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Theorem 1.1. Let k be a field of characteristic =26 , 3, and let L be a finite-dimensional central simple Lie algebra over k. Then the following are equivalent. (1) L is an isotropic simple Lie algebra of Chevalley type. (2) L has a non-trivial 5-grading. (3) There is a central simple structurable algebra A over k such that L =∼ K(A).

The implication (3 ) =⇒ (2 ) of Theorem 1.1 is obvious. The implication (2 ) =⇒ (1 ) is proved in Theorem 3.1 of § 3. The implication (1 ) =⇒ (3 ) is obtained in the very end of § 5 as a corollary of Theorem 5.10. Apart from the characteristic 0 case, Theorem 1.1 has been known in the following cases. If A is a central simple structurable algebra with trivial involution, then it is a Jordan algebra, and K(A) is 3-graded. A theorem of A. Premet [Pre83, Str04] establishes that any central simple 3-graded Lie algebra over a field of characteristic > 3 is a Chevalley type Lie algebra. Alternatively, the fact that K(A) is of Chevalley type follows from O. Loos’ theory of algebraic groups defined by Jordan pairs [Loo79-1, Loo79-2]. This theory, moreover, extends to arbitrary characteristic, provided one uses quadratic Jordan algebras instead of the usual ones. Assuming k is a field of characteristic =6 2, 3, 5, B. Allison and O. Smirnov [All78, Smi90] classified all central simple structurable k-algebras by direct methods, partially relying on the classification of simple Jordan algebras but not on the classification of simple Lie algebras in positive characteristic. Given that the resulting list of central simple structurable algebras is uniform in all characteristics, it is possible to conclude via case-by-case analysis that the central simple Lie algebras K(A) are exactly the central simple Lie algebras of Chevalley type, although this was not explicitly done in the above works. In [Sta09, BDMS] we gave a uniform, classification-free proof that any 5-graded central simple Lie algebra over an arbitrary field of characteristic =26 , 3, 5 is of Chevalley type, extending the argument used in the above-mentioned theorem of A. Premet. The same result also follows from [GGLN11, Theorem 3.1], which shows that if char(k) =26 , 3, 5 then the Lie algebras K(A) are non-degenerate, and the well-known fact (seemingly, due to A. I. Kostrikin and A. Premet; see e.g. [Wil76] and [Str09, Corollary 12.4.7]) that the only non-degenerate Lie algebras in the Block–Wilson–Premet–Strade classification are the Chevalley type Lie algebras. 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTORPAIRS 3 Apart from the above result, we have obtained in [BDMS, Theorem 4.3.1] a more precise version of Theorem 1.1 for structurable division algebras. Namely, we established a bijective correspondence between isotopy classes of central simple structurable division algebras over any field k of characteristic =6 2, 3, and classes of graded-isomorphic central simple Chevalley type Lie algebras of isotropic rank 1 over k. This correspondence does not extend to all central simple structurable algebras, since Chevalley type Lie algebras of isotropic rank > 1 may have several distinct 5-gradings corresponding to non-isomorphic structurable algebras. This is evidenced by the second main result of the present paper, Theorem 5.10, that classifies all 5-gradings on central simple Chevalley type Lie algebras L over algebraically closed fields of characteristic =6 2, 3 that give rise to an isomorphism L ∼= K(A). This classification may serve as a first step towards an alternative proof of the Allison–Smirnov classification theorem, eliminating the characteristic =56 restriction. Structurable algebras and 5-graded Lie algebras are also closely related to Kantor pairs [Ka72, AF99, AFS17, GGLN11]. Namely, every Kantor pair (K+,K−) identifies with the pair of sub- spaces (G−1, G1) of a 5-graded Lie algebra G = G(K+,K−), called the standard graded embed- ding of (K+,K−) in [AF99], with a trilinear operation corresponding to the triple [[x, y], z] in G; see § 4 for the details. This makes our results applicable to the study of simple Kanor pairs. In particular, as a corollary of the implication (2 ) =⇒ (1 ) of Theorem 1.1, we prove in Corollary 4.12 that every simple Kantor pair over a field of characteristic =6 2, 3 is non-degenerate, removing the restriction of invertibility of 5 in [GGLN11, Theorem 3.1]. All commutative rings we consider are assumed to be associative and unital. All algebras are finite-dimensional over the respective scalars unless explicitly mentioned otherwise. 2. Preliminaries on Z-graded Lie algebras Definition 2.1. Z Let R be a commutative ring, and let L = Li∈Z Li be a -graded Lie algebra over R. We say that L is (2n +1)-graded, if Li =0 for all i ∈ Z such that |i| > n. The grading is non-trivial, if L= 6 L0. To simplify the notation, we will denote by Aut(L) the group of R-Lie algebra automor- phisms and by Der(L) the ring of R-Lie derivations of L, omitting the reference to R. We also occasionally consider Aut(L) and Der(L) as functors on the category of R-algebras R′, so that ′ ′ ′ ′ Aut(L)(R ) = AutR′ (L ⊗R R ), Der(L)(R ) = DerR′ (L ⊗R R ). Note that Aut(L) and Der(L) are naturally represented by closed R-subschemes of the affine R-scheme of R-linear endomorphisms of L. Z Definition 2.2. Let L = L Li be a -graded Lie algebra over a commutative ring R. Consider i∈Z ∼ ′ the 1-dimensional split R-subtorus S = Gm of Aut(L) defined as follows: for any R-algebra R , ′ ′ i any t ∈ Gm(R ), and any i ∈ Z, v ∈Li ⊗R R , we set t · v = t v. We call S the grading torus of L. Z Definition 2.3. L = L Li be a -graded Lie algebra over a commutative ring R. The grading i∈Z derivation on L is the derivation ζ ∈ Der(L) such that one has

ζ(x)= i · x for any i ∈ Z and any x ∈Li. If L contains an element ζ such that ad(ζ) is the grading derivation, we occasionally call ζ a grading derivation of L by abuse of language. 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTORPAIRS 4 Definition 2.4. Let L be a finite-dimensional 5-graded Lie algebra over a commutative ring R 4 such that 2, 3 ∈ R×. We say that x ∈L is algebraic, if the endomorphism exp(x)= 1 ad(x)i P i! i=0 is an automorphism of L as an R-Lie algebra. We say that L is algebraic, if all elements x ∈Li, i =06 , are algebraic. Remark 2.5. By [BDMS, Lemma 3.1.7] any 3-graded Lie algebra, and hence any Jordan algebra over R with 2, 3 ∈ R× is algebraic; any 5-graded Lie algebra, and hence any structurable algebra over R with 2, 3, 5 ∈ R× is algebraic. (Despite that lemma is stated for algebras over a field, the proof is also valid over commutative rings.) By [BDMS, Theorem 4.2.8] any central simple structurable division algebra over a field of characteristic =26 , 3 is algebraic. Definition 2.6. Let L be a Lie algebra. An element x ∈L is called an absolute zero divisor, or a sandwich, if [x, [x, L]] = 0. The Lie algebra L is called non-degenerate, if it contains no non-zero absolute zero divisors. The following lemma elaborates on the proof of [GGLN11, Theorem 3.1]. Lemma 2.7. Let L be a simple 5-graded Lie algebra over a commutative ring R such that 2, 3 ∈ R×. If L is algebraic, it is non-degenerate. Proof. Since L is simple, R is in fact a field. Let U ⊆ L be a linear span of all absolute zero divizors. Then U is invariant under all Lie algebra automorphisms of L. It is straightforward to see that U is a Lie subalgebra. We claim that U is a graded Lie subalgebra. Indeed, by [GGLN11, Theorem 2.3] we have

U =(U ∩ (L−1 ⊕L1)) ⊕ (U ∩ (L−2 ⊕L0 ⊕L2)). Since the characteristic of R is =6 2, 3, it contains at least 4 distinct invertible elements. Con- sidering the action of the grading torus on U ∩ (L−1 ⊕L1) and U ∩ (L−2 ⊕L0 ⊕L2), one readily sees that they contain homogeneous components of all their elements. 4 Since L is algebraic, exp(x) = 1 ad(x)i is a Lie automorphism of L for all x ∈L , i =6 0. P i! i i=0 Considering the homogeneous components of of exp(x)(u) for an u ∈ Ui, −2 ≤ i ≤ 2, we see that [x, u] ∈ U. Since L is simple, it is generated by Li, i =06 (see e.g. [BDMS, Lemma 4.1.5]). Hence U is an ideal of L, and hence U = 0 or U = L. By [Zel80, Theorem 1] U is nilpotent, hence U =0.  The following results relate algebraicity to the classification of simple Lie algebras. Theorem 2.8. [BDMS, Theorem 4.1.8] Let L be an algebraic central simple 5-graded Lie algebra over a field k of characteristic different from 2, 3, such that L= 6 L0. Then the algebraic k-group G = Aut(L)◦ is an adjoint absolutely simple group of k-rank ≥ 1, satisfying L = [Lie(G), Lie(G)]. Lemma 2.9. [BDMS, Lemma 4.2.4 (i)] Let k be a field, char k =6 2, 3. Let G be an adjoint 2 simple algebraic group over k. Let L = Lie(G) be its Lie algebra. Let L = L Li be any i=−2 5-grading on L such that L1 ⊕L−1 =06 . The 5-graded Lie algebra L is algebraic. Theorem 2.10. Let L be a central simple 5-graded Lie algebra over a field k of characteristic different from 2, 3, such that L= 6 L0. Then L is of Chevalley type if and only if L is algebraic. 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTORPAIRS 5 Proof. By [BDMS, Lemma 3.1.6] a 5-graded Lie algebra L over k is algebraic, if and only if L ⊗k k¯ is algebraic, where k¯ is the algebraic closure of k. By definition, a Lie algebra L is ¯ of Chevalley type if and only if L ⊗k k is of Chevalley type. Thus, we can assume from the start that k is algebraically closed. Over an algebraically closed field k, simple Lie algebras of Chevalley type coincide with [Lie(G), Lie(G)], where G is an adjoint simple algebraic group over k (cf. [Sel67, Ch. III] and [BDMS, Lemma 4.1.6]), and all these Lie algebras are algebraic by Lemma 2.9. The converse holds by Theorem 2.8. 

3. 5-Graded simple Lie algebras In this section we establish the implication (2 ) =⇒ (1 ) of Theorem 1.1. Namely, we prove the following theorem. Theorem 3.1. Let L be a central simple 5-graded Lie algebra over a field k of characteristic different from 2, 3, such that L= 6 L0. Then L is of Chevalley type. The proof relies on the Block–Wilson–Premet–Strade classification of simple Lie algebras over algebraically closed fields of characteristic =26 , 3, and in what follows we use the notation of [Str04] for the classes of central simple Lie algebras that arise in that classification.

Lemma 3.2. Let L 6= L0 be a simple 5-graded Lie algebra over a field k of characteristic S s Z =26 , 3, and let be the corresponding grading torus. Let L = Li=−r L[i] be another -grading −1 on L such that Li=−r L[i] is generated as a Lie algebra by L[−1] =6 0. If the second grading is preserved by S(k), then L[−1] 6⊆ L0. Proof. Since S preserves the second grading, we have 2 L[i] = M (L[i] ∩Lj) j=−2 for all −r ≤ i ≤ s. Assume that L[−1] ⊆L0. Then L[−i] ⊆L0 for all 1 ≤ i ≤ r. For any x ∈Lj, −2 ≤ j ≤ 2, we have t · x = tjx for any t ∈ S(k). Since char(k) =26 , 3, there is t ∈ k such that ±1 ±2 t =6 1 and t =6 1. Then Lj ⊆ Li≥0 L[i] for all j =6 0. Since L is simple, it is generated by  Lj, j =06 , hence L = Li≥0 L[i]. However, this contradicts L[−1] =06 . Lemma 3.3. Let L be one of the simple graded Cartan type Lie algebras X(m, n)(2), X ∈ {W,S,H,K}, in the notation of [Str04], over an algebraically closed field k of characteristic 5. Then L does not have a non-trivial 5-grading. 2 s Proof. Let L = Li=−2 Li be a non-trivial 5-grading on L, and let L = Li=−r L[i] be the standard grading on L. Note that r = 1 for X = W,S,H, and r = 2, dim(L[−2])=1 for X = K. By [Str04, Theorem 7.4.1] we can assume that the grading torus S of the 5-grading preserves the standard grading. Then by Lemma 3.2 the 5-grading on L[−1] is non-trivial. Since L[−1] is contained in the restricted subalgebra X(m, 1)(2) of L, the induced 5-grading on the restricted subalgebra is non-trivial, and hence we can assume that L = X(m, 1)(2) from the start. s+1 We have L[−1] = ad(L[−1]) (L[s]) (see e.g. [SF88, Ch. 4, proofs of Theorems 2.4, 3.5, 4.5, 5.5]). We claim that L[−1] contains an element y =06 of non-zero 5-grading such that 4 s−3 ad(y) ad(L[−1]) (L[s])=0. 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTORPAIRS 6 4 Since any y ∈ L−2 ⊕L2 satisfies ad(y) = 0, we only need to consider the case where L[−1] ⊆ L−1 ⊕L0 ⊕L1. Since the 5-grading on L[−1] is non-trivial, there is a non-zero element y ∈ 4 4 s−3 L[−1] ∩ (L−1 ⊕L1). Then ad(y) (L) ⊆ L±2, hence ad(y) ad(L[−1]) (L[s]) ⊆L[−1] ∩L±2 = 0, as required. Assume without loss of generality that y ∈ L1 or y ∈ L2. By [Wil75, Theorem 2] the space L[−1] is an irreducible representation for the group of Lie algebra automorphisms of L preserving the standard grading. Hence L[−1] has a basis consisting of elements x1 = y, 4 s−3 x2,...,xdim(L[−1]) such that ad(xi) ad(L[−1]) (L[s]) =0 for all i. Any z ∈ L[−1] can be dim(L[−1]) s+1 s+1 written as z = Pi=1 λixi. Any endomorphism Qk=1 ad(zk) ∈ ad(L[−1]) , zk ∈ L[−1], is a linear combination of products of basic derivations ad(xi) of length s +1. Assume first that X = W,S,H, so that r = 1, and L[−1] is abelian. Then we can reorder the terms of such a product of basic derivations so that all occurences of the same basic derivation are brought 4 s−3 together. Since ad(xi) ad(L[−1]) (L[s])=0, we conclude that every non-zero such product contains ≤ 3 occurences of each ad(xi). In all three cases X = W,S,H we have dim(L[−1])= m. If X = W , then m ≥ 1 and s = 4m − 1. If X = S, then m ≥ 2 and s = 4m − 2. If X = H, then s =4m − 3 and m ≥ 2. Therefore, s +1 ≥ 3m, hence the only possible non-zero product of length s +1 is the one that contains exactly 3 occurences of each ad(xi). Consequently, 3 L[−1] ⊆ ad(y) (L) ⊆ L1 ⊕L2. But then the total 5-grading of a product of s +1 derivations ad(x), x ∈ L[−1], is at least s +1. In all cases s +1 ≥ 4. If s +1 > 4, then this implies s+1 s+1 ad(L[−1]) = 0, a contradiction. If s +1=4, then ad(L[−1]) (L[s]) ⊆ L2, so that all s+1 x ∈L[−1] have degree 2, and then again ad(L[−1]) (L[s])=0, a contradiction. It remains to consider the case X = K, where L[−1] is not abelian, r =2, dim(L[−2])=1. In this case we have m ≥ 3, dim(L[−1])= m−1, and s =4m, if m+3 ≡ 0 (mod 5), and s =4m+1 otherwise. As before, consider a product of basic derivations ad(xi) of length s+1. We can still reorder all basic derivations in any way we want, however, in the process we add new products where some pairs ad(xj) · ad(xi) will be replaced by ad([xi, xj]) ∈ ad(L[−2]). Since L[−2] is 1-dimensional, we may obtain ≤ 4 such new terms ad([xi, xj]), and after that the new products will always be 0. These additional products thus contain ≥ s+1−2·4= s−7 basic derivations ad(xi). If m +3 ≡ 0 (mod 5), then m ≥ 7, and s − 7=4m − 7=3(m − 1)+ m − 4 > 3(m − 1), so such a product of ≥ s − 7 derivaions is necessarily zero. If m + 3 6≡ 0 (mod 5), then s − 7=4m − 6=3(m − 1) + m − 3 may still be equal 3(m − 1), if m =3. Then we conclude as in the previous case that L[−1] ⊆ L1 ⊕L2 (and automatically L[−2] ⊆ L2). Hence the total s+1 degree of a product of basic derivations is ≥ s − 7 ≥ 3(m − 1)=6, hence ad(L[−1]) (L[s])=0, a contradiction. 

Theorem 3.4. Let L be a central simple 5-graded Lie algebra over an algebraically closed field k of characteristic different from 2, 3, such that L= 6 L0. Then L is a simple Lie algebra of Chevalley type.

Proof. If char k =6 5, then L is a Chevalley Lie algebra by Remark 2.5 combined with Theo- rem 2.10. Assume char k = 5. According to the Block–Wilson–Premet–Strade classification theorem [Str04], it is enough to check that L is not of Cartan or Melikian type. Assume first that L is a simple Lie algebra of Cartan type [Str04, Definition 4.2.4]. Let L = L(−r) ⊇ ... ⊇ L(s) be a standard filtration of L. By [Str04, Theorem 4.2.7 (3)] the standard filtration is invariant under all automorphisms of L. In particular, it is invariant 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTORPAIRS 7 under the grading torus S of the 5-grading. Let s Gr(L)= M Gr(L)[i] i=−r be the associated graded Lie algebra. Hence Gr(L) carries an induced 5-grading. The induced F× S 5-grading is non-trivial, since there is 0 =6 x ∈ Li, i =6 0, and t ∈ 5 ⊆ (k) such that t · x = tix =6 x, and hence t acts non-trivially on the image of x in Gr(L). The derived series of Gr(L) also inherits the 5-grading, hence Gr(L)(∞) is 5-graded. We show that the induced 5-grading on Gr(L)(∞) is also non-trivial. Indeed, if it were trivial, then (∞) Gr(L) ⊆ Gr(L)0. Since Gr(L) is also a Lie algebra of Cartan type in the sense of [Str04, Definition 4.2.4], Gr(L)(∞) is simple by [Str04, Theorem 4.2.7 (1)]. Then we have (∞) (∞) (∞) Gr(L) ⊆ (Gr(L)0) ⊆ Gr(L) , (∞) (∞) which implies Gr(L) = (Gr(L)0) . Then by [Str04, Lemma 4.2.5] we have Gr(L) = Gr(L)0, which contradicts the non-triviality of the 5-grading on Gr(L). By [Str04, Theorem 4.2.7 (2)] we have Gr(L)(∞) = X(m, n)(2), X = W,S,H,K. By Lemma 3.3 these algebras do not have non-trivial 5-gradings. Now let L = M(2, n1, n2) be a simple Lie algebra of Melikian type. By [BKM15, Theorem 1.2] we can assume that the grading torus corresponding to the 5-grading under consideration s preserves the standard granding L = Li=−3 L[i] of L. Then the grading derivation ζ corre- sponding to the 5-grading is a homogenous derivation of L. Let W (2, n1, n2) be the standard simple Witt subalgebra of L. Then by [Str04, Theorem 7.1.4] any homogeneous derivation of L acts non-trivially on W (2, n1, n2), and hence this subalgebra carries a non-trivial 5-grading induced by ζ. However, this is not possible by Lemma 3.3.  ¯ ¯ Proof of Theorem 3.1. Let k be the algebraic closure of k. Then L ⊗k k is a central simple Lie algebra over k¯ with a non-trivial 5-grading. Then by Theorem 3.4 L ⊗k k¯ is a Lie algebra of Chevalley type. Then it is algebraic by Theorem 2.10. Hence L is algebraic, since L embeds into L ⊗k k¯. Again by Theorem 2.10 we conclude that L is of Chevalley type.  4. Simple structurable algebras and Kantor pairs Definition 4.1. A structurable algebra over a field k of characteristic not 2 or 3 is a finite- dimensional, unital k-algebra with involution (A,¯) such that

(1) [Vx,y,Vz,w]= V{x,y,z},w − Vz,{y,x,w} for x,y,z,w ∈ A, where the left hand side denotes the Lie bracket of the two operators, and where Vx,yz := {x y z} := (xy)z +(zy)x − (zx)y.

For all x, y, z ∈ A, we write Ux,yz := Vx,zy and Uxy := Ux,xy. The trilinear map (x, y, z) 7→ {x y z} is called the triple product of the structurable algebra. In [All78] and [All79], a structurable algebra is defined as an algebra with involution such that

(2) [Tz,Vx,y]=VTzx,y − Vx,Tzy for all x, y, z ∈A with Tx := Vx,1. The equivalence of (1) and (2) follows from [All79, Corollary 5.(v)]. 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTORPAIRS 8 Definition 4.2. Let (A,¯) be a structurable algebra; then A = H ⊕ S for H = {h ∈A| h = h} and S = {s ∈A| s = −s}. The elements of H are called hermitian elements, the elements of S are called skew-hermitian elements or briefly skew elements. As usual, the commutator and the associator are defined as [x, y]= xy − yx, [x, y, z]=(xy)z − x(yz), for all x, y, z ∈A. For each s ∈ S, we define the operator Ls : A→A by

Lsx := sx. The following map is of crucial importance in the study of structurable algebras: ψ : A×A→S : (x, y) 7→ xy − yx. Definition 4.3. An ideal of A is a two-sided ideal stabilized by . A structurable algebra (A, ) is simple if its only ideals are {0} and A, and it is called central if its center Z(A, ) = Z(A) ∩ H = {c ∈A| [c, A] = [c, A, A] = [A, c, A] = [A, A,c]=0} ∩ H is equal to k1. Recall the generalization of the Tits–Kantor–Koecher construction that associates to any structurable algebra A a 5-graded Lie algebra K(A). Let End(A) be the ring of k-linear maps from A to A. For each A ∈ End(A), we define new k-linear maps ǫ A = A − LA(1)+A(1), δ A = A + RA(1), where Lx and Rx denote left and right multiplication by an element x ∈A, respectively. Construction 4.4. [All79, §3] Let (A,¯) be a structurable algebra over a field k of characteristic not 2 or 3. Consider two copies A+ and A− of A with corresponding isomorphisms A → A+ : x 7→ x+ and A→A− : x 7→ x−, and let S+ ⊆ A+ and S− ⊆ A− be the corresponding subspaces of skew elements. Let Inn(A) be the k-subspace of End(A) spanned by all Vx,y, x, y ∈A. The 5-graded Lie algebra K(A) associated to A is the vector space

K(A)= S− ⊕A− ⊕ Inn(A) ⊕A+ ⊕ S+ with the 5-graded components

K(A)−2 = S−, K(A)−1 = A−, K(A)0 = Inn(A), K(A)1 = A+, K(A)2 = S+, and the following Lie bracket.

• [Inn,K(A)]

[Va,b,Va′,b′ ] := Va,bVa′,b′ − Va′,b′ Va,b = V{a,b,a′},b′ − Va′,{b,a,b′}∈ Inn(A) ǫ [Va,b, x+] := (Va,bx)+ ∈A+ [Va,b,y−] := (Va,by)− =(−Vb,ay)− ∈A− δ ǫδ [Va,b,s+] := (Va,bs)+ = −ψ(a, sb)+ ∈ S+ [Va,b, t−] := (Va,bt)− = ψ(b, ta)− ∈ S− 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTORPAIRS 9

• [S±, A±]

[s+, x+] := 0 [t−,y−] := 0

[s+,y−] := (sy)+ ∈A+ [t−, x+] := (tx)− ∈A−

• [A±, A±] ′ ′ [x+,y−] := Vx,y ∈ Inn(A)[y−,y−] := ψ(y,y )− ∈ S− ′ ′ [x+, x+] := ψ(x, x )+ ∈ S+

• [S±, S±] ′ ′ [s+,s+] := 0, [t−, t−] := 0

[s+, t−] := LsLt ∈ Inn(A) ′ ′ ′ ′ for all x, x ,y,y ∈A, all s,s , t, t ∈ S, and all Va,b,Va′,b′ ∈ Inn(A). In the case where A is a Jordan algebra, we have S = 0, and thus the Lie algebra K(A) has a 3-grading; in this case K(A) is exactly the Tits–Kantor–Koecher construction of a Lie algebra from a Jordan algebra. It is shown in [All79, §5] that the structurable algebra A is simple if and only if K(A) is a simple Lie algebra, and that A is central if and only if K(A) is central. Definition 4.5. A structurable algebra A is called algebraic, if K(A) is algebraic. Definition 4.6. Let A be a structurable algebra over a field k of characteristic =6 2, 3. An element x ∈ A is called an absolute zero divisor if Uxy = 0 for any y ∈ A. The algebra A is called non-degenerate if it has no non-trivial absolute zero divisors.

If an element x ∈ K(A)σ = Aσ is an absolute zero divisor of K(A), then it is represented by an absolute zero divisor of A; this follows from the fact that by Definition 4.4,

[xσ, [xσ,y−σ]] = −Vx,yx ∈Aσ for all x, y ∈A. The following theorem strengthens [BDMS, Theorem 4.1.1]. Theorem 4.7. Let A be a central simple structurable algebra over a field k of characteristic different from 2, 3. Then A is algebraic and non-degenerate. The algebraic k-group G = Aut(K(A))◦ is an adjoint absolutely simple group of k-rank ≥ 1, and K(A) = [Lie(G), Lie(G)]. Proof. The Lie algebra K(A) is a central simple Lie algebra with a non-trivial 5-grading, hence it is a Lie algebra of Chevalley type by Theorem 3.1. Hence it is algebraic by Theorem 2.10. By Lemma 2.7 K(A) is non-degenerate, and then A is non-degenerate. The remaining claims were established in [BDMS, Theorem 4.1.1] under the additional assumption that A is algebraic.  5-Graded Lie algebras and structurable algebras are closely related to Kantor pairs.

Definition 4.8 ([AF99]). A Kantor pair is a pair of finite-dimensional vector spaces (K+,K−) over k equipped with a trilinear product

{·, ·, ·}: Kσ × K−σ × Kσ → Kσ, σ ∈ {−1, 1}, satisfying the following two identities: 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTOR PAIRS 10

(KP1) [Vx,y,Vz,w]= V{x,y,z},w − Vz,{y,x,w};

(KP2) Ka,bVx,y + Vy,xKa,b = KKa,bx,y; where Vx,yz := {x, y, z} and Ka,bz := {a, z, b}−{b, z, a}. 2 If L = Li=−2 Li is a 5-graded Lie algebra and ζ is its grading derivation, then the pair (L1, L−1) is a Kantor pair with respect to the triple product {x, y, z} = −[[x, y], z] for all x, z ∈ Lσ, y ∈ L−σ (alternatively, one may set {x, y, z} = [[x, y], z], see e.g. [GGLN11]). Conversely, by [AF99, Theorem 7 and p. 535], for any Kantor pair (K+,K−) there exists a unique up to isomorphism 5-graded Lie algebra G = G(K+,K−), called the standard embedding of (K+,K−), such that the associated Kantor pair (G+, G−) is isomorphic to (K+,K−), G2σ = [Gσ, Gσ], G0 = kζ + [Gσ, G−σ], and ad : G−2 ⊕ G0 ⊕ G2 → End(G1 ⊕ G−1) is injective. If A is a structurable algebra over k, then the pair (A, A) with the triple product {x, y, z} = 2{x, y, z}A is a Kantor pair (we double the triple product in accordance with the conventions of [AF99]). It is easy to see that the corresponding standard embedding G(A, A) is graded- isomorphic to K(A)+ kζ [BDMS, § 3.2].

Definition 4.9. An ideal of a Kantor pair is a pair of subspaces Iσ ⊆ Kσ such that

{Iσ,V−σ,Vσ} + {Vσ,I−σ,Vσ} + {Vσ,V−σ,Iσ} ⊆ Iσ for σ = ±1. A Kantor pair with non-zero product is called simple, if it has no non-trivial ideals. The centroid of a Kantor pair is the set of all (c+,c−) ∈ End(K+) ⊕ End(K−) such that

cσ({xσ,y−σ, zσ})= {cσ(xσ),y−σ, zσ} = {xσ,c−σ(y−σ), zσ} = {xσ,y−σ,cσ(zσ)} for all xσ, zσ ∈ Kσ, y−σ ∈ K−σ. A Kantor pair is called central, if the centroid coincides with the set of scalar pairs (c,c), c ∈ k. According to the following result, every 5-grading on a simple Lie algebra of Chevalley type over a field of characteristic =6 2, 3 arises from a Kantor pair. In particular, every isotropic adjoint simple algberaic group can be constructed from a simple Kantor pair. Lemma 4.10. [BDMS, Lemma 4.3.3] Let k be a field of characteristic different from 2, 3. Let G be an adjoint simple algebraic group over k. Let L = Lie(G) be its Lie algebra, and let 2 L = L Li be any 5-grading on L such that L1 ⊕L−1 =06 . Then (L1, L−1) is a central simple i=−2 Kantor pair with respect to the triple product operation Lσ ×L−σ ×Lσ →Lσ given by {x, y, z} = −[[x, y], z], and its standard 5-graded embedding G = G(L1, L−1) is canonically isomorphic to the graded Lie subalgebra [L, L]+ kζ of L.

Proof. By [BDMS, Lemma 4.3.3] (L1, L−1) is a Kantor pair, and G(L1, L−1) is as required. It remains to prove centrality and simplicity. By [BDMS, Lemma 4.1.6] the k-Lie algebra [L, L] is ′ central simple, and differs from L only in the grading 0 component. Let (L1, L−1) be the Kantor ′ pair which differs from (L1, L−1) by the sign of the triple product, i.e. {x, y, z} = [[x, y], z]. ′ Then [L, L] envelopes the Kantor pair (L1, L−1) in the sense of [AFS17]. Then by [AFS17, ′ Theorem 4.20] (L1, L−1) is central simple. Clearly, this is equivalent to (L1, L−1) being central simple.  Now we can also establish the converse, namely, that any simple Kantor pair over a field of characteristic =26 , 3 arises from an isotropic simple algebraic group. 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTOR PAIRS 11

Theorem 4.11. Let (K+,K−) be a central simple Kantor pair over a field k of characteristic =26 , 3, and let G = G(K+,K−) be its standard 5-graded embedding. Then the algebraic k-group G = Aut(G)◦ is an adjoint absolutely simple group of k-rank ≥ 1, and G ∼= [Lie(G), Lie(G)]+kζ. ′ Proof. By definition, we have G1 = K+, G−1 = K−. Let (G1, G−1) be the Kantor pair which ′ differs from (G1, G−1) by the sign of the triple product, i.e. {x, y, z} = −{x, y, z} = [[x, y], z]. ′ ′ Since (G1, G−1) is central simple, (G1, G−1) is also central simple. Let K = K((G1, G−1) ) be the ′ 5-graded Lie algebra associated to (G1, G−1) in [AFS17, § 4.3]. By construction, G = kζ + K. By [AFS17, Corollary 4.14] the Lie algebra K is central simple. Then by Theorem 3.1 K is of Chevalley type. Then by Theorem 2.10 it is algebraic. Then by [BDMS, Theorem 4.1.8] G = Aut(K)◦ is an adjoint absolutely simple group of k-rank ≥ 1, and K ∼= [Lie(G), Lie(G)]. Then by [BDMS, Lemma 4.1.6] we have Aut(G)◦ ∼= G. 

Corollary 4.12. Let (K+,K−) be a simple Kantor pair over a commutative ring k such that × 2, 3 ∈ k . Then (K+,K−) is non-degenerate in the sense of [GGLN11].

Proof. Since (K+,K−) is simple, k does not have non-trivial ideals, and hence is a field. Ex- tending k, we can assume that (K+,K−) is central simple. Then by Theorem 4.11 its standard 5-graded embedding L is a central simple Lie algebra of Chevalley type, and, in particular, al- gebraic. Then L is non-degenerate by Lemma 2.7. Then by [GGLN11, Corollary 2.5] (K+,K−) is non-degenerate.  The following lemmas provides a criterion that a Kantor pair (or, equivalently, its 5-graded standard embedding) originates from a structurable algebra via Allison’s construction. It is a combination of [AF99, Corollaries 14 and 15]. This result is crucial for the classification of 5-gradings on simple Lie algebras of Chevalley type that correspond to structurable algebras (Theorems 5.7 and 5.10 below). Lemma 4.13. Let R be a commutative ring with 2, 3 ∈ R×. Let G be a 5-graded Lie algebra over R which is the standard 5-graded embedding of a Kantor pair (G−1, G1) over R. Then (G−1, G1) is a Kantor pair associated to a structurable algebra A over R if and only if there exist u ∈ G1, v ∈ G−1 such that ζ = [u, v] is the grading derivation of G.

Proof. Assume first that the Kantor pair (G−1, G1) is isomorphic to (A, A), and G is graded- isomorphic to K(A)+ kζ. Let 1 ∈ G1, 1ˆ ∈ G−1 denote the images of 1 ∈A. By the definition of K(A), the element V1,1 = [1+, 1−] ∈ K(A)0 acts as the grading derivation on K(A). Then its image [1, 1]ˆ in G coincides with ζ. Conversely, assume that ζ = [u, v] for some u ∈ G1, v ∈ G−1. By [AF99, Corollary 14] an element (x, 0) ∈ G1 ⊕ G2 is 1-invertible if and only if there is xˆ ∈ G−1 such that Vx,xˆ = 2idG1 ,

Vx,xˆ = 2idG−1 . Since [u, v] = ζ, we have Vu,v = −idG1 and Vv,u = −idG−1 . Then x = u is 1-invertible with xˆ = −2v. Since (G1, G−1) contains a 1-invertible element of the form (x, 0), by [AF99, Corollary 15] it is isomorphic to a Kantor pair associated with a structurable algebra.  Lemma 4.14. Let k be a field of characteristic different from 2, 3. Let G be an adjoint simple 2 algebraic group over k. Let L = Lie(G) be its Lie algebra, and let L = L Li be any 5-grading i=−2 on L such that L1 ⊕L−1 =6 0, and let ζ ∈ L0 be the grading derivation. Then ζ = [u, v] for some u ∈L1, v ∈L−1 if and only if there is a structurable algebra A over k such that [L, L] is graded-isomorphic to K(A). 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTOR PAIRS 12

Proof. Assume first that L is graded-isomorphic to K(A). Let 1 ∈ L1, 1ˆ ∈ L−1 denote the ∼ images of 1 ∈ A. Then [1, 1]ˆ ∈ L0 acts as the grading derivation ζ on [L, L]. Since L = Der([L, L]) by [BDMS, Lemma 4.1.6], [1, 1]ˆ = ζ. Conversely, assume that ζ = [u, v] for some u ∈ L1, v ∈ L−1. Consider the Kantor pair (L−1, L1) with the triple product operation Lσ ×L−σ ×Lσ →Lσ given by {x, y, z} = −[[x, y], z].

By Lemma 4.10 its standard 5-graded embedding G = G(L1, L−1) is canonically isomorphic to the graded subalgebra [L, L]+ kζ of L. Then by Lemma 4.13 the pair (L−1, L1) is associated to a structurable algebra A. Then K(A) is graded-isomorphic to [G, G] ∼= [L, L]. 

5. Classification of 5-gradings that correspond to structurable algebras Definition 5.1. Let G be an algebraic group over a field k and T ⊆ G be a split n-dimensional k-subtorus of G. Let X∗(T ) ∼= Zn be the group of characters of T , and let

Lie(G)= M Lie(G)α α∈X∗(T ) be the Zn-grading on Lie(G) induced by the adjoint action of T . We call ∗ Φ(T,G)= {α ∈ X (T ) | Lie(G)α =06 } the set of roots of G with respect to T . If G is a reductive algebraic group over k and T is a maximal split k-subtorus of G, then Φ(T,G) \{0} is a root system in the sense of Bourbaki [BT65, Bou]. By abuse of language, we call Φ(T,G) a root system of G. Let Φ be a root system and Π ⊆ Φ be a system of simple roots. For any α ∈ Φ we write

α = X mβ(α)β, β∈Π where the coefficients mβ(α) are all non-negative, or all non-positive. Once Π is fixed, we denote the corresponding sets of positive and negative roots by Φ±. Recall that two parabolic k-subgroups P± of G are called opposite, if P+ ∩P− is their common Levi subgroup. If this is the case, then there is a maximal split k-subtorus T of G such that

T ⊆ P+ ∩ P−, and a system of simple roots Π ⊆ Φ and a non-empty subset J ⊆ Π such that, if one defines + Z Ψ = Φ ∪ Φ ∩ (Π \ J), then

(3) Lie(P+)= M Lie(G)α, Lie(P−)= M Lie(G)α. α∈Ψ α∈−Ψ Conversely, if T is maximal split k-subtorus of G, Φ=Φ(T,G), and Ψ ⊆ Φ is a subset such that 0 ∈ Ψ, Ψ is closed under the (partially defined) addition of elements of Φ, and Ψ ∪ (−Ψ)=Φ, then there is a unique pair of opposite parabolic k-subgroups P± of G satisfying (3) [DG70b, + Z Exp. XXVI, Prop. 6.1]. In particular, every such subset Ψ has the form Ψ = Φ ∪Φ∩ (Π\J) as above. 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTOR PAIRS 13

Definition 5.2. If k is an algebraically closed field, then the set t(P+) = Π \ J is called the type of P+; it is a system of simple roots of the root system of the Levi subgroup P+ ∩ P− of ¯ P+. If k is not algebraically closed, and k is an algebraic closure of k, then we define t(P+) to be the type of (P+)k¯. A subset Ψ ⊆ Φ such that 0 ∈ Ψ, Ψ is closed under addition, and Ψ ∪ (−Ψ) =Φ arises, in particular, if we are given a non-trivial Z-grading on Lie(G). Namely, one takes

Ψ= {α ∈ Φ(T,G) | Lie(G)α ⊆ M Lie(G)i}. i≥0 Thus, to any such grading one may associate a pair of opposite parabolic subgroups. In order to classify 5-gradings that correspond to structurable algebras, we rely on the theory of nilpotent orbits in Lie algebras of simple algebraic groups over an algebraically closed field. This theory originates from the work of E. Dynkin [Dyn52], with further developements by B. Kostant, G. E. Wall, R. W. Richardson, T.A. Springer, R. Steinberg, G. B. Elkington, P. Bala and R. Carter, K. Pommerening, and many others. As a result, the specific statements that we need are seriously scattered in the literature, and we cite them according to more recent sources where they are stated in a more explicit form. We recall the essence of Dynkin’s classification of nilpotent elements in complex simple Lie algebras. Let LC be a simple Lie algebra over C, and let e ∈ LC be a nilpotent element. By the Jacobson–Morozov theorem, LC contains an sl2-triple of the form {e, h, f}. Let H≤LC be a Cartan subalgebra of LC containing h. Let Φ be the root system of LC, and let eα, and hα = [eα, e−α], α ∈ Φ, be the standard root vectors in a Chevalley basis of LC with respect to H. There is a choice of a system of simple roots Π ⊆ Φ such that for any α ∈ Π one has α(h) ≥ 0. Dynkin established that, morever, α(h) ∈{0, 1, 2}. The Dynkin diagram of Φ with the integers α(h) associated to the nodes corresponding to roots α ∈ Π is called a weighted Dynkin diagram of e. It is uniquely determined by e, and the weighted diagrams of two nilpotents e, e′ coincide ′ if and only if e and e are conjugate by an inner automorphism of LC [Dyn52, Theorems 8.1 and 8.3]. Let LZ be the Z-Lie subalgebra of LC generated by all eα and hα, α ∈ Φ. Then LZ is a sc Z-form of LC, i.e. LC = LZ ⊗Z C. Let k be any algebraically closed field, and let G be a ad simply connected simple algebraic group over k of the same type Φ as LC, and let G be the sc corresponding adjoint group. Then Lie(G ) ∼= LZ ⊗Z k, see e.g. [Mil17, Theorem 23.72]. Define ad sc α(h) the cocharacter λ : Gm → G ≤ Aut(Lie(G )) in such a way that λ(t) · eα = t eα and × λ(t) · hα = hα for any α ∈ Π ∪ (−Π) and t ∈ k . Thus, we associate to any weighted Dynkin ad diagram a k-cocharacter λ : Gm → G .

Notation 5.3. Let G be a reductive algebraic group over a field k, and let λ : Gm → G be a cocharacter of G over k. Then λ induces a Z-grading on Lie(G), and we denote by Lie(G)(λ, i) the i-th component of this grading, i.e. i × Lie(G)(λ, i)= {v ∈ Lie(G) | λ(t) · v = t v for any t ∈ Gm(k)= k }. We denote by P (λ) and P (−λ) the unique pair of (not necessarily proper) opposite parabolic subgroups in G such that

Lie(P (λ)) = M Lie(G)(λ, i) and Lie(P (−λ)) = M Lie(G)(λ, −i). i≥0 i≥0 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTOR PAIRS 14

We denote by CG(λ) the centralizer of λ(Gm) in G; this is a Levi subgroup P (λ) ∩ P (−λ) of P (λ). The classification of weighted Dynkin diagrams corresponding to nilpotents involves the no- tion of a distinguished parabolic subgroup. Note that the classification of types of distinguished parabolic subgroups, as defined below, is independent of the characteristic of the base field. Definition 5.4. [LS12, § 2.6] Let G be a semisimple algebraic group over an algebraically closed field k, and let Φ = Φ(T,G) be the root system of G. A parabolic subgroup P of G is called distinguished if

(4) dim LP = dim M Lie(G)α, α∈Φ: P mβ (α)=1 β∈J where LP is a Levi subgroup of P , and Π \ J is the type of P in a system of simple roots Π of Φ. Theorem 5.5. [Pre03, LS12] Let G be an adjoint simple algebraic group over an algebraically closed field k of type Φ such that char(k) =6 2 if Φ = Bl,Cl (l ≥ 2) or Φ = Dl (l ≥ 4), and char(k) =26 , 3 if Φ= E6, E7, E8,G2, F4.

(1) Let λ : Gm → G be a k-cocharacter of G corresponding to a weighted Dynkin diagram of a nilpotent element in a complex simple Lie algebra of the same type as G. Then (i) CG(λ) has a unique dense open orbit V in Lie(G)(λ, 2). (ii) For any e ∈ V (k), CG(e) ≤ P (λ). ◦ (iii) For any e ∈ V (k), let C(λ, e) = CG(λ) ∩ CG(e). Then C(λ, e) is a reductive subgroup of G, and for any maximal torus S of C(λ, e) the parabolic subgroup Q = P (λ) ∩ H is a distinguished parabolic subgroup of the semisimple group H = [CG(S),CG(S)]. Moreover, λ(Gm) ≤ H, and e lies in the dense open orbit of CH (λ) in Lie(H)(λ, 2). (2) Conversely, for any nilpotent element e ∈ Lie(G) there is a cocharacter λ : Gm → G as above, such that e belongs to the unique dense open orbit of CG(λ) in Lie(G)(λ, 2).

Proof. (1) Set L = Lie(G), P = P (λ), LP = CG(λ), and LQ = CH (λ) for short. Clearly, LP is a Levi subgroup of P and LQ is a Levi subgroup of Q. Assume first that G is not of type E8 if char(k)=5. Then the characteristic of k is good for G. There is reductive group G˜ over k such that [G,˜ G˜] is the simply connected group isogenous to G, and Lie(G˜) admits a non-degenerate G˜-invariant form, see [Pre03, 2.3]. To prove our theorem, clearly, we can replace G by G˜; this makes other results of Premet applicable. By the discussion before [Pre03, Theorem 2.3] LP has a dense open orbit V in L(λ, 2), and for any e ∈ V (k), CG(e) ≤ P . By [Pre03, Theorem 2.3 (iii)] C(λ, e) is a reductive subgroup of G. By [Pre03, Theorem 2.3 (ii)] CG(e) ≤ P . The remaining statements of (iii) follow from [Pre03, Proposition 2.5]. Assume that G is of type E8 and char(k) =6 2, 3. Consider the table [LS12, Table 22.1.1]. By [LS12, Theorem 15.1] the weighted diagrams of complex nilpotent elements are exactly the ones in the 2nd column of this table, and, conversely, for any λ corresponding to such a diagram, and any field k as above, there is a nilpotent element e ∈ L(λ, 2) such that eP is LP dense in Li≥2 L(λ, i) and CG(e) ≤ P . This implies that e = V is a dense open orbit of LP 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTOR PAIRS 15 in L(λ, 2). Since any two such orbits would intersect, this orbit is unique. Clearly, it is enough to establish (iii) for this particular element e ∈ V (k). By [LS12, Theorem 1 (c)] CG(e) = C(λ, e)Ru(CG(e)), where Ru(CG(e)) is the unipotent ◦ radical of CG(e) and C(λ, e) is a reductive group (note that, contrary to our conventions, in [LS12] reductive groups are not required to be connected). Let S be a maximal torus of C(λ, e), then CG(S) is a Levi subgroup of a parabolic subgroup of G by [LS12, Lemma 2.2]. Set H = [CG(S),CG(S)]. Clearly, e ∈ L(λ, 2) ∩ Lie(H), since e is nilpotent and belongs to Lie(CG(S)). By the proof of [LS12, Lemma 2.13] e is a distinguished element of H = ◦ [CG(S),CG(S)], i.e. CH (e) is a unipotent group. Since S ≤ CG(e), this implies that S = ◦ Cent(CG(S)) . On the other hand, by the actual statement of [LS12, Lemma 2.13], CG(S) is conjugate to the Levi subgroup L¯ of a parabolic subgroup of G used in the original construction ◦ of the 1-dimensional torus λ(Gm) = T given in [LS12, Lemma 15.3 (i)]. Let S¯ = Cent(L¯) , then S¯ ≤ CG(e). Since S and S¯ are conjugate in G, they have the same dimension, and hence ◦ ◦ they are both maximal tori in CG(e). Moreover, they are both contained in C(λ, e) ≤ CG(e) , hence they are conjugate in this group, i.e. by an element centralizing T . Then the remaining statements of our claim (iii) follow from the corresponding properties of T with respect to L¯ stated in [LS12, Lemma 15.3 (i)]. (2) Let e ∈ Lie(G) be any nilpotent. The classification of nilpotent classes [LS12, Theorem 1 c)] implies that there is a cocharacter λ : Gm → G, that corresponds to a weighted Dynkin diagram, and such that e belongs to the dense open orbit of CG(λ) in Lie(G)(λ, 2). Moreover, the explicit classification of occurring weighted Dynkin diagrams [LS12, Theorem 3.1; Tables 22.1.1-22.1.5] is independent of the ground field under the assumption char k =26 , 3, hence any such weighted Dynkin diagram is a diagram of a complex nilpotent.  Lemma 5.6. In the setting of Theorem 5.5 (1) (iii), assume moreover that H is not of type E8 if char(k)=5. Then [Lie(H)(λ, −2), e] = Lie(H)(λ, 0). Proof. Let Ψ be the root system of H, let Σ be a system of simple roots of Ψ, and let J ⊂ Σ be the set of simple roots corresponding to the parabolic subgroup Q = P (λ) ∩ H of H. Since Q is distinguished, one has λ(α)=2 for all α ∈ J, see [LS12, Lemma 10.3]. Hence one has dimLie(H)(λ, 2) = dimLie(H)(λ, −2) = dimLie(H)(λ, 0). Hence it remains to prove that [u, e] =6 0 for any 0 =6 u ∈ Lie(H)(λ, −2). If (Ψ, char(k)) =6 (E8, 5), then this follows from [Pre03, Theorem 2.3 (iv)].  From now and until the end of this section, assume that k is a field of characteristic =26 , 3, and G is an adjoint simple algebraic group over k. Denote by k¯ an algebraic closure of k. Let Z L = Lie(G) be the Lie algebra of G, and let L = L Li be any -grading on L such that L= 6 L0. i∈Z By [BDMS, Lemma 4.1.6] one has G = Aut(Lie(G))◦, hence there is a unique closed embedding of a 1-dimensional split k-torus λ : Gm → G, such that Li = Lie(G)(λ, i) for all i ∈ Z. 2 Theorem 5.7. Let L = L Li be a 5-grading on the Lie algebra L = Lie(G) of G such that i=−2 L1 ⊕L−1 =06 , and let λ : Gm → G be the corresponding cocharacter of G. Let ∆ be the weighted Dynkin diagram of the same root system type as Gk¯, such that simple roots in t(P (λ)) have weight 0, and roots in Π \ t(P (λ)) have weight 2. Then [L, L] is graded-isomorphic to K(A) for a structurable algebra A over k if and only if ∆ is a weighted Dynkin diagram of a nilpotent element in a complex simple Lie algebra of the same type. 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTOR PAIRS 16

Proof. Consider the cocharacter 2λ : Gm → G, so that Li = Lie(G)(2λ, 2i), i ∈ Z. Then ∆ is the weighted Dynkin diagram corresponding to 2λ over k¯. Assume the condition on ∆ is satisfied. We show that there are u ∈ L−1, e ∈ L1 such that [u, e]= ζ, the grading derivation of L; then [L, L] is graded-isomorphic to K(A) by Lemma 4.14. ∼ Let 1+ ǫ ∈ Gm(k[ǫ]) be the unit element of Lie(Gm)(k) = k. Then 2ζ =2λ(1 + ǫ) ∈ Lie(G) is the grading derivation of Lie(G) with respect to 2λ. We have

2ζ ∈ Lie(G)(2λ, 0) ∩ Lie(2λ(Gm)) ⊆ Lie(H)(2λ, 0), where H is as in Theorem 5.5. By Lemma 5.6 there is a dense open orbit V ⊆L1 = Lie(G)(2λ, 2) of LP , such that for any e ∈ V (k) one has [Lie(H)(2λ, −2), e] = Lie(H)(2λ, 0). In particular, ζ ∈ [Lie(G)(2λ, −2), e] = [L−1, e]. Now let k be not necessarily algebraically closed, and let k¯ be its algebraic closure. Note that P (±λ) and CG(λ) = P (λ) ∩ P (−λ) are defined over k. Since CG(λ) ×k k¯ has a unique dense open orbit V in L1 ⊗k k¯, the open subvariety V of the affine space L1 is defined over k (although the action of CG(k) on it does not have to be transitive). If k is infinite, then V (k) =6 ∅ just because V is an open subvariety of an affine space. If k is finite, then V (k) =6 ∅ by [SS70, 2.7], since V is a homogeneous space for CG(λ). Thus, there is a k-point e ∈ V (k). Then ζ ∈ [L−1, e], since the same holds over k¯. Next, assume that [L, L] is graded-isomorphic to K(A), and show that ∆ is a weighted Dynkin diagram of a nilpotent element in a complex simple Lie algebra of type Φ. We can assume that k is algebraically closed without loss of generality. Let e =1+ ∈L1 and f =1− ∈L−1 be the elements representing the unit of the structurable algebra A. By the very definition of K(A), we conclude that [e, f]= ζ ∈L0. Furthermore, for any 0 =6 x ∈A one has V1,x(1) = 2¯x−x =06 , since 2¯x = x implies 2x =x ¯ and 3¯x =0, whence x =0. ′ By Theorem 5.5 there is a cocharacter λ : Gm → G that corresponds to a weighted Dynkin ′ diagram of complex nilpotent, and such that e belongs to the dense open orbit of CG(λ ) in Lie(G)(λ′, 2). Let ζ′ be the grading derivation of L corresponding to λ′. Then [ζ′, e]=2e. ′ Assume for the moment that the subgroup H corresponding to λ and e is not of type E8 if char k = 5. Then by Lemma 5.6 there is f ′ ∈ Lie(G)(λ′, −2) such that [e, f ′] = ζ′. We have ′ ′ λ(Gm),λ (Gm) ≤ NG(k · e). Hence after conjugating λ (Gm) by an element of CG(e)(k) ≤ ′ NG(k · e)(k) we can assume that λ (Gm) and λ(Gm) lie in the same maximal torus of NG(k · e), ′ and thus centralize each other. In particular, ζ ∈ L0, and hence without loss of generality ′ ′ ′ ′ ′ f ∈L−1. Then 2ζ − ζ = [e, 2f − f ] and [2ζ − ζ , e]=0. In other words, [[e, 2f − f ], e]=0. ′ ′ However, [e, 2f − f ] acts on L1 as V1,2f−f ′ acts on A, whence 2f − f = 0 by the above computation. Thus 2ζ = ζ′, and we are done. ′ Assume that H = G has type E8; then P (λ ) is a distinguished parabolic subgroup of G. We claim that this case cannot occur in our setting. The types of distinguished parabolic subgroups ′ are listed in [LS12, Table 13.2]. Denote by L = Li∈Z L[i] the grading on L induced by λ . In all cases, one readily sees that L[2i] =6 0 for all 1 ≤ i ≤ k, where k ≥ 5. Then by [LS12, proof ′ of Propositions 13.4 and 13.5], under the assumption char k =26 , 3 there is a nilpotent e ∈L[2] such that e′ lies in the dense orbit of C (λ′) in L , and one has ad(e′)| 5 =06 . Clearly, G [2] Li≥0 L[i]  ′ ′ 5 e and e are CG(λ )-conjugate. However, since e ∈ L1, and L is 5-graded, one has ad(e) = 0, hence ad(e′)5 =0 as well, a contradiction.  Lemma 5.8. 2 Let L = Li=−2 Li be a 5-grading on the Lie algebra L = Lie(G) of G such that L1 ⊕L−1 =06 , and let λ : Gm → G be the corresponding cocharacter of G. Let α˜ be the root of 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTOR PAIRS 17 maximal height in Φ with respect to Π. Let J = Π \ t(P (λ)), then J satisfies (a) J = {α1} or

(b) J = {α1,α2}. If (a) holds, then mα1 (˜α)=1 or 2. If (b) holds, then mα1 (˜α)= mα2 (˜α)=1. × In both cases λ(t) · eα = teα for any α ∈ J and t ∈ k .

Proof. By [BDMS, Lemma 4.2.2] the root system Φ ∩ ZJ is of type A1, BC1, A2, or A1 × A1, and for any −2 ≤ i ≤ 2 we have

(5) Li = M Lie(G)α. α∈Φ: P mβ (α)=i β∈J

Then λ(t)eα = teα for α ∈ J by the definition of λ. Since Pα∈J mα(˜α) ≤ 2, the remaining claims are clear.  Definition 5.9. Z Assume that the -grading Lie(G) = Li∈Z Lie(G)i of the Lie algebra of a reductive algebraic group G corresponds to the cocharacter λ : Gm → G. We define the type of the grading to be the type t(P (λ)) of the corresponding positive parabolic subgroup P (λ). Theorem 5.10. Let G be an adjoint simple algebraic group over a field k, char k =6 2, 3. Let 2 L = L Li be a 5-grading on the Lie algebra L = Lie(G) of G such that L1 ⊕L−1 =06 . Then i=−2 [L, L] is graded-isomorphic to K(A) for a structurable k-algebra A if and only if the type of grading is the complement of the set J of simple roots of the root system Φ of G listed in the following table.

Φ J

Al, l ≥ 1 {αi,αl+1−i}, 1 ≤ i ≤ (l + 1)/3;

{αl+1/2}, if l is odd

Bl, l ≥ 2 {αi}, 1 ≤ i ≤ (2l + 1)/3

Cl, l ≥ 3 {αi}, 1 ≤ i ≤ 2l/3, i is even; {αl}

Dl, l ≥ 4 {αi}, 1 ≤ i ≤ 2l/3;

{αl−1} and {αl}, if l is even

E6 {α1,α6}; {α2}

E7 {α1}; {α2}; {α6}; {α7}

E8 {α1}; {α8}

F4 {α1}; {α4}

G2 {α2}

Proof. Let λ : Gm → G be the cocharacter of G cooresponding to the grading. By Theorem 5.7 it remains to check if 2λ is a k-cocharacter corresponding to a weighted Dynkin diagram of a 2 nilpotent element in the complex case. By Lemma 5.8 one has 2λ(t) · eα = t eα for all α ∈ J 0 and 2λ(t) · eα = t eα for all α ∈ Π \ J. If Φ is of exceptional type, then one readily checks that in all cases 2λ is as required, since its weighted Dynkin diagram occurs in the Tables 22.1.1–22.1.5 of nilpotent conjugacy classes in [LS12]. 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTOR PAIRS 18 Assume Φ is of classical type. We use the descriptions of weighted Dynkin diagrams of nilpotent orbits given in [CM93]. Case Φ= Al. In the notation of [CM93, § 3.6], we have n = l+1 and h1 −hn =2|J|∈{2, 4}. n We need to describe suitable non-negative partitions n = Pi=1 di. By definition, di ≥ 1 for 1 ≤ di ≤ k and di =0 for k +1 ≤ i ≤ n. If h1 − hn = 2, then di ∈ {1, 2} for all 1 ≤ i ≤ k. Then the sequence h1 ≥ h2 ≥ ... ≥ hn contains numbers 1 and −1 repeated m ≥ 1 times each, and k − m zeroes. Since hi − hi+1 =2 only for one i, we have k−m =0 and n =2k. In particular, l is odd, and J = {αk} = {α(l+1)/2}. If h1 − hn = 4, then di ∈{1, 2, 3} for all 1 ≤ i ≤ k. Then the sequence h1 ≥ h2 ≥ ... ≥ hn contains numbers 2, 0, −2 repeated m ≥ 1 times each, numbers 1, −1 repeated m′ times each, ′ ′ and k − m − m zeroes. Since hi − hi+1 =2 for exactly two i’s, we conclude that m =0. Then l +1= n =3m +(k − m), hence l +1 ≥ 3m and J = {αm,αl+1−m}, as required. Case Φ = Bl. In the notation of [CM93, Lemma 5.3.3], we have n = l. Nilpotent orbits 2n+1 are classified by partitions 2n +1= Pi=1 di of 2n +1 in which even parts occur with even multiplicity [CM93, Theorem 5.1.2]. In [CM93, Lemma 5.3.3], h1 is the sum of labels of the weighted Dynkin diagram, hence h1 =2. Then di ∈{1, 2, 3}. If hn =2, then h1 = h2 = ... = hn =2, which is not possible, since there are not enough zeroes. Hence the label 2 occurs only for one i between 1 and n − 1, and h1 = ... = hi = 2, hi+1 = ... = hn = 0. Then partition consists of i numbers 3 and 2n +1 − 3i numbers 1. The condition on parity is void. The set {0, ±h1,..., ±hn} contains i + (2n +1 − 3i) zeroes, therefore, 2(n − i) ≥ i − 1, which implies 3i ≤ 2n +1. Case Φ = Cl. In the notation of [CM93], we have n = l. Nilpotent orbits are classified by 2n partitions 2n = Pi=1 di of 2n in which odd parts occur with even multiplicity [CM93, Theorem 5.1.3]. In [CM93, Lemma 5.3.1], h1 + hn = 2 is the sum of labels of the weighted Dynkin diagram. Since 2hn is the label of αl, we have hn =0 or hn =1. If hn =1, then h1 =1, and hence hi =1 for all i. Then di =2 for all non-zero di, hence the partition is 2n =2+ ... +2. The condition on odd parts is satisfied, hence J = {αl} is valid. If hn = 0, then h1 = 2 and di ∈ {1, 2, 3} for 1 ≤ i ≤ 2n. Since hi − hi+1 = 2 only for one i between 1 and n − 1, we conclude that h1 = ... = hi =2 and hi+1 = ... = hn−1 =0. Then 2n is partitioned into the sum of i times number 3, and 2n − 3i ≥ 0 times number 1, where i is even. Summing up, J = {αi} with even i =2m, 1 ≤ m ≤ l/3, or J = {αl}. Case Φ = Dl. In our notation, l = n. Nilpotent orbits are classified by partitions 2n 2n = Pi=1 di of 2n in which even parts occur with even multiplicity, except that "very even" partitions with only even parts (each having even multiplicity) correspond to two different or- bits [CM93, Theorem 5.1.4]. Since all weights of the Dynkin diagram in our seting are even, all numbers h1,...,hn have the same parity. Assume first that the partition is not very even, i.e. the numbers di are odd, and the numbers hi are even. By [CM93, Lemma 5.3.4] the sum of labels on the Dynkin diagram equals h1 + hn−1 ∈ {2, 4}. Since h1 ≥ hn−1, we have h1 = 2, hn−1 = 0 or h1 = hn−1 = 2. If h1 = hn−1 = 2, then h1 = h2 = ... = hn−1 = 2 and hn = 0, which is not possible (not enough zeroes). Hence h1 =2, hn−1 =0. Then hn =0 as well. The partition consists of i times 3, where i ≥ 1, and 2n − 3i numbers 1. The multiplicity condition is void. Since numbers 3 produce triples 2, 0, −2, one has 2n ≥ 3i. This case corresponds to J = {αi}, 1 ≤ i ≤ 2n/3. Assume the partition is very even. Then by [CM93, Lemma 5.3.5] the sum of labels of vertices αn and αn−1 equals 2. Then either all other labels are 0, or there is label 2 at α1. In 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTOR PAIRS 19 the first case we have h1 = h2 = ... = hn−1, and since all di are even, this implies that the partition is n numbers 2. Since it is very even, n is even. In the second case h1 − h2 = 2, h2 = ... = hn−1. Since h1 ≥ h2 ≥ hn, then d1 = h1 +1 can only have multiplicity one, which is wrong. Therefore, this case does not take place in our setting. Summing up, very even partitions occur with J = {αn−1} and J = {αn} for n even.  Proof of Theorem 1.1. The implication (3 ) =⇒ (2 ) is obvious. The implication (2 ) =⇒ (1 ) follows from Theorem 3.1. It remains to prove (1 ) =⇒ (3 ). By assumption, L is a central simple Chevalley type Lie algebra with a non-trivial Z-grading. Let S be the grading torus of L, and let ϕ : S → Aut(L) be the natural embedding of algebraic k-groups. The algebraic group G = Aut(L)◦ is an adjoint simple algebraic group, and L = [Lie(G), Lie(G)] (cf. [BDMS, Lemma 4.1.6]). Since S is a 1-dimensional split torus, it follows that G contains a 1-dimensional split torus, i.e. is isotropic. The possible types of minimal parabolic k-subgroups of G are given by the Tits index of G. Considering the classification of possible Tits indices [Tit66, PSt], one readily sees that in all cases there is a parabolic subgroup type Π \ J listed in Theorem 5.10 that is preserved by the ∗-action and contains the type of a minimal parabolic k-subgroup of G. Then G contains at least one parabolic k-subgroup P (λ) whose type is Π \ J [Tit66, 2.5.4]. Then by Theorem 5.7 L is graded-isomorphic to K(A) for a structurable algebra A over k. 

References

[All78] B. N. Allison, A class of nonassociative algebras with involution containing the class of Jordan alge- bras, Math. Ann. 237 (1978), no. 2, 133–156. [All79] B. N. Allison, Models of isotropic simple Lie algebras, Comm. Algebra 7 (1979), no. 17, 1835–1875. [AF99] B. N. Allison, J. R. Faulkner, Elementary groups and invertibility for Kantor pairs, Comm. Algebra 27 (1999), no. 2, 519–556. [AFS17] B. Allison, J. Faulkner, and O. Smirnov, Weyl images of Kantor pairs, Canad. J. Math. 69 (2017), no. 4, 721–766. [BKM15] Yu. Bahturin, M. Kochetov, and J. McGraw, Gradings by groups on Cartan type Lie algebras, Lie algebras and related topics, Contemp. Math., vol. 652, Amer. Math. Soc., Providence, RI, 2015, pp. 15–46. [BDMS] L. Boelaert, T. De Medts, and A. Stavrova, Moufang sets and structurable division algebras, Mem. Amer. Math. Soc. 259 (2019), no. 1245, v+90. [BT65] A. Borel and J. Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. (1965), no. 27, 55–150. [Bor91] A. Borel, Linear algebraic groups, second ed., Graduate Texts in Mathematics, vol. 126, Springer- Verlag, New York, 1991. [Bou] N. Bourbaki, Groupes et algèbres de Lie. Chapitres 4–6. Hermann, Paris, 1968. [CM93] D. H. Collingwood and W. M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. [DG70b] M. Demazure and A. Grothendieck, Schémas en groupes (SGA 3), Lecture Notes in Mathematics, vol. 151–153, Springer-Verlag, Berlin-Heidelberg-New York, 1970. [Dyn52] E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349–462 (3 plates). [GGLN11] E. García, M. Gómez Lozano, and E. Neher, Nondegeneracy for Lie triple systems and Kantor pairs, Canad. Math. Bull. 54 (2011), no. 3, 442–455. [Ka72] I.L.Kantor, Certain generalizations of Jordan algebras, Trudy Sem. Vektor. Tenzor. Anal. 16 (1972), 407–499. [LS12] M. W. Liebeck and G. M. Seitz, Unipotent and nilpotent classes in simple algebraic groups and Lie algebras, Mathematical Surveys and Monographs, vol. 180, American Mathematical Society, Providence, RI, 2012. [Loo79-1] O. Loos, On algebraic groups defined by Jordan pairs, Nagoya Math. J. 74 (1979), 23–66. 5-GRADED SIMPLE LIE ALGEBRAS, STRUCTURABLE ALGEBRAS, AND KANTOR PAIRS 20 [Loo79-2] O. Loos, Homogeneous algebraic varieties defined by Jordan pairs, Monatsh. Math. 86 (1978/79), no. 2, 107–129. [Mil17] J. S. Milne, Algebraic groups, Cambridge Studies in Advanced Mathematics, vol. 170, Cambridge University Press, Cambridge, 2017, The theory of group schemes of finite type over a field. [PSt] V. Petrov, A. Stavrova, Tits indices over semilocal rings, Transf. Groups 16 (2011), 193–217. [Pre83] A. A. Premet, Algebraic groups associated with Lie p-algebras of Cartan type, Mat. Sb. (N.S.) 122(164) (1983), no. 1, 82–96. [Pre03] A. Premet, Nilpotent orbits in good characteristic and the Kempf-Rousseau theory, J. Algebra 260 (2003), no. 1, 338–366, Special issue celebrating the 80th birthday of Robert Steinberg. [PS08b] A. Premet and H. Strade, Simple Lie algebras of small characteristic. VI. Completion of the classi- fication, J. Algebra 320 (2008), no. 9, 3559–3604. [Sel67] G. B. Seligman, Modular Lie algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 40, Springer-Verlag New York, Inc., New York, 1967. [Smi90] O. N. Smirnov, Simple and semisimple structurable algebras, Algebra i Logika 29 (1990), no. 5, 571–596, 627. [SS70] T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69), Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 167–266. MR 0268192 [Sta09] A. K. Stavrova, The structure of isotropic reductive groups, Ph. D. thesis, St. Petersburg State University, 2009 (Russian). [SF88] H. Strade and R. Farnsteiner, Modular Lie algebras and their representations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 116, Marcel Dekker, Inc., New York, 1988. [Str04] H. Strade, Simple Lie algebras over fields of positive characteristic. I, De Gruyter Expositions in Mathematics, vol. 38, Walter de Gruyter & Co., Berlin, 2004, Structure theory. [Str09] H. Strade, Simple Lie algebras over fields of positive characteristic. II, de Gruyter Expositions in Mathematics, vol. 42, Walter de Gruyter & Co., Berlin, 2009, Classifying the absolute toral rank two case. [Tit66] J. Tits, Classification of algebraic semisimple groups, Algebraic groups and discontinuous subgroups, Proc. Sympos. Pure Math. 9, Amer. Math. Soc., Providence RI, 1966, 33–62. [Wil75] R. L. Wilson, Automorphisms of graded Lie algebras of Cartan type, Comm. Algebra 3 (1975), no. 7, 591–613. [Wil76] R. L. Wilson, A structural characterization of the simple Lie algebras of generalized Cartan type over fields of prime characteristic, J. Algebra 40 (1976), no. 2, 418–465. [Zel80] E. I. Zel’manov, Absolute zero-divisors in Jordan pairs and Lie algebras, Mat. Sb. (N.S.) 112(154) (1980), no. 4(8), 611–629.

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