Triality and algebraic groups of type ^{3}D_{4}

Dedicated to Sasha Merkurjev on the occasion of his 60th birthday,

in fond memory of the time spent writing the Book of Involutions

Max-Albert Knus and Jean-Pierre Tignol^{1}

Received: September 8, 2014 Revised: April 12, 2014

Abstract. We determine which simple algebraic groups of type ^{3}D_{4} over arbitrary fields of characteristic different from 2 admit outer automorphisms of order 3, and classify these automorphisms up to conjugation. The criterion is formulated in terms of a representation of the group by automorphisms of a trialitarian algebra: outer automorphisms of order 3 exist if and only if the algebra is the endomorphism algebra of an induced cyclic composition; their conjugacy classes are in one-to-one correspondence with isomorphism classes of symmetric compositions from which the induced cyclic composition stems.

2010 Mathematics Subject Classification: 20G15, 11E72, 17A75. Keywords and Phrases: Algebraic group of outer type ^{3}D_{4}, triality, outer automorphism of order 3, composition algebra, symmetric composition, cyclic composition, octonions, Okubo algebra.

1The second author is supported in part by the Fonds de la Recherche Scientifique–FNRS under grant n^{∘} 1.5054.12. Both authors acknowledge the hospitality of the Fields Institute, Toronto, where this research was initiated. The first author also acknowledges the hospitality of the Université catholique de Louvain and enlightening conversations with Michel Racine.

Documenta Mathematica · Extra Volume Merkurjev (2015) 387–405

1. Introduction

Let G_{0} be an adjoint Chevalley group of type D_{4} over a field F. Since the automorphism group of the Dynkin diagram of type D_{4} is isomorphic to the symmetric group 𝔖_{3}, there is a split exact sequence of algebraic groups

(1)  \xymatrix{1 \ar[r] & G_{0} \ar[r]^{Int} & Aut(G_{0}) \ar[r]^{π} & 𝔖_{3} \ar[r] & 1.}

Thus, Aut(G_{0}) ≅ G_{0} ⋊ 𝔖_{3}; in particular G_{0} admits outer automorphisms of order 3, which we call trialitarian automorphisms. Adjoint algebraic groups of type D_{4} over F are classified by the Galois cohomology set H^{1}(F, G_{0} ⋊𝔖_{3}) and the map induced by π in cohomology

π_{∗}: H^{1}(F, G_{0} ⋊ 𝔖_{3}) → H^{1}(F, 𝔖_{3})

Triality and groups of type ^{3}D_{4} 389

associates to any group G of type D_{4} the isomorphism class of a cubic étale F -algebra L. The group G is said to be of type ^{1}D_{4} if L is split, of type ^{2}D_{4} if L ≅ F ×Δ for some quadratic separable field extension Δ/F, of type ^{3}D_{4} if L is a cyclic field extension of F and of type ^{6}D_{4} if L is a non-cyclic field extension. An easy argument given in Proposition 4.2 below shows that groups of type 2D_{4} and ^{6}D_{4} do not admit trialitarian automorphisms defined over the base field. Trialitarian automorphisms of groups of type ^{1}D_{4} were classified in [3], and by a different method in [2]: the adjoint groups of type ^{1}D_{4} that admit trialitarian automorphisms are the groups of proper projective similitudes of 3-fold Pfister quadratic spaces; their trialitarian automorphisms are shown in [3, Th. 5.8] to be in one-to-one correspondence with the symmetric composition structures on the quadratic space. In the present paper, we determine the simple groups of type ^{3}D_{4} that admit trialitarian automorphisms, and we classify those automorphisms up to conjugation. Our main tool is the notion of a trialitarian algebra, as introduced in [9, Ch. X]. Since these algebras are only defined in characteristic different from 2, we assume throughout (unless specifically mentioned) that the characteristic of the base field F is different from 2. In view of [9, Th. (44.8)], every adjoint simple group G of type D_{4} can be represented as the automorphism group of a trialitarian algebra T = (E, L, σ, α). In the datum defining T, L is the cubic étale F -algebra given by the map π_{∗} above, E is a central simple L-algebra with orthogonal involution σ, known as the Allen invariant of G (see [1]), and α is an isomorphism relating (E, σ) with its Clifford algebra C(E, σ) (we refer to [9, §43] for details). We show in Proposition 4.2 that if G admits an outer automorphism of order 3 modulo inner automorphisms, then L is either split (i.e., isomorphic to F × F × F), or it is a cyclic field extension of F (so G is of type ^{1}D_{4} or ^{3}D_{4}), and the Allen invariant E of G is a split central simple L-algebra. This implies that T has the special form T = End Γ for some cyclic composition Γ. We further show in Theorem 4.3 that if G carries a trialitarian automorphism, then the cyclic composition Γ is induced, which means that it is built from some symmetric composition over F, and we establish a one-to-one correspondence between trialitarian automorphisms of G up to conjugation and isomorphism classes of symmetric compositions over F from which Γ is built. Note that we only consider outer automorphisms of order 3, hence we do not investigate the weaker property considered by Garibaldi in [6], about the existence of outer automorphisms whose third power is inner. Nevertheless, our Theorem 4.3 has bearing on it, in view of a result recently announced by Garibaldi and Petersson [7], establishing the existence of outer automorphisms whose third power is inner for any group of type ^{3}D_{4} with trivial Allen invariant. If Γ is a cyclic composition that is not induced (examples are given in Remark 2.1), the group of automorphisms of End Γ does not admit trialitarian automorphisms, but the Garibaldi–Petersson result shows that it has outer automorphisms whose third power is inner. The notions of symmetric and cyclic compositions are recalled in §2. Trialitarian algebras are discussed in §3, which contains the most substantial part of the argument: we determine the trialitarian algebras that have semilinear automorphisms of order 3 (Theorem 3.1) and we classify these automorphisms up to conjugation (Theorem 3.5). The group-theoretic results follow easily in §4 by using the correspondence between groups of type D_{4} and trialitarian algebras. Notation is generally as in the Book of Involutions [9], which is our main reference. For an algebraic structure S defined over a field F, we let Aut(S) denote the group of automorphisms of S, and write Aut(S) for the corresponding group scheme over F. We gratefully thank Vladimir Chernousov, Alberto Elduque, and Sasha Merkurjev for their help during the preparation of this paper. We are also grateful to Skip Garibaldi for his comments on a preliminary version of the paper, and to an anonymous referee for their careful reading.

2. Cyclic and symmetric compositions

Cyclic compositions were introduced by Springer in his 1963 Göttingen lecture notes ([11], [12]) to get new descriptions of Albert algebras. We recall their definition from [12]^{1} and [9, §36.B], restricting to the case of dimension 8. Let F be an arbitrary field (of any characteristic). A cyclic composition (of dimension 8) over F is a 5-tuple Γ = (V, L, Q, ρ, ∗) consisting of

• a cubic étale F -algebra L;

• a free L-module V of rank 8;

• a quadratic form Q: V → L with nondegenerate polar bilinear form b_{Q};

• an F -automorphism ρ of L of order 3;

• an F -bilinear map ∗: V × V → V with the following properties: for all x, y, z ∈ V and λ ∈ L,

(xλ) ∗ y = (x ∗ y)ρ(λ), x ∗ (yλ) = (x ∗ y)ρ^{2}(λ),

1A cyclic composition is called a normal twisted composition in [11] and [12].

Documenta Mathematica · Extra Volume Merkurjev (2015) 387–405

Q(x ∗ y) = ρ(Q(x)) · ρ^{2}(Q(y)),
b_{Q}(x ∗ y, z) = ρ(b_{Q}(y ∗ z, x)) = ρ^{2}(b_{Q}(z ∗ x, y)).

These properties imply the following (see [9, §36.B] or [12, Lemma 4.1.3]): for all x, y ∈ V,

(2)  (x ∗ y) ∗ x = yρ^{2}(Q(x)) and x ∗ (y ∗ x) = yρ(Q(x)).

Since the cubic étale F -algebra L has an automorphism of order 3, L is either a cyclic cubic field extension of F, and ρ is a generator of the Galois group, or we may identify L with F × F × F and assume ρ permutes the components cyclically. We will almost exclusively restrict to the case where L is a field; see however Remark 2.3 below.

Let Γ′ = (V′, L′, Q′, ρ′, ∗′) be also a cyclic composition over F. An isotopy^{2} Γ → Γ′ is defined to be a pair (ν, f) where ν: (L, ρ) →^{∼} (L′, ρ′) is an isomorphism of F -algebras with automorphisms (i.e., ν ∘ ρ = ρ′ ∘ ν) and f: V →^{∼} V′ is a ν-semilinear isomorphism for which there exists μ ∈ L^{×} such that

Q′(f(x)) = ν(ρ(μ)ρ^{2}(μ) · Q(x)) and f(x) ∗′ f(y) = f(x ∗ y)ν(μ)

for x, y ∈ V. The scalar μ is called the multiplier of the isotopy. Isotopies with multiplier 1 are isomorphisms. When the map ν is clear from the context, we write simply f for the pair (ν, f), and refer to f as a ν-semilinear isotopy.

Examples of cyclic compositions can be obtained by scalar extension from symmetric compositions over F, as we now show. Recall from [9, §34] that a symmetric composition (of dimension 8) over F is a triple Σ = (S, n, ⋆) where (S, n) is an 8-dimensional F -quadratic space (with nondegenerate polar bilinear form b_{n}) and ⋆: S × S → S is a bilinear map such that for all x, y, z ∈ S

n(x ⋆ y) = n(x)n(y) and b_{n}(x ⋆ y, z) = b_{n}(x, y ⋆ z).

If Σ′ = (S′, n′, ⋆′) is also a symmetric composition over F, an isotopy Σ → Σ′ is a linear map f: S → S′ for which there exists λ ∈ F^{×} (called the multiplier) such that

n′(f(x)) = λ^{2}n(x) and f(x) ⋆′ f(y) = f(x ⋆ y)λ for x, y ∈ S.

Note that if f: Σ → Σ′ is an isotopy with multiplier λ, then λ^{−1}f: Σ → Σ′ is an isomorphism. Thus, symmetric compositions are isotopic if and only if they are isomorphic. For an explicit example of a symmetric composition, take a Cayley (octonion) algebra (C, ·) with norm n and conjugation map . Letting x ⋆ y = \overline{x} · \overline{y} for x, y ∈ C yields a symmetric composition C̃ = (C, n, ⋆), which is called a para-Cayley composition (see [9, §34.A]). Given a symmetric composition Σ = (S, n, ⋆) and a cubic étale F -algebra L with an automorphism ρ of order 3, we define a cyclic composition Σ ⊗ (L, ρ) as follows:

Σ ⊗ (L, ρ) = (S ⊗_{F} L, L, n_{L}, ρ, ∗)

2The term used in [9, p. 490] is similarity.

Documenta Mathematica · Extra Volume Merkurjev (2015) 387–405

where n_{L} is the scalar extension of n to L and ∗ is defined by extending ⋆ linearly to S ⊗_{F} L and then setting

x ∗ y = (Id_{S} ⊗ρ)(x) ⋆ (Id_{S} ⊗ρ^{2})(y) for x, y ∈ S ⊗_{F} L.

(See [9, (36.11)].) Clearly, every isotopy f: Σ → Σ′ of symmetric compositions extends to an isotopy of cyclic compositions (Id_{L}, f): Σ ⊗ (L, ρ) → Σ′ ⊗ (L, ρ). Observe for later use that the map ρ̂ = Id_{S} ⊗ρ ∈ End_{F} (S ⊗_{F} L) defines a ρ-semilinear automorphism

(3)  ρ̂: Σ ⊗ (L, ρ) →^{∼} Σ ⊗ (L, ρ)

such that ρ^{3} = Id. We call a cyclic composition that is isotopic to Σ ⊗ (L, ρ) for some symmetric composition Σ induced. Cyclic compositions induced from para-Cayley symmetric compositions are called reduced in [12].

Remark 2.1. Induced cyclic compositions are not necessarily reduced. This can be shown by using the following cohomological argument. We assume for simplicity that the field F contains a primitive cube root of unity ω. There is a cohomological invariant g_{3}(Γ) ∈ H^{3}(F, ℤ/3ℤ) attached to any cyclic composition Γ. The cyclic composition Γ is reduced if and only if g_{3}(Γ) = 0 (we refer to [12, §8.3] or [9, §40] for details). We construct an induced cyclic composition Γ with g_{3}(Γ) ≠ 0. Let a, b ∈ F^{×} and let A(a, b) be the F -algebra with generators α, β and relations α^{3} = a, β^{3} = b, βα = ωαβ. The algebra A(a, b) is central simple of dimension 9 and the space A^{0} of elements of A(a, b) of reduced trace zero admits the structure of a symmetric composition Σ(a, b) = (A^{0}, n, ⋆) (see [9, (34.19)]). Such symmetric compositions are called Okubo symmetric compositions. From the Elduque–Myung classification of symmetric compositions [5, p. 2487] (see also [9, (34.37)]), it follows that symmetric compositions are either para-Cayley or Okubo. Let L = F (γ) with γ^{3} = c ∈ F^{×} be a cubic cyclic field extension of F, and let ρ be the F -automorphism of L such that γ ↦ ωγ. We may then consider the induced cyclic composition Γ(a, b, c) = Σ(a, b) ⊗ (L, ρ). Its cohomological invariant g_{3}(Γ(a, b, c)) can be computed by the construction in [12, §8.3]: Using ω, we identify the group μ_{3} of cube roots of unity in F with ℤ/3ℤ, and for any u ∈ F^{×} we write [u] for the cohomology class in H^{1}(F, ℤ/3ℤ) corresponding to the cube class uF^{×3} under the isomorphism F^{×}/F^{×3} ≅ H^{1}(F, μ_{3}) arising from the Kummer exact sequence (see [9, p. 413]). Then g_{3}(Γ(a, b, c)) is the cup-product [a] ∪ [b] ∪ [c] ∈ H^{3}(F, ℤ/3ℤ). Thus any cyclic composition Γ(a, b, c) with [a] ∪ [b] ∪ [c] ≠ 0 is induced but not reduced. Another cohomological argument can be used to show that there exist cyclic compositions that are not induced. We still assume that F contains a primitive cube root of unity ω. There is a further cohomological invariant of cyclic compositions f_{3}(Γ) ∈ H^{3}(F, ℤ/2ℤ) which is zero for any cyclic composition induced by an Okubo symmetric composition^{3} and is given by the class in H^{3}(F, ℤ/2ℤ) of the 3-fold Pfister form which is the norm of C̃ if Γ is induced from the para-Cayley C̃ (see for example [9, §40]). Thus a cyclic composition Γ with f_{3}(Γ) ≠ 0 and g_{3}(Γ) ≠ 0 is not induced. Such examples can be given with the help of the Tits process used for constructing Albert algebras (see [9, §39 and §40]). However, for example, cyclic compositions over finite fields, p-adic fields or algebraic number fields are reduced, see [12, p. 108].

3The fact that F contains a primitive cubic root of unity is relevant for this claim.

Documenta Mathematica · Extra Volume Merkurjev (2015) 387–405

Examples 2.2. (i) Let F = 𝔽_{q} be the field with q elements, where q is odd and q ≡ 1 mod 3. Thus F contains a primitive cube root of unity and we are in the situation of Remark 2.1. Let L = 𝔽_{q^{3}} be the (unique, cyclic) cubic field extension of F, and let ρ be the Frobenius automorphism of L/F. Because H^{3}(F, ℤ/3ℤ) = 0, every cyclic composition over F is reduced; moreover every 3-fold Pfister form is hyperbolic, hence every Cayley algebra is split. Therefore, up to isomorphism there is a unique cyclic composition over F with cubic algebra (L, ρ), namely Γ = C̃ ⊗ (L, ρ) where C̃ is the split para-Cayley symmetric composition. If Σ denotes the Okubo symmetric composition on 3 × 3 matrices of trace zero with entries in F, we thus have Γ ≅ Σ ⊗ (L, ρ), which means that Γ is also induced by Σ. By the Elduque–Myung classification of symmetric compositions, every symmetric composition over F is isomorphic either to the Okubo composition Σ or to the split para-Cayley composition C̃. Therefore, Γ is induced by exactly two symmetric compositions over F up to isomorphism.

(ii) Assume that F contains a primitive cube root of unity and that F carries an anisotropic 3-fold Pfister form n. Let C be the non-split Cayley algebra with norm n and let C̃ be the associated para-Cayley algebra. For any cubic cyclic field extension (L, ρ) the norm n_{L} of the cyclic composition C̃ ⊗ (L, ρ) is anisotropic. Thus it follows from the Elduque–Myung classification that any symmetric composition Σ such that Σ ⊗ (L, ρ) is isotopic to C̃ ⊗ (L, ρ) must be isomorphic to C̃.

(iii) Finally, we observe that the cyclic compositions of type Γ(a, b, c), described in Remark 2.1, have invariant g_{3} equal to zero if c = a. Since the f_{3}-invariant is also zero, they are all isotopic to the cyclic composition induced by the split para-Cayley algebra. Thus we can get (over suitable fields) examples of many mutually non-isomorphic symmetric compositions Σ(a, b) that induce isomorphic cyclic compositions Γ(a, b, c).

Of course, besides this construction of cyclic compositions by induction from symmetric compositions, we can also extend scalars of a cyclic composition: if Γ = (V, L, Q, ρ, ∗) is a cyclic composition over F and K is any field extension of F, then Γ_{K} = (V ⊗_{F} K, L ⊗_{F} K, Q_{K}, ρ ⊗ Id_{K}, ∗_{K}) is a cyclic composition over K.

Remark 2.3. Let Γ = (V, L, Q, ρ, ∗) be an arbitrary cyclic composition over F with L a field. Write θ for ρ^{2}. We have an isomorphism of L-algebras

ν: L ⊗_{F} L →^{∼} L × L × L given by ℓ_{1} ⊗ ℓ_{2} ↦ (ℓ_{1}ℓ_{2}, ρ(ℓ_{1})ℓ_{2}, θ(ℓ_{1})ℓ_{2}).

Therefore, the extended cyclic composition Γ_{L} over L has a split cubic étale algebra. To give an explicit description of Γ_{L}, note first that under the isomorphism ν the automorphism ρ ⊗ Id_{L} is identified with the map ρ̃ defined by ρ̃(ℓ_{1}, ℓ_{2}, ℓ_{3}) = (ℓ_{2}, ℓ_{3}, ℓ_{1}). Consider the twisted L-vector spaces ^{ρ}V, ^{θ}V defined by

ρV = {^{ρ}x | x ∈ V }, ^{θ}V = {^{θ}x | x ∈ V }

with the operations

ρ(x+y) = ^{ρ}x+^{ρ}y, ^{θ}(x+y) = ^{θ}x+^{θ}y, and ^{ρ}(xλ) = (^{ρ}x)ρ(λ), ^{θ}(xλ) = (^{θ}x)θ(λ)

for x, y ∈ V and λ ∈ L. Define quadratic forms ^{ρ}Q: ^{ρ}V → L and ^{θ}Q: ^{θ}V → L by

ρQ(^{ρ}x) = ρ(Q(x)) and ^{θ}Q(^{θ}x) = θ(Q(x)) for x ∈ V ,

and L-bilinear maps

∗_{Id} : ^{ρ}V ×^{θ}V → V, ∗_{ρ}: ^{θ}V × V → ^{ρ}V, ∗_{θ}: V ×^{ρ}V → ^{θ}V

by

ρx ∗_{Id} ^{θ}y = x ∗ y, ^{θ}x ∗_{ρ} y = ^{ρ}(x ∗ y), x ∗_{θ} ^{ρ}y = ^{θ}(x ∗ y) for x, y ∈ V.

We may then consider the quadratic form

Q ×^{ρ}Q ×^{θ}Q: V ×^{ρ}V ×^{θ}V → L × L × L

and the product ⋄: (V ×^{ρ}V ×^{θ}V) × (V ×^{ρ}V ×^{θ}V) → (V ×^{ρ}V ×^{θ}V) defined by

(x,^{ρ}x,^{θ}x) ⋄ (y,^{ρ}y,^{θ}y) = (^{ρ}x ∗_{Id} ^{θ}y,^{θ}x ∗_{ρ} y, x ∗_{θ} ^{ρ}y).

Straightforward calculations show that the F -vector space isomorphism f: V ⊗_{F} L → V ×^{ρ}V ×^{θ}V given by

f(x ⊗ ℓ) = (xℓ, (^{ρ}x)ℓ, (^{θ}x)ℓ) for x ∈ V and ℓ ∈ L

defines with ν an isomorphism of cyclic compositions

Γ_{L} →^{∼} (V ×^{ρ}V ×^{θ}V, L × L × L, Q ×^{ρ}Q ×^{θ}Q, ρ̃, ⋄).

3. Trialitarian algebras

In this section, we assume that the characteristic of the base field F is different from 2. Trialitarian algebras are defined in [9, §43] as 4-tuples T = (E, L, σ, α) where L is a cubic étale F -algebra, (E, σ) is a central simple L-algebra of degree 8 with an orthogonal involution, and α is an isomorphism from the Clifford algebra C(E, σ) to a certain twisted scalar extension of E. We just recall in detail the special case of trialitarian algebras of the form End Γ for Γ a cyclic composition, because this is the main case for the purposes of this paper.

Let Γ = (V, L, Q, ρ, ∗) be a cyclic composition (of dimension 8) over F, with L a field, and let θ = ρ^{2}. Let also σ_{Q} denote the orthogonal involution on End_{L} V adjoint to Q. We will use the product ∗ to see that the Clifford algebra C(V, Q) is split and the even Clifford algebra C_{0}(V, Q) decomposes into a direct product of two split central simple L-algebras of degree 8. Using the notation of Remark 2.3, to any x ∈ V we associate L-linear maps

ℓ_{x}: ^{ρ}V → ^{θ}V and r_{x}: ^{θ}V → ^{ρ}V

defined by

ℓ_{x}(^{ρ}y) = x ∗_{θ} ^{ρ}y = ^{θ}(x ∗ y) and r_{x}(^{θ}z) = ^{θ}z ∗_{ρ} x = ^{ρ}(z ∗ x)

for y, z ∈ V. From (2) it follows that for x ∈ V the L-linear map

α_{∗}(x) = ( \begin{matrix}0 & r_{x} \\ ℓ_{x} & 0 \end{matrix} ): ^{ρ}V ⊕^{θ}V → ^{ρ}V ⊕^{θ}V given by (^{ρ}y,^{θ}z) ↦ (r_{x}(^{θ}z), ℓ_{x}(^{ρ}y))

satisfies α_{∗}(x)^{2} = Q(x) Id. Therefore, there is an induced L-algebra homomorphism

(4)  α_{∗}: C(V, Q) → End_{L}(^{ρ}V ⊕^{θ}V).

This homomorphism is injective because C(V, Q) is a simple algebra, hence it is an isomorphism by dimension count. It restricts to an L-algebra isomorphism

α_{∗0} : C_{0}(V, Q) →^{∼} End_{L}(^{ρ}V) × End_{L}(^{θ}V),

see [9, (36.16)]. Note that we may identify End_{L}(^{ρ}V) with the twisted algebra ρ(End_{L} V) (where multiplication is defined by ^{ρ}f_{1} ·^{ρ}f_{2} = ^{ρ}(f_{1} ∘ f_{2})) as follows: for f ∈ End_{L} V, we identify ^{ρ}f with the map ^{ρ}V → ^{ρ}V such that ^{ρ}f(^{ρ}x) = ρ(f(x)) for x ∈ V. On the other hand, let σ_{Q} be the orthogonal involution on End_{L} V adjoint to Q. The algebra C_{0}(V, Q) is canonically isomorphic to the Clifford algebra C(End_{L} V, σ_{Q}) (see [9, (8.8)]), hence it depends only on End_{L} V and σ_{Q}. We may regard α_{∗0} as an isomorphism of L-algebras

α_{∗0} : C(End_{L} V, σ_{Q}) →^{∼} ^{ρ}(End_{L} V) ×^{θ}(End_{L} V).

Thus, α_{∗0} depends only on End_{L} V and σ_{Q}. The trialitarian algebra End Γ is the 4-tuple

End Γ = (End_{L} V, L, σ_{Q}, α_{∗0}).

An isomorphism of trialitarian algebras End Γ →^{∼} End Γ′, for Γ′ = (V′, L′, Q′, ρ′, ∗′) a cyclic composition, is defined to be an isomorphism of F - algebras with involution φ: (End_{L} V, σ_{Q}) →^{∼} (End_{L′} V′, σ_{Q′}) subject to the following conditions:

(i) the restriction of φ to the center of End_{L} V is an isomorphism

φ|_{L}: (L, ρ) →^{∼} (L′, ρ′), and

(ii) the following diagram (where θ′ = ρ′^{2}) commutes:

\xymatrix{C(End_{L} V, σ_{Q}) \ar[r]^{α_{∗0}} \ar[d]_{C(φ)} & ^{ρ}(End_{L} V) ×^{θ}(End_{L} V) \ar[d]^{^{ρ}φ×^{θ}φ} \\ C(End_{L′} V′, σ_{Q′}) \ar[r]^{α_{∗′0}} & ^{ρ′}(End_{L′} V′) ×^{θ′}(End_{L′} V′)}

For example, it is straightforward to check that every isotopy (ν, f): Γ → Γ′ induces an isomorphism End Γ → End Γ′ mapping g ∈ End_{L} V to f ∘ g ∘ f^{−1} ∈ End_{L′} V′. As part of the proof of the main theorem below, we show that every isomorphism End Γ →^{∼} End Γ′ is induced by an isotopy; see Lemma 3.4. (A cohomological proof that the trialitarian algebras End Γ, End Γ′ are isomorphic if and only if the cyclic compositions Γ, Γ′ are isotopic is given in [9, (44.16)].)

We show that the trialitarian algebra End Γ admits a ρ-semilinear automorphism of order 3 if and only if Γ is induced. More precisely:

Theorem 3.1. Let Γ = (V, L, Q, ρ, ∗) be a cyclic composition over F, with L a field.

(i) If Σ is a symmetric composition over F and f: Σ ⊗ (L, ρ) → Γ is an L-linear isotopy, then the automorphism τ_{(Σ,f)} = Int(f ∘ρ̂∘f^{−1})|_{End_{L} V} of End Γ, where ρ̂ is defined in (3), is such that τ_{(Σ,f)}^{3} = Id and τ_{(Σ,f)}|_{L}= ρ. The automorphism τ_{(Σ,f)} only depends, up to conjugation in Aut_{F} (End Γ), on the isomorphism class of Σ.

(ii) If End Γ carries an F -automorphism τ such that τ|_{L} = ρ and τ^{3} = Id, then Γ is induced. More precisely, there exists a symmetric composition Σ over F and an L-linear isotopy f: Σ ⊗ (L, ρ) → Γ such that τ = τ_{(Σ,f)}.

Proof. (i) It is clear that τ_{(Σ,f)}^{3} = Id and τ_{(Σ,f)}|_{L}= ρ. For the last claim, note that if g: Σ ⊗ (L, ρ) → Γ is another L-linear isotopy, then f ∘ g^{−1} is an isotopy of Γ, hence Int(f ∘ g^{−1}) is an automorphism of End Γ, and

τ_{(Σ,f)} = Int(f ∘ g^{−1}) ∘ τ_{(Σ,g)} ∘ Int(f ∘ g^{−1})^{−1}.

The proof of claim (ii) relies on three lemmas. Until the end of this section, we fix a cyclic composition Γ = (V, L, Q, ρ, ∗), with L a field. We start with some general observations on ρ-semilinear automorphisms of End_{L} V. For this, we consider the inclusions

L ↪ End_{L} V ↪ End_{F} V.

The field L is the center of End_{L} V, hence every automorphism of End_{L} V restricts to an automorphism of L.

Lemma 3.2. Let ν ∈ {Id_{L}, ρ, θ} be an arbitrary element in the Galois group Gal(L/F). For every F -linear automorphism φ of End_{L} V such that φ|_{L} = ν, there exists an invertible transformation u ∈ End_{F} V such that φ(f) = u∘f∘u^{−1} for all f ∈ End_{L} V. The map u is uniquely determined up to a factor in L^{×}; it is ν-semilinear, i.e., u(xλ) = u(x)ν(λ) for all x ∈ V and λ ∈ L. Moreover, if φ ∘ σ_{Q} = σ_{Q} ∘ φ, then there exists μ ∈ L^{×} such that

Q(u(x)) = ν(μ · Q(x)) for all x ∈ V.

Proof. The existence of u is a consequence of the Skolem–Noether theorem, since End_{L} V is a simple subalgebra of the simple algebra End_{F} V: the automorphism φ extends to an inner automorphism Int(u) of End_{F} V for some invertible u ∈ End_{F} V. Uniqueness of u up to a factor in L^{×} is clear because L is the centralizer of End_{L} V in End_{F} V, and the ν-semilinearity of u follows from the equation φ(f) = u ∘ f ∘ u^{−1} applied with f the scalar multiplication by an element in L. Now, suppose φ commutes with σ_{Q}, hence for all f ∈ End_{L} V

(5)  u ∘ σ_{Q}(f) ∘ u^{−1} = σ_{Q}(u ∘ f ∘ u^{−1}).

Let Tr_{∗}(Q) denote the transfer of Q along the trace map Tr_{L/F}, so Tr_{∗}(Q): V → F is the quadratic form defined by Tr_{∗}(Q)(x) = Tr_{L/F} (Q(x)). The adjoint involution σ_{Tr_{∗}(Q)} coincides on End_{L} V with σ_{Q}, hence from (5) it follows that σ_{Tr_{∗}(Q)}(u)u centralizes End_{L} V. Therefore, σ_{Tr_{∗}(Q)}(u)u = μ for some μ ∈ L^{×}. We then have b_{Tr_{∗}(Q)}(u(x), u(y)) = b_{Tr_{∗}(Q)}(x, yμ) for all x, y ∈ V, which means that

(6)  Tr_{L/F} (b_{Q}(u(x), u(y))) = Tr_{L/F} (μb_{Q}(x, y)).

Now, observe that since u is ν-semilinear, the map c: V × V → L defined by c(x, y) = ν^{−1}(b_{Q}(u(x), u(y))) is L-bilinear. From (6), it follows that c − μb_{Q} is a bilinear map on V that takes its values in the kernel of the trace map. But the value domain of an L-bilinear form is either L or {0}, and the trace map is not the zero map. Therefore, c − μb_{Q} = 0, which means that

ν^{−1}(b_{Q}(u(x), u(y))) = μb_{Q}(x, y) for all x, y ∈ V,

hence Q(u(x)) = ν(μ · Q(x)) for all x ∈ V. □

Note that the arguments in the preceding proof apply to any quadratic space (V, Q) over L. By contrast, the next lemma uses the full cyclic composition structure: Let again ν ∈ {Id_{L}, ρ, θ}. Given an invertible element u ∈ End_{F} V and μ ∈ L^{×} such that for all x ∈ V and λ ∈ L

u(xλ) = u(x)ν(λ) and Q(u(x)) = ν(μ · Q(x)),

we define an L-linear map β_{u}: ^{ν}V → End_{L}(^{ρ}V ⊕^{θ}V) by

β_{u}(^{ν}x) = (\begin{matrix}0 & ν(μ)^{−1}r_{u(x)} \\ ℓ_{u(x)} & 0 \end{matrix}) ∈ End_{L}(^{ρ}V ⊕^{θ}V) for x ∈ V.

Then from (2) we get β_{u}(x)^{2} = ν(Q(x)) = ^{ν}Q(^{ν}x). Therefore, the map β_{u} extends to an L-algebra homomorphism

β_{u}: C(^{ν}V,^{ν}Q) → End_{L}(^{ρ}V ⊕^{θ}V).

Just like α_{∗} in (4), the homomorphism β_{u} is an isomorphism. We also have an isomorphism of F -algebras C(^{ν}·): C(V, Q) → C(^{ν}V,^{ν}Q) induced by the F - linear map x ↦ ^{ν}x for x ∈ V, so we may consider the F -automorphism ψ_{u} of End_{L}(^{ρ}V ⊕^{θ}V) that makes the following diagram commute:

\begin{matrix}C(V, Q) & α_{∗} & [figure] & C(^{ν}·) & [figure] & End_{L}(^{ρ}V ⊕^{θ}V) & ψ_{u} & [figure] & C(^{ν}V,^{ν}Q) & β_{u} & [figure] & End_{L}(^{ρ}V ⊕^{θ}V) \\ (7) \end{matrix}

Lemma 3.3. The F -algebra automorphism ψ_{u} restricts to an F -algebra automorphism ψ_{u0} of End_{L}(^{ρ}V) × End_{L}(^{θ}V). The restriction of ψ_{u0} to the center L × L is either ν × ν or (ν × ν) ∘ ε where ε is the switch map (ℓ_{1}, ℓ_{2}) ↦ (ℓ_{2}, ℓ_{1}). Moreover, if ψ_{u0}|_{L×L} = ν × ν, then there exist invertible ν-semilinear transformations u_{1}, u_{2} ∈ End_{F} V such that

ψ_{u}(f) = ( \begin{matrix}^{ρ}u_{1} & 0 \\ 0 & ^{θ}u_{2} \end{matrix} ) ∘ f ∘ ( \begin{matrix}^{ρ}u_{1}^{−1} & 0 \\ 0 & ^{θ}u_{2}^{−1} \end{matrix} ) for all f ∈ End_{L}(^{ρ}V ⊕^{θ}V).

For any pair (u_{1}, u_{2}) satisfying this condition, we have

u_{2}(x∗y) = u(x)∗u_{1}(y) and u_{1}(x∗y) = (u_{2}(x)∗u(y))θν(μ)^{−1} for all x, y ∈ V.

Proof. The maps α_{∗} and β_{u} are isomorphisms of graded L-algebras for the usual (ℤ/2ℤ)-gradings of C(V, Q) and C(^{ν}V,^{ν}Q), and for the “checker-board” grading of End_{L}(^{ρ}V ⊕^{θ}V) defined by

End_{L}(^{ρ}V ⊕^{θ}V)_{0} = End_{L}(^{ρ}V) × End_{L}(^{θ}V)

and

End_{L}(^{ρ}V ⊕^{θ}V)_{1} = (\begin{matrix}0 & Hom_{L}(^{θ}V,^{ρ}V) \\ Hom_{L}(^{ρ}V,^{θ}V) & 0 \end{matrix}) .

Therefore, ψ_{u} also preserves the grading, and it restricts to an automorphism ψ_{u0} of the degree 0 component. Because the map C(^{ν}·) is ν-semilinear, the map ψ_{u} also is ν-semilinear, hence its restriction to the center of the degree 0 component is either ν × ν or (ν × ν) ∘ ε. Suppose ψ_{u0}|_{L×L} = ν ×ν. By Lemma 3.2 (applied with ^{ρ}V ⊕^{θ}V instead of V), there exists an invertible ν-semilinear transformation v ∈ End_{F} (^{ρ}V ⊕^{θ}V) such that ψ_{u}(f) = v ∘ f ∘ v^{−1} for all f ∈ End_{F} (^{ρ}V ⊕^{θ}V). Since ψ_{u0} fixes ( \begin{matrix}Id_{^{ρ}V} & 0 \\ 0 & 0 \end{matrix} ), the element v centralizes ( \begin{matrix}Id_{^{ρ}V} & 0 \\ 0 & 0 \end{matrix} ), hence v = ( \begin{matrix}^{ρ}u_{1} & 0 \\ 0 & ^{θ}u_{2} \end{matrix} ) for some invertible u_{1}, u_{2} ∈ End_{F} V. The transformations u_{1} and u_{2} are ν-semilinear because v is ν-semilinear. From the commutativity of (7) we have v ∘ α_{∗}(x) = β_{u}(^{ν}x) ∘ v = α_{∗}(u(x)) ∘ v for all x ∈ V. By the definition of α_{∗}, it follows that

u_{1}(z ∗ x) = θν^{−1}(μ)(u_{2}(z) ∗ u(x)) and u_{2}(x ∗ y) = u(x) ∗ u_{1}(y) for all y, z ∈ V.

□

Lemma 3.4. Let ν ∈ {Id_{L}, ρ, θ}. For every F -linear automorphism φ of End Γ such that φ|_{L} = ν, there exists an invertible transformation u ∈ End_{F} V, uniquely determined up to a factor in L^{×}, such that φ(f) = u ∘ f ∘ u^{−1} for all f ∈ End_{L} V. Every such u is a ν-semilinear isotopy Γ → Γ.

Proof. The existence of u, its uniqueness up to a factor in L^{×}, and its νsemilinearity, were established in Lemma 3.2. It only remains to show that u is an isotopy. Since φ is an automorphism of End Γ, it commutes with σ_{Q}, hence Lemma 3.2 yields μ ∈ L^{×} such that Q(u(x)) = ν(μ·Q(x)) for all x ∈ V. We may therefore consider the maps β_{u} and ψ_{u} of Lemma 3.3. Now, recall from [9, (8.8)] that C_{0}(V, Q) = C(End_{L} V, σ_{Q}) by identifying x · y for x, y ∈ V with the image in C(End_{L} V, σ_{Q}) of the linear transformation x ⊗ y defined by z ↦ x · b_{Q}(y, z) for z ∈ V. We have

φ(x ⊗ y) = u ∘ (x ⊗ y) ∘ u^{−1} : z ↦ u(x · b_{Q}(y, u^{−1}(z))) for x, y, z ∈ V.

Since u is ν-semilinear and Q(u(x)) = ν(μ · Q(x)) for all x ∈ V, it follows that

u(x · b_{Q}(y, u^{−1}(z))) = u(x) · ν(b_{Q}(y, u^{−1}(z))) = u(x) · ν(μ)^{−1}b_{Q}(u(y), z).

Therefore, φ(x ⊗ y) = u(x) ⊗ u(y)ν(μ)^{−1} for x, y ∈ V, hence the following diagram (where β_{u} and C(^{ν}·) are as in (7)) is commutative:

\xymatrix{C_{0}(V, Q) \ar[r]^{C(^{ν}·)|_{C_{0}(V,Q)}} \ar[d]_{C(φ)} & C_{0}(^{ν}V,^{ν}Q) \ar[d]^{β_{u}|_{C_{0}(^{ν} V,^{ν} Q)}} \\ C_{0}(V, Q) \ar[r]^{α_{∗0}} & End_{L}(^{ρ}V) × End_{L}(^{θ}V)}

On the other hand, the following diagram is commutative because φ is an automorphism of End Γ:

\xymatrix{C_{0}(V, Q) \ar[r]^{α_{∗0}} \ar[d]_{C(φ)} & End_{L}(^{ρ}V) × End_{L}(^{θ}V) \ar[d]^{^{ρ}φ×^{θ}φ} \\ C_{0}(V, Q) \ar[r]^{α_{∗0}} & End_{L}(^{ρ}V) × End_{L}(^{θ}V)}

Therefore, β_{u}|_{C_{0}(^{ν}V,^{ν}Q)} ∘ C(^{ν}·)|_{C_{0}(V,Q)} = (^{ρ}φ ×^{θ}φ) ∘ α_{∗0}. By comparing with (7), we see that ψ_{u0} = ^{ρ}φ ×^{θ}φ, hence ψ_{u0}|_{L×L} = ν × ν. Lemma 3.3 then yields ν-semilinear transformations u_{1}, u_{2} ∈ End_{F} V such that

ψ_{u}(f) = ( \begin{matrix}^{ρ}u_{1} & 0 \\ 0 & ^{θ}u_{2} \end{matrix} ) ∘ f ∘ ( \begin{matrix}^{ρ}u_{1}^{−1} & 0 \\ 0 & ^{θ}u_{2}^{−1} \end{matrix} ) for all f ∈ End_{L}(^{ρ}V ⊕^{θ}V),

hence ψ_{u0} = Int(^{ρ}u_{1})×Int(^{θ}u_{2}). But we have ψ_{u0} = ^{ρ}φ×^{θ}φ = Int(^{ρ}u)×Int(^{θ}u). Therefore, multiplying (u_{1}, u_{2}) by a scalar in L^{×}, we may assume u = u_{1} and u_{2} = uζ for some ζ ∈ L^{×}. Lemma 3.3 then gives

u(x ∗ y)ζ = u(x) ∗ u(y) and u(x ∗ y) = ((ζu(x)) ∗ u(y))θν(μ)^{−1} for all x, y ∈ V.

The second equation implies that u(x ∗ y) = (u(x) ∗ u(y))ρ(ζ)θν(μ)^{−1}. By comparing with the first equation, we get ρ(ζ)θν(μ)^{−1} = ζ^{−1}, hence ν(μ) = ρ(ζ)θ(ζ). Therefore, (ν, u) is an isotopy Γ → Γ with multiplier ν^{−1}(ζ). □

We start with the proof of claim (ii) of Theorem 3.1. Suppose τ is an F - automorphism of End Γ such that τ|_{L} = ρ and τ^{3} = Id. By Lemma 3.4, we may find an invertible ρ-semilinear transformation t ∈ End_{F} V such that τ(f) = t ∘ f ∘ t^{−1} for all f ∈ End_{L} V, and every such t is an isotopy of Γ. Since τ^{3} = Id, it follows that t^{3} lies in the centralizer of End_{L} V in End_{F} V, which is L. Let t^{3} = ξ ∈ L^{×}. We have ρ(ξ) = tξt^{−1} = ξ, hence ξ ∈ F^{×}. The F -subalgebra of End_{F} V generated by L and t is a crossed product (L, ρ, ξ); its centralizer is the F -subalgebra (End_{L} V)^{τ} fixed under τ, and we have

End_{F} V ≅ (L, ρ, ξ) ⊗_{F} (End_{L} V)^{τ} .

Now, deg(L, ρ, ξ) = 3 and deg(End_{L} V)^{τ} = 8, hence (L, ρ, ξ) is split. Therefore ξ = N_{L/F} (η) for some η ∈ L^{×}. Substituting η^{−1}t for t, we get t^{3} = Id_{V}, and t is still a ρ-linear isotopy of Γ. Let μ ∈ L^{×} be the corresponding multiplier, so that for all x, y ∈ V

(8)  Q(t(x)) = ρ(ρ(μ)θ(μ)Q(x)) and t(x) ∗ t(y) = t(x ∗ y)ρ(μ).

From the second equation we deduce that t^{3}(x) ∗ t^{3}(y) = t^{3}(x ∗ y)N_{L/F} (μ) for all x, y ∈ V, hence N_{L/F} (μ) = 1 because t^{3} = Id_{V}. By Hilbert’s Theorem 90, we may find ζ ∈ L^{×} such that μ = ζθ(ζ)^{−1}. Define Q′ = ρ(ζ)θ(ζ)Q and let x ∗′ y = (x ∗ y)ζ for x, y ∈ V. Then Id_{V} is an isotopy Γ → Γ′ = (V, L, Q′, ρ, ∗′) with multiplier ζ, and (8) implies that

Q′(t(x)) = ρ(Q′(x)) and t(x) ∗′ t(y) = t(x ∗′ y) for all x, y ∈ V.

Now, observe that because t is ρ-semilinear and t^{3} = Id_{V}, the Galois group of L/F acts by semilinear automorphisms on V, hence we have a Galois descent (see [9, (18.1)]): the fixed point set S = {x ∈ V | t(x) = x} is an F -vector space such that V = S ⊗_{F} L. Moreover, since Q′(t(x)) = ρ(Q′(x)) for all x ∈ V, the restriction of Q′ to S is a quadratic form n: S → F, and we have Q′ = n_{L}. Also, because t(x ∗′ y) = t(x) ∗′ t(y) for all x, y ∈ V, the product ∗′ restricts to a product ⋆ on S, and Σ = (S, n, ⋆) is a symmetric composition because Γ′ is a cyclic composition. The canonical map f: S⊗_{F} L → V yields an isomorphism of cyclic compositions f: Σ ⊗ (L, ρ) →^{∼} Γ′, hence also an isotopy f: Σ ⊗ (L, ρ) → Γ. We have t = f ∘ ρ̂ ∘ f^{−1}, hence τ is conjugation by f ∘ ρ̂∘ f^{−1}. □

Theorem 3.5. The assignment Σ ↦ τ_{(Σ,f)} induces a bijection between the isomorphism classes of symmetric compositions Σ for which there exists an L-linear isotopy f: Σ ⊗ (L, ρ) → Γ and conjugacy classes in Aut_{F} (End Γ) of automorphisms τ of End Γ such that τ^{3} = Id and τ|_{L} = ρ.

Proof. We already know by Theorem 3.1 that the map induced by Σ ↦ τ_{(Σ,f)} is onto. Therefore, it suffices to show that if the automorphisms τ_{(Σ,f)} and τ_{(Σ′,f′)} associated to symmetric compositions Σ and Σ′ are conjugate, then Σ and Σ′ are isomorphic. Assume τ_{(Σ′,f′)} = φ ∘ τ_{(Σ,f)} ∘ φ^{−1} for some φ ∈ Aut_{F} (End Γ), and let t = f ∘ ρ̂ ∘ f^{−1}, t′ = f′ ∘ ρ̂ ∘ f′^{−1} ∈ End Γ be the ρ-semilinear transformations such that τ_{(Σ,f)} = Int(t)|_{End_{L} V} and τ_{(Σ′,f′)} = Int(t′)|_{End_{L} V}. By Lemma 3.4 we may find an isotopy (ν, u): Γ → Γ such that φ = Int(u)|_{End_{L} V}. The equation τ_{(Σ′,f′)} = φ ∘ τ_{(Σ,f)} ∘ φ^{−1} then yields Int(t′)|_{End_{L} V} = Int(u ∘ t ∘ u^{−1})|_{End_{L} V}, hence there exists ξ ∈ L^{×} such that u ∘ t ∘ u^{−1} = ξt′. Because t^{3} = t′^{3} = Id_{V}, we have N_{L/F} (ξ) = 1, hence Hilbert’s Theorem 90 yields η ∈ L^{×} such that ξ = ρ(η)η^{−1}. Then η^{−1}u: Γ → Γ is a ν-semilinear isotopy such that (η^{−1}u) ∘ t ∘ (η^{−1}u)^{−1} = t′, and we have a commutative diagram

\xymatrix{Σ ⊗ (L, ρ) \ar[d]_{ρ̂} \ar[r]^{f′^{−1}∘(η^{−1}u)∘f} & Σ′ ⊗ (L, ρ) \ar[d]^{ρ̂} \\ Σ ⊗ (L, ρ) \ar[r]^{f′^{−1}∘(η^{−1}u)∘f} & Σ′ ⊗ (L, ρ)}

The restriction of f′^{−1} ∘(η^{−1}u)∘f to Σ is an isotopy of symmetric compositions Σ → Σ′; a scalar multiple of this map is an isomorphism Σ →^{∼} Σ′. □

4. Trialitarian automorphisms of groups of type D_{4}

Let F be a field of characteristic different from 2. By [9, (44.8)], for every adjoint simple group G of type D_{4} over F there is a trialitarian algebra T = (E, L, σ, α) such that G is isomorphic to Aut_{L}(T).

Proposition 4.1. The natural map Φ : Aut_{F} (T) → Aut(G) induced by conjugation is an isomorphism of group schemes.

Proof. The group G is the connected component of Aut_{F} (T) by construction. By [4, Exp. XXIV, Th. 1.3], the group Aut(G) is a smooth algebraic group scheme, and the conjugation homomorphism Φ is a homomorphism of algebraic groups. Since G is adjoint semisimple the restriction of Φ to the connected component is an injective homomorphism G → Aut(G), hence by [9, (22.2)] the differential dΦ is injective. On the other hand, since the correspondence between trialitarian algebras and adjoint simple groups of type D_{4} is actually shown in [9, (44.8)] to be an equivalence of groupoids, over an algebraic closure F_{alg} the map Φ_{alg} : Aut_{F} (T)(F_{alg}) → Aut(G)(F_{alg}) is an isomorphism. By [9, (22.5)] it follows that Φ is an isomorphism of group schemes. □

We thus have a commutative diagram with exact rows:

[figure: 1; Aut_{L}(T); Aut_{F} (T); Φ; Aut_{F} (L); 1; 1; G; Aut(G); π; (𝔖_{3})_{L}; 1]

(9)

where (𝔖_{3})_{L} is a (non-constant) twisted form of the symmetric group 𝔖_{3}. Here Aut_{F} (L) is the group scheme given by Aut_{F} (L)(R) = Aut_{R-alg}(L ⊗_{F} R) for any commutative F -algebra R. Thus, the type of the group G is related as follows to the type of L and to Aut_{F} (L):

(i) type ^{1}D_{4}: L ≅ F × F × F and Aut_{F} (L)(F) ≅ 𝔖_{3}; (ii) type ^{2}D_{4}: L ≅ F × Δ (with Δ a quadratic field extension of F) and

Aut_{F} (L)(F) ≅ 𝔖_{2}; (iii) type ^{3}D_{4}: L a cyclic cubic field extension of F and Aut_{F} (L)(F) ≅

ℤ/3ℤ; (iv) type ^{6}D_{4}: L a non-cyclic cubic field extension of F and Aut_{F} (L)(F) =

1.

Proposition 4.2. Let G be an adjoint simple group of type D_{4} over F. If Aut(G)(F) contains an outer automorphism φ such that φ^{3} is inner, then G is of type ^{1}D_{4} or ^{3}D_{4}, and in the trialitarian algebra T = (E, L, σ, α) such that G ≅ Aut_{L}(T), the central simple L-algebra E is split.

Proof. The exactness of the bottom row of (9) implies the exactness of

(10)  \xymatrix{1 \ar[r] & G(F) \ar[r] & Aut(G)(F) \ar[r]^{π} & (𝔖_{3})_{L}(F)}

Since the image π(φ) ∈ (𝔖_{3})_{L}(F) has order 3, Aut_{F} (L)(F) must be isomorphic to 𝔖_{3} or to ℤ/3ℤ and hence the cases ^{2}D_{4} and ^{6}D_{4} can be ruled out from the characterization of the various types above. Therefore the type of G is ^{1}D_{4} or 3D_{4}. If G is of type ^{1}D_{4}, then the algebra E is split by [6, Example 17] or by [2, Theorem 13.1]. If G is of type ^{3}D_{4}, then after scalar extension to L the group G_{L} has type ^{1}D_{4}, so E ⊗_{F} L is split. Therefore, the Brauer class of E has 3-torsion since it is split by a cubic extension. But it also has 2-torsion since E carries an orthogonal involution, hence E is split. □

For the rest of this section, we focus on trialitarian automorphisms (i.e., outer automorphisms of order 3) of groups of type ^{3}D_{4}. Let G be an adjoint simple group of type^{3}D_{4} over F, and let L be its associated cyclic cubic field extension of F. Thus,

(𝔖_{3})_{L}(F) = Gal(L/F) ≅ ℤ/3ℤ.

If G carries a trialitarian automorphism φ defined over F, then the map π: Aut(G)(F) → Gal(L/F) is a split surjection, hence Aut(G)(F) ≅ G(F) ⋊ (ℤ/3ℤ). Therefore, it is easy to see that for any other trialitarian automorphism φ′ of G defined over F, the elements φ and φ′ are conjugate in Aut(G)(F) if and only if there exists g ∈ G(F) such that φ′ = Int(g) ∘ φ ∘ Int(g)^{−1}. When this occurs, we have π(φ) = π(φ′).

Theorem 4.3. (i) Let G be an adjoint simple group of type ^{3}D_{4} over F. The group G carries a trialitarian automorphism defined over F if and only if the trialitarian algebra T = (E, L, σ, α) (unique up to isomorphism) such that G ≅ Aut_{L}(T) has the form T ≅ End Γ for some induced cyclic composition Γ.

(ii) Let G = Aut_{L}(End Γ) for some induced cyclic composition Γ. Every trialitarian automorphism φ of G has the form φ = Int(τ) for some uniquely determined F -automorphism τ of End Γ such that τ^{3} = Id and τ|_{L} = π(φ). For a given nontrivial ρ ∈ Gal(L/F), the assignment Σ ↦ Int(τ_{(Σ,f)}) defines a bijection between the isomorphism classes of symmetric compositions for which there exists an L-linear isotopy f: Σ ⊗ (L, ρ) → Γ and conjugacy classes in Aut(G)(F) of trialitarian automorphisms φ of G such that π(φ) = ρ.

Proof. Suppose first that φ is a trialitarian automorphism of G, and let G = Aut_{L}(T) for some trialitarian algebra T = (E, L, σ, α). Proposition 4.2 shows that the central simple L-algebra E is split, hence by [9, (44.16), (36.12)], we have T = End Γ for some cyclic composition Γ = (V, L, Q, ρ, ∗) over F. Substituting φ^{2} for φ if necessary, we may assume π(φ) = ρ. The preimage of φ under the isomorphism Φ_{F} : Aut_{F} (T)(F) →^{∼} Aut(G)(F) (from (9)) is an F -automorphism τ of T such that φ = Int(τ), τ^{3} = Id, and τ|_{L} = ρ. Since Φ_{F} is a bijection, τ is uniquely determined by φ. By Theorem 3.1(ii), the existence of τ implies that the cyclic composition Γ is induced. Conversely, if Γ is induced, then by Theorem 3.1(i), the trialitarian algebra End Γ carries automorphisms τ such that τ^{3} = Id and τ|_{L} ≠ Id_{L}. For any such τ, conjugation by τ is a trialitarian automorphism of G. The last statement in (ii) readily follows from Theorem 3.5 because trialitarian automorphisms Int(τ), Int(τ′) are conjugate in Aut(G)(F) if and only if τ, τ′ are conjugate in Aut_{F} (End Γ). □

The following proposition shows that the algebraic subgroup of fixed points under a trialitarian automorphism of the form Int(τ_{(Σ,f)}) is isomorphic to Aut(Σ), hence in characteristic different from 2 and 3 it is a simple adjoint group of type G_{2} or A_{2}, in view of the classification of symmetric compositions (see [3, §9]).

Proposition 4.4. Let G = Aut_{L}(End(Σ ⊗ (L, ρ))) for some symmetric composition Σ = (S, n, ⋆) over F and some cyclic cubic field extension L/F with nontrivial automorphism ρ. The subgroup of G fixed under the trialitarian automorphism Int(ρ̂) is canonically isomorphic to Aut(Σ).

Proof (Sketch). Mimicking the construction of the map α_{∗} in (4), we may use the product ⋆ to construct an F -algebra isomorphism

α_{⋆}: C(S, n) →^{∼} End_{F} (S ⊕ S)

such that α_{⋆}(x)(y, z) = (z ⋆ x, x ⋆ y) for x, y, z ∈ S. This isomorphism restricts to an isomorphism

α_{⋆0} : C_{0}(S, n) →^{∼} (End_{F} S) × (End_{F} S).

Let Aut(End Σ) be the group scheme whose rational points are the F -algebra automorphisms φ of (End_{F} S, σ_{n}) that make the following diagram commute:

\xymatrix{C(End_{F} S, σ_{n}) \ar[r]^{α_{⋆0}} \ar[d]_{C(φ)} & (End_{F} S) × (End_{F} S) \ar[d]^{φ×φ} \\ C(End_{F} S, σ_{n}) \ar[r]^{α_{⋆0}} & (End_{F} S) × (End_{F} S)}

Arguing as in Lemma 3.4, one proves that every such automorphism has the form Int(u) for some isotopy u of Σ. But if u is an isotopy of Σ with multiplier μ, then μ^{−1}u is an automorphism of Σ. Therefore, mapping every automorphism u of Σ to Int(u) yields an isomorphism Aut(Σ) →^{∼} Aut(End Σ). The extension of scalars from F to L yields an isomorphism

PGL(S) →^{∼} R_{L/F} (PGL(S ⊗_{F} L))^{Int(ρ̂)},

which carries the subgroup Aut(End Σ) to G^{Int(ρ̂)}. □

To conclude, we briefly mention without proof the analogue of Theorem 4.3 for simply connected groups, which we could have considered instead of adjoint groups. (Among simple algebraic groups of type D_{4}, only adjoint and simply connected groups may admit trialitarian automorphisms.)

Theorem 4.5. (i) For any cyclic composition Γ = (V, L, Q, ρ, ∗) over F, with L a field, the group Aut_{L}(Γ) is simple simply connected of type 3D_{4}, and there is an exact sequence of algebraic groups

\xymatrix{1 \ar[r] & μ_{2}^{2} \ar[r] & Aut_{L}(Γ) \ar[r]^{Int} & Aut_{L}(End Γ) \ar[r] & 1.}

(ii) A simple simply connected group of type ^{3}D_{4} admits trialitarian automorphisms defined over F if and only if it is isomorphic to the automorphism group of an induced symmetric composition Γ = (V, L, Q, ρ, ∗), with L a field. Conjugacy classes of trialitarian automorphisms of Aut_{L}(Γ) defined over F are in bijection with isomorphism classes of symmetric compositions Σ for which there is an isotopy Σ⊗(L, ρ) → Γ.

Theorems 4.3 and 4.5 apply in particular to show that over a finite field of characteristic different from 2 and 3, every simple adjoint or simply connected group of type ^{3}D_{4} admits trialitarian automorphisms. This follows because the Allen invariant is trivial and cyclic compositions are reduced, see [12, §4.8]. Note that the property holds without restriction on the characteristic (needed for the arguments in [12, §4.8]), and is a particular case of a more general result: every simple adjoint or simply connected linear algebraic group over a finite field is quasi-split by a theorem of Lang [10, Prop. 6.1], and therefore Aut(G) is a semidirect product, see [4, Exp. XXIV, 3.10] or [9, (31.4)].^{4}

Examples 4.6. (i) Let F = 𝔽_{q} be the field with q elements, where q is odd and q ≡ 1 mod 3. As observed in Example 2.2(i), every symmetric composition over F is isomorphic either to the Okubo composition Σ or to the split para-Cayley composition C̃, and (up to isomorphism) there is a unique cyclic composition Γ ≅ C̃ ⊗ (L, ρ) ≅ Σ ⊗ (L, ρ) with cubic algebra (L, ρ). Therefore, the simply connected group Aut_{L}(Γ) and the adjoint group Aut_{L}(End Γ) have exactly two conjugacy classes of trialitarian automorphisms defined over F. See also [8, (9.1)].

4We are indebted to Skip Garibaldi for this observation.

Documenta Mathematica · Extra Volume Merkurjev (2015) 387–405

(ii) Example 2.2(ii) describes a cyclic composition induced by a unique (up to isomorphism) symmetric composition. Its automorphism group is a group of type ^{3}D_{4} admitting a unique conjugacy class of trialitarian automorphisms.

(iii) In contrast to (i) and (ii) we get from Example 2.2(iii) examples of groups of type ^{3}D_{4} with many conjugacy classes of trialitarian automorphisms.

References

[1] Bruce N. Allison. Lie algebras of type D_{4} over number fields. Pacific J. Math., 156(2):209–250, 1992.

[2] Vladimir Chernousov, Alberto Elduque, Max-Albert Knus, and Jean-Pierre Tignol. Algebraic groups of type D_{4}, triality, and composition algebras. Documenta Math., 18:413–468, 2013.

[3] Vladimir Chernousov, Max-Albert Knus, and Jean-Pierre Tignol. Conjugate classes of trialitarian automorphisms and symmetric compositions. J. Ramanujan Math. Soc., 27:479–508, 2012.

[4] Michel Demazure and Alexander Grothendieck. Schémas en groupes (SGA 3). Tome III. Documents Mathématiques (Paris), 8, Soc. Math. France, Paris, 2011. Revised and annotated edition of the 1970 original, edited by Philippe Gille and Patrick Polo.

[5] Alberto Elduque and Hyo Chul Myung. On flexible composition algebras. Comm. Algebra, 21(7):2481–2505, 1993.

[6] Skip Garibaldi. Outer automorphisms of algebraic groups and determining groups by their maximal tori. Michigan Math. J., 61(2):227–237, 2012.

[7] Skip Garibaldi and Holger Petersson. Paper in preparation on outer automorphisms of algebraic groups.

[8] Daniel Gorenstein and Richard Lyons. The local structure of finite groups of characteristic 2 type. Mem. Amer. Math. Soc., 42(276):vii+731, 1983.

[9] Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol. The Book of Involutions. Number 44 in American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, R.I., 1998. With a preface in French by J. Tits.

[10] Vladimir Platonov and Andrei Rapinchuk. Algebraic groups and number theory. Pure and Applied Mathematics, 139, Academic Press, Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen.

[11] Tonny A. Springer. Oktaven, Jordan-Algebren und Ausnahmegruppen. Mathematisches Institut der Universität Göttingen, 1963. Vorlesungsausarbeitung von P. Eysenbach, 101 S.

[12] Tonny A. Springer and Ferdinand D. Veldkamp. Octonions, Jordan algebras and exceptional groups. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2000.

Max-Albert Knus Department Mathematik ETH Zentrum CH-8092 Zürich Switzerland knus@math.ethz.ch Jean-Pierre Tignol ICTEAM Institute Université catholique de Louvain B-1348 Louvain-la-Neuve Belgium jean-pierre.tignol@uclouvain.be 406

Documenta Mathematica · Extra Volume Merkurjev (2015)