∈ Z m , A ∈ GL m ( Z ) (cid:27) . For the case m = 3, recall from [EK13] that if Γ C is the fine Z m -grading on C , then W (Γ C ) ∼ = Aut( Z m ) ∼ = GL m ( Z ). By restriction of the automorphisms, the sameholds for the cases m = 1 , V C with m = 3, so that C = C isthe Cayley algebra. Since the automorphisms of C extend to V C , we can identify W (Γ C ) ≤ W (Γ V ). Therefore, W (Γ V ) has a subgroup corresponding to the blockstructure G C := (cid:26)(cid:18) A (cid:19) | A ∈ GL ( Z ) (cid:27) ∼ = GL ( Z ) . Now, fix a traceless element x ∈ C of norm 1 that is homogeneous in V + C . Then, x is also homogeneous in V − C and, by Lemma 3.3, the map L x is a homogeneousautomorphism of Γ V . Moreover, L x induces an element of W (Γ V ) corresponding toa block of the form M = (cid:18) a A (cid:19) for some 0 = a ∈ Z , A ∈ GL ( Z ). It is easy to see that the group generated by M and G C is W , which implies the inclusion W . W (Γ V ). Furthermore, we have W (Γ V ) ≤ { ϕ ∈ Aut( Z × Z ) | ϕ (Supp Γ σ V ) = Supp Γ σ V , σ = ±} ≡ W . We have proven the isomorphism W (Γ V ) ∼ = W .The same arguments above prove the result for the Kantor pairs in the cases m = 1 ,
2, and also that W . W (Γ T ) in all the cases for triple systems. Sincewe have the natural inclusion W (Γ T ) ≤ W (Γ V ) ∼ = W , the result follows for triplesystems too. (cid:3) Theorem 5.3.
Let C be a Hurwitz algebra with dim C = 2 m > . Then, W (Γ Z ( V C )) ∼ = W (Γ Z ( T C )) ∼ = Sym( Z m − ) × Sym( b Z ) ∼ = Sym(2 m − ) × Z . Aut Φ , where Φ denotes the root system of K ( C ) .Here, an element ( ρ, τ ) ∈ Sym( Z m − ) × Sym( b Z ) corresponds to the permutationof subspaces F v αg F v τ ( α ) ρ ( g ) in T C or V + C .Proof. We already know that Weyl groups extend as subgroups when extendinggradings from Kantor triple systems to Kantor pairs, and from Kantor pairs totheir Kantor-Lie algebras, so that we have W (Γ Z ( T C )) ≤ W (Γ Z ( V C )) ≤ Aut Φ.
INE GRADINGS ON KANTOR SYSTEMS OF HURWITZ TYPE 31
From the automorphisms given in [EK13, Proof of Th. 4.17], it follows that foreach ρ ∈ Sym( Z m − \ { ¯0 } ) and τ ∈ Sym( b Z ) there exists ϕ ∈ W (Γ Z ( T C )) acting via ϕ ( F v αg ) = F v τ ( α ) ρ ( g ) . Hence Sym( Z m − \ { ¯0 } ) × Sym( b Z ) is a subgroup of W (Γ Z ( T C )).Since L x g is a homogeneous automorphism of T C for g ∈ Z m − , it follows from(1.20) that W := Sym( Z m − ) × Sym( b Z ) is a subgroup of W (Γ Z ( T C )).Let ϕ ∈ W (Γ Z ( V C )). We claim that ϕ ∈ W . If we compose ϕ with an element of W , we can assume that ϕ fixes the subspace ( F v ) + = F e +1 , and therefore ϕ also fixesthe subspace with opposite degree, which is the subspace ( F v ω ) − = F e − . Then, thePeirce subspaces in Proposition 4.2 associated to the homogeneous idempotent e are fixed too. Consequently, ϕ fixes the homogeneous component ( F v ω ) + = F e +2 andpermutes the homogeneous components of L = g ∈ Z m − ( F v g ) + = L i =1 F u + i . Then,composing ϕ again with an element of W , we can assume that ϕ fixes each of thesubspaces ( F v g ) + for g ∈ Z m − . Again, since the Peirce subspaces are preserved,it follows that each of the subpaces ( F v ωg ) + for g ∈ Z m − are fixed too, so that ϕ isthe identity map. We have proven that W (Γ Z ( V C )) ≤ W . Consequently, we havethat W ≤ W (Γ Z ( T C )) ≤ W (Γ Z ( V C )) ≤ W , and the result follows. (cid:3) Remark . Let C be a Hurwitz algebra. Consider the root decomposition of K ( V C ) associated to the Cartan grading Γ Z ( V C ) on V C . Let x α , x β be elements ofthe Cartan basis of V C with associated roots α , β . By Theorem 5.3, the action of theWeyl group on the homogenous components is transitive; thus all the roots relatedto the homogeneous components of V σC have the same length, so that k α k = k β k .Hence, Proposition 1.15 shows that the left Peirce constant relating x α and x β isexactly cos( α, β ). Since the left Peirce constants appearing in Proposition 4.2 areexactly 1, 1 /
2, 0 and − /
2, the corresponding angles appearing between their rootsare 0 ◦ , 60 ◦ , 90 ◦ and 120 ◦ , respectively.6. Induced fine gradings via the Kantor construction
In this section we give a summary of the fine gradings on Lie algebras obtained,using the Kantor construction, from the fine gradings on Kantor pairs of Hurwitztype. We will first determine the associated Kantor-Lie algebra for each case.Recall from [Al79] that the Kantor construction can be regarded as K ( A ) = S − ⊕ A − ⊕ ( T A ⊕ Der( A )) ⊕ A + ⊕ S + , where T A = { T x := V x, | x ∈ A } .Let ( C, − ) be a Hurwitz algebra, and recall that in this case Der( C ) = Inder( C ).By [Al79, Corollary 6], K ( C ) is a simple Lie algebra. Recall also that if the dimen-sion of C is 1, 2, 4 or 8, then the dimension of Der( C ) is 0, 0, 3 or 14, respectively.Therefore the dimension of K ( C ) is 3, 8, 21 or 52, respectively. Consequently, in thecase that dim C equals 1, 2 or 8, the simple Lie algebra K ( C ) must be isomorphicto a = sl , a = psl or f , respectively. On the other hand, for the case dim C = 4there are two simple Lie algebras of dimension 21 (of types B and C ), but sincethe type of the main grading on K ( C ) is (0 , , , , , ,
1) and there is no gradingon o ( B ) with such type (whereas there is one for the Lie algebra of type C )(see [EK13] and references therein), we can conclude that K ( C ) is isomorphic to sp = c . Note that the Lie algebras we obtained are the ones appearing in the firstrow (or column) of the well-known Freudenthal magic square. Recall from sections above that, for the Hurwitz Kantor pair V C in the case withchar F = 3 and dim C = 2, the results related to automorphisms and gradings aredifferent. The reason of this is that K ( C ) = a is exceptional in this case, and it isalso well-known that Aut( a ) is an exceptional Lie group of type G [Ste61, § A required a different treating in the case of a . Also, recall that a = sl if char F = 3, but a = sl /Z ( sl ) with Z ( sl ) = F I if char F = 3. Proposition 6.1.
Let C be a Hurwitz algebra with dim C = 2 m > , and either char F = 3 or dim C = 2 . The Cayley-Dickson grading on the Kantor pair V C extends to a fine grading on K ( V C ) with universal group Z × Z m . For each pos-sible case, dim C = 2 , , , the type of the grading is (8) , (15 , and (31 , , ,respectively.Proof. The first part follows from Prop. 1.6. Consider the case dim C = 8. Denote V = V C . Recall that both subspaces K ( V ) = V + and K ( V ) − = V − consist ofeight 1-dimensional homogeneous components. Also, note that dim K ( V ) = 22and dim K ( V ) ± = 7.For x g , x h in the Cayley-Dickson basis B CD ( C ) of C , we have that K ( x + g , x + h ) = L ψ ( x g ,x h ) where ψ ( x, y ) := x ¯ y − y ¯ x . Therefore, K ( x + g , x + h ) = 0 if g = h , thatis, K ( V ) ,e ) = 0, where e is the neutral element of Z . Hence, Supp K ( V ) = { (2 , g ) | e = g ∈ Z } . The Weyl group in Theorem 5.2 shows that the homoge-neous components of Supp K ( V ) are in the same orbit under the action by au-tomorphisms, so that they have the same dimension, and consequently they are1-dimensional. The same arguments hold for Supp K ( V ) − .Note that for x g ∈ B CD ( C ) we have that D σ ( x g , x g ) = id V σ (that is, x − g is theconjugate inverse of x + g ), so that dim K ( V ) ,e ) = 1. Again, the Weyl group inTheorem 5.2 shows that the homogeneous components with degrees { (0 , g ) | e = g ∈ Z } are in the same orbit under the action by automorphisms, so that theymust have the same dimension, which must be 3. It follows that the type of thegrading on K ( V ) is (31 , ,
7) (note that this coincides with the type of the grading on f given in [EK13, Corollary 5.40]). The proof is analogous in the cases dim C = 2,4. (cid:3) Proposition 6.2.
Let C be a Hurwitz algebra with dim C = 2 m . The Cartangrading on the Kantor pair V C extends to a fine grading on K ( V C ) with universalgroup Z m +1 , that is, the Cartan grading on K ( V C ) .Proof. Consequence of the correspondence given in Prop. 1.6 and the fact thatCartan gradings on Lie algebras are produced by maximal tori. (cid:3)
In Figure 2 we summarize the general information for each fine Kantor-compatiblegrading on the Kantor-Lie algebras of our study, including the type and universalgroup. (The types of the gradings follow from the classification of fine gradings onthe classical simple Lie algebras [EK13].) Recall that we denote Cartan gradingsand Cayley-Dickson gradings by Γ Z and Γ CD , respectively. Acknowledgements
The authors are very thankful to Alberto Elduque forproviding important references and some advice.
INE GRADINGS ON KANTOR SYSTEMS OF HURWITZ TYPE 33 dim C K ( C ) a a c f U (Γ Z ) Z Z Z Z Type(Γ Z ) (3) (6 ,
1) (18 , ,
1) (48 , , , U (Γ CD ) − Z × Z (char F = 3) Z × Z Z × Z Type(Γ CD ) − (8) (15 ,
3) (31 , , Figure 2.
Fine gradings obtained via the Kantor constructionfrom Kantor pairs of Hurwitz type
References [Al78] B.N. Allison,
A class of nonassociative algebras with involution containing the class ofJordan algebras , Math. Ann. (1978), 133–156.[Al79] B. Allison,
Models of isotropic simple Lie algebras , Communications in Algebra, 7:17,1835-1875 (1979).[Ara17] D. Aranda-Orna,
Fine gradings on simple exceptional Jordan pairs and triple systems ,J. Algebra (2017) 517–572.[AF84] B. Allison and J. Faulkner,
A Cayley–Dickson process for a class of structurable algebras ,Trans. Amer. Math. Soc. (1984), no. 1, 185–210.[AF92] B.N. Allison and J.R. Faulkner,
Norms on structurable algebras , Comm. Algebra (1992), no. 1, 155–188.[AF99] B. Allison and J. Faulkner, Elementary groups and invertibility for Kantor pairs , Comm.Algebra (1999), 519–556.[AFS17] B. Allison, J. Faulkner and O. Smirnov, Weyl images of Kantor pairs , Canadian Journalof Mathematics (2017) 721–766.[AM99] H. Albuquerque and S. Majid, Quasialgebra structure of the octonions , J. Algebra (1999), no. 1, 188–224.[DETpr] A. Daza-Garca, A. Elduque and L. Tang,
Cross products, automorphisms, and gradings .Preprint: arXiv:2006.10324[Eld00] A. Elduque,
On triality and automorphisms and derivations of composition algebras ,Linear Algebra Appl. (2000), no. 1–3, 49–74.[Eld96] A. Elduque,
On a class of ternary composition algebras , J. Korean Math. Soc. (1996),no. 1, 183–203.[Eld98] A. Elduque, Gradings on octonions , J. Algebra (1998), no. 1, 342–354.[EK13] A. Elduque and M. Kochetov,
Gradings on simple Lie algebras , Mathematical Surveysand Monographs , American Mathematical Society, Providence, RI, 2013.[F94] J.R. Faulkner,
Structurable triples, Lie triples, and symmetric spaces , Forum Math. (1994), no. 5, 637–650.[H78] J. E. Humphreys, Introduction to Lie algebras and representation theory , Second printing,revised. Graduate Texts in Mathematics, 9. Springer-Verlag, New York-Berlin, 1978.[J89] N. Jacobson,
Basic algebra II , Second Edition, W. H. Freeman and Company, New York,(1989).[K72] I.L. Kantor,
Certain generalizations of Jordan algebras (Russian), Trudy Sem. Vektor.Tenzor. Anal. (1972), 407–499.[K73] I.L. Kantor, Models of the exceptional Lie algebras , Soviet Math. Dokl. (1973), 254–258.[KK03] I. Kantor, N. Kamiya, A Peirce Decomposition for Generalized Jordan Triple Systems ofSecond Order , Comm. Algebra (2003), no. 12, 5875–5913.[L75] O. Loos, Jordan Pairs , Lecture Notes in Mathematics, Vol. . Springer-Verlag, Berlin-New York, 1975.[MC03] K. McCrimmon,
A Taste of Jordan Algebras (2003), Springer-Verlag.[Smi92] O.N. Smirnov,
Simple and semisimple structurable algebras , Proceedings of the Inter-national Conference on Algebra, Part 2 (Novosibirsk, 1989), Contemp. Math. , Part 2,Amer. Math. Soc., Providence, RI, 1992, pp. 685–694.[Ste61] R. Steinberg,
Automorphisms of classical Lie algebras , Pacific J. Math. (1961), 1119–1129. [ZSSS82] K.A. Zhevlakov, A.M. Slin’ko, I.P. Shestakov, A.I. Shirshov, Rings that are nearly as-sociative , Academic Press, 1982.
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