Quadratic Quaternion Forms,

Involutions and Triality

Max-Albert Knus and Oliver Villa

Received: May 31, 2001

Communicated by Ulf Rehmann

Abstract. Quadratic quaternion forms, introduced by Seip-Hornix (1965), are special cases of generalized quadratic forms over algebras with involutions. We apply the formalism of these generalized quadratic forms to give a characteristic free version of different results related to hermitian forms over quaternions:

1) An exact sequence of Lewis

2) Involutions of central simple algebras of exponent 2.

3) Triality for 4-dimensional quadratic quaternion forms.

1991 Mathematics Subject Classification: 11E39, 11E88

Keywords and Phrases: Quadratic quaternion forms, Involutions, Triality

1. Introduction

Let F be a field of characteristic not 2 and let D be a quaternion division algebra over F. It is known that a skew-hermitian form over D determines a symmetric bilinear form over any separable quadratic subfield of D and that the unitary group of the skew-hermitian form is the subgroup of the orthogonal group of the symmetric bilinear form consisting of elements which commute with a certain semilinear mapping (see for example Dieudonné [3]). Quadratic forms behave nicer than symmetric bilinear forms in characteristic 2 and Seip-Hornix developed in [9] a complete, characteristic-free theory of quadratic quaternion forms, their orthogonal groups and their classical invariants. Her theory was subsequently (and partly independently) generalized to forms over algebras (even rings) with involution (see [11], [10], [1], [8]). Similitudes of hermitian (or skew-hermitian) forms induce involutions on the endomorphism algebra of the underlying space. To generalize the case where only similitudes of a quadratic form are considered, the notion of a quadratic pair was worked out in [6]. Relations between quadratic pairs and generalized quadratic forms were first discussed by Elomary [4].

The aim of this paper is to apply generalized quadratic forms to give a characteristic free presentation of some results on forms and involutions. After briefly recalling in Section 2 the notion of a generalized quadratic form (which, following the standard literature, we call an (ε, σ)-quadratic form) we give in Section 3 a characteristic-free version of an exact sequence of Lewis (see [7], [8, p. 389] and the appendix to [2]), which connects Witt groups of quadratic and quaternion algebras. The quadratic quaternion forms of Seip-Hornix are the main ingredient. Section 4 describes a canonical bijective correspondence between quadratic pairs and (ε, σ)-quadratic forms and Section 5 discusses the Clifford algebra. In particular we compare the definitions given in [10] and in [6]. In Section 6 we develop triality for 4-dimensional quadratic quaternion forms whose associated forms (over a separable quadratic subfield) are 3-Pfister forms. Any such quadratic quaternion form θ is an element in a triple (θ_{1}, θ_{2}, θ_{3}) of forms over 3 quaternions algebras D_{1}, D_{2} and D_{3} such that [D_{1}][D_{2}][D_{3}] = 1 in the Brauer group of F. Triality acts as permutations on such triples.

2. Generalized quadratic forms

Let D be a division algebra over a field F with an involution σ : x ↦ \overline{x}. Let V be a finite dimensional right vector space over D. An F -bilinear form

k : V × V → D

is sesquilinear if k(xa, yb) = \overline{a}k(x, y)b for all x, y ∈ V, a, b ∈ D. The additive group of such maps will be denoted by Sesq_{σ}(V, D). For any k ∈ Sesq_{σ}(V, D) we write

k^{∗}(x, y) = \overline{k(y, x)}.

Let ε ∈ F^{×} be such that ε\overline{ε} = 1. A sesquilinear form k such that k = εk^{∗} is called ε-hermitian and the set of such forms on V will be denoted by Herm_{σ}^{ε}(V, D). Elements of

Alt_{σ}^{ε}(V, D) = {g = f − εf^{∗} | f ∈ Sesq_{σ}(V, D)}.

are ε-alternating forms. We obviously have Alt_{σ}^{−ε}(V, D) ⊂ Herm_{σ}^{ε}(V, D). We set

Q_{σ}^{ε}(V, D) = Sesq_{σ}(V, D)/ Alt_{σ}^{ε}(V, D)

and refer to elements of Q_{σ}^{ε}(V, D) as (ε, σ)-quadratic forms. We recall that (ε, σ)-quadratic forms were introduced by Tits [10], see also Wall [11], Bak [1] or Scharlau [8, Chapter 7]. For any algebra A with involution τ, let Sym^{ε}(A, τ) = {a ∈ A | a = ετ(a)} and Alt^{ε}(A, τ) = {a ∈ A | a = c − ετ(c), c ∈ A}. To any class θ = [k] ∈ Q_{σ}^{ε}(V, D), represented by k ∈ Sesq_{σ}(V, D), we associate a quadratic map

q_{θ} : V → D/ Alt^{ε}(D, σ), q_{θ}(x) = [k(x, x)]

where [d] denotes the class of d in D/ Alt_{σ}^{ε}(D). The ε-hermitian form

b_{θ}(x, y) = k(x, y) + εk^{∗}(x, y) = k(x, y) + ε\overline{k(y, x)}

depends only on the class θ of k in Q_{σ}^{ε}(V, D). We say that b_{θ} is the polarization of q_{θ}.

Proposition 2.1. The pair (q_{θ}, b_{θ}) satisfies the following formal properties:

\begin{aligned}q_{θ}(x + y) & = & q_{θ}(x) + q_{θ}(y) + [b_{θ}(x, y)] \\ q_{θ}(xd) & = & \overline{d}q_{θ}(x)d \\ b_{θ}(x, x) & = & q_{θ}(x) + ε\overline{q_{θ}(x)} \end{aligned} (1)

for all x, y ∈ V, d ∈ D. Conversely, given any pair (q, b), q : V → D/ Alt^{ε}(D, σ), b ∈ Herm_{σ}^{ε}(V, D) satisfying (1), there exist a unique θ ∈ Q_{σ}^{ε}(V, D) such that q = q_{θ}, b = b_{θ}.

Proof. The formal properties are straightforward to verify. For the converse see [11, Theorem 1].

Example 2.2. Let D = F, σ = Id_{F} and ε = 1. Then sesquilinear forms are F -bilinear forms, Alt^{ε}(D, σ) = 0 and a (σ, ε)-quadratic form is a (classical) quadratic form. We denote the set of bilinear forms on V by Bil(V, F). Accordingly we speak of ε-symmetric bilinear forms instead of ε-hermitian forms.

Example 2.3. Let D be a division algebra with involution σ and let D be a finite dimensional (right) vector space over D. We use a basis of V to identify V with D^{n} and End_{D}(V) with the algebra M_{n}(D) of (n × n)-matrices with entries in D. For any (n×m)-matrix x = (x_{ij}), let x^{∗} = \overline{x}^{t}, where t is transpose and \overline{x} = (\overline{x}_{ij}). In particular the map a ↦ a^{∗} is an involution of A = M_{n}(D). If we write elements of D^{n} as column vectors x = (x_{1}, …, x_{n})^{t} any sesquilinear form k over D^{n} can be expressed as k(x, y) = x^{∗}ay, with a ∈ M_{n}(D), and k^{∗}(x, y) = x^{∗}a^{∗}y. We write Alt_{n}(D) = {a = b − εb^{∗}} ⊂ M_{n}(D), so that Q_{σ}^{ε}(V, D) = M_{n}(D)/ Alt_{n}(D).

Example 2.4. Let D be a quaternion division algebra, i.e. D is a central division algebra of dimension 4 over F. Let K be a maximal subfield of D which is a quadratic Galois extension of F and let σ : x ↦ \overline{x} be the nontrivial automorphism of K. Let j ∈ K \ F be an element of trace 1, so that K = F (j) with j^{2} = j + λ, λ ∈ F. Let ℓ ∈ D be such that ℓxℓ^{−1} = \overline{x} for x ∈ K, ℓ^{2} = μ ∈ F^{×}. The elements {1, j, ℓ, ℓj} form a basis of D and D = K ⊕ ℓK is also denoted [K, μ). The F -linear map σ : D → D, σ(d) = Trd_{D}(d) − d = \overline{d} is an involution of D (the “conjugation”) which extends the automorphism σ of K. The element N(d) = dσ(d) = σ(d)d is the reduced norm of d. We have Alt_{σ}^{−1}(D) = F and (σ, −1)-quadratic forms correspond to the quadratic quaternion forms introduced by Seip-Hornix in [9]. Accordingly we call (σ, −1)- quadratic forms quadratic quaternion forms.

The restriction of the involution τ to the center Z of A is either the identity (involutions of the first kind) or an automorphism of order 2 (involutions of the second kind). If the characteristic of F is different from 2 or if the involution is of second kind there exists an element j ∈ Z such that j + σ(j) = 1. Under such conditions the theory of (σ, ε)-quadratic forms reduces to the theory of ε-hermitian forms:

Proposition 2.5. If the center of D contains an element j such that j+σ(j) = 1, then Herm_{σ}^{−ε}(V, D) = Alt_{σ}^{ε}(V, D) and a (σ, ε)-quadratic form is uniquely determined by its polar form b_{θ}.

Proof. If k = −εk^{∗} ∈ Herm_{σ}^{−ε}(V, D), then k = 1k = jk + \overline{j}k = jk − \overline{j}εk^{∗} ∈ Alt_{σ}^{ε}(V, D). The last claim follows from the fact that polarization induces an isomorphism Sesq_{σ}(V, D)/ Herm_{σ}^{−ε}(V, D) ⟶^{∼} Q_{σ}^{ε}(V, D).

For any left (right) D-space V we denote by ^{σ}V the space V viewed as right (left) D-space through the involution σ. If ^{σ}x is the element x viewed as an element of ^{σ}V, we have ^{σ}xd =^{σ} (σ(d)x). Let V^{∗} be the dual ^{σ}Hom_{D}(V, D) as a right D-module, i.e., (^{σ}fd)(x) = ^{σ}(\overline{d}f)(x), x ∈ V, d ∈ D. Any sesquilinear form k ∈ Sesq_{σ}(V, D) induces a D-module homomorphism k̂ : V → V^{∗}, x ↦ k(x, −). Conversely any homomorphism g : V → V^{∗} induces a sesquilinear form k ∈ Sesq_{σ}(V, D), k(x, y) = g(x)(y) and the additive groups Sesq_{σ}(V, D) and Hom_{D}(V, V^{∗}) can be identified through the map h ↦ k̂. For any f : V → V′, let f^{∗} : V′^{∗} → V^{∗} be the transpose, viewed as a homomorphisms of right vector spaces. We identify V with V^{∗∗} through the map v ↦ v^{∗∗}, v^{∗∗}(f) = \overline{f(v)}. Then, for any f ∈ Hom_{D}(V, V^{∗}), f^{∗} is again in Hom_{D}(V, V^{∗}) and k̂^{∗} = k̂^{∗}. A (σ, ε)-quadratic form q_{θ} is called nonsingular if its polar form b_{θ} induces an isomorphism b̂_{θ}. A pair (V, q_{θ}) with q_{θ} nonsingular is called a (σ, ε)- quadratic space. For any vector space W, the hyperbolic space V = W ⊕ W^{∗} equipped with the quadratic form q_{θ}, θ = [k] with

k((p, q), (p′, q′)) = q(p′),

is nonsingular. There is an obvious notion of orthogonal sum V ⊥ V′ and a quadratic space decomposes whenever its polarization does. Most of the classical theory of quadratic spaces extends to (σ, ε)-quadratic spaces. For example Witt cancellation holds and any (σ, ε)-quadratic space decomposes uniquely (up to isomorphism) as the orthogonal sum of its anisotropic part with a hyperbolic space. Moreover, if we exclude the case σ = 1 and ε = −1, any (σ, ε)-quadratic space has an orthogonal basis. A similitude of (σ, ε)-quadratic spaces t : (V, q) ⟶^{∼} (V′, q′) is a D-linear isomorphism V ⟶^{∼} V′ such that q′(tx) = μ(t)q(x) for some μ(t) ∈ F^{×}. The element μ(t) is called the multiplier of the similitude. Similitudes with multipliers equal to 1 are isometries. As in the classical case there is a notion of Witt equivalence and corresponding Witt groups are denoted by W^{ε}(D, σ).

3. An exact sequence of Lewis

Let D be a quaternion division algebra. We fix a representation D = [K, μ) = K ⊕ ℓK, with ℓ^{2} = μ, as in (2.4). Let V be a vector space over D. Any sesquilinear form k : V × V → D can be decomposed as

k(x, y) = P (x, y) + ℓR(x, y)

with P : V × V → K and R : V × V → K. The following properties of P and R are straightforward.

Lemma 3.1. 1) P ∈ Sesq_{σ}(V, K), R ∈ Sesq_{1}(V, K) = Bil(V, K).

2) k^{∗} = P^{∗} − ℓR^{t}, where P^{∗}(x, y) = \overline{P (y, x)} and R^{t}(x, y) = R(y, x).

The sesquilinearity of k implies the following identities:

\begin{aligned}R(xℓ, y) & = & −P (x, y), & R(x, yℓ) & = & \overline{P (x, y)} \\ P (xℓ, y) & = & −μR(x, y), & P (x, yℓ) & = & μ\overline{R(x, y)} \\ P (xℓ, yℓ) & = & −μ\overline{P (x, y)}, & R(xℓ, yℓ) & = & −μ\overline{R(x, y)} \end{aligned} (2)

Let V^{0} be V considered as a (right) vector space over K (by restriction of scalars) and let T : V^{0} → V^{0}, x ↦ xℓ. The map T is a K-semilinear automorphism of V^{0} such that T^{2} = μ. Conversely, given a vector space U over K, together with a semilinear automorphism T such that T^{2} = μ ∈ F^{×}, we define the structure of a right D-module on U, D = [K, μ), by putting xℓ = T (x).

Lemma 3.2. Let V be a vector space over D. 1) Let f_{1} : V^{0} × V^{0} → K be a sesquilinear form over K. The form

f(x, y) = f_{1}(x, y) − ℓμ^{−1}f_{1}(T x, y)

is sesquilinear over D if and only if f_{1}(T x, T y) = −μ\overline{f_{1}(x, y)}.

2) Let f_{2} : V^{0} × V^{0} → K be a bilinear form over K. The form

f(x, y) = −f_{2}(T x, y) + ℓf_{2}(x, y)

is sesquilinear over D if and only if f_{2}(T x, T y) = −μ\overline{f_{2}(x, y)}.

Proof. The two claims follow from the identities (2).

Let f be a bilinear form on a space U over K and let λ ∈ K^{×}. A semilinear automorphism t of U such that f(tx, ty) = λ\overline{f(x, y)} for all x ∈ U is a semilinear similitude of (U, f), with multiplier λ. In particular T x = xℓ is a semilinear similitude of R on V^{0}, such that T^{2} = μ and with multiplier −μ. The following nice observation of Seip-Hornix [9, p. 328] will be used later:

Proposition 3.3. Let R be a K-bilinear form over U and let T be a semilinear similitude of U with multiplier λ ∈ K^{×} and such that T^{2} = μ. Then:

1) μ ∈ F,

2) For any ξ ∈ K and x ∈ U, let ρ_{ξ}(x) = xξ. There exists ν ∈ K^{×} such that T′ = ρ_{ν} ∘ T satisfies T′^{2} = μ′ and R(T′x, T′y) = −μ′\overline{R(x, y)}.

Proof. The first claim follows from μ = λ\overline{λ}. For the second we may assume that λ ≠ μ (if λ = μ replace T by T ∘ ρ_{k} for an appropriate k). For ν = (1 − μλ^{−1}) we have μ′ = 2μ − λ − \overline{λ}.

Assume that k ∈ Sesq_{σ}(V, D) defines a (σ, ε)-quadratic space [k] on V over D. It follows from (3.1) that P defines a (σ, ε)-quadratic space [P] on V^{0} over K and R a (Id, −ε)-quadratic space [R] on V^{0} over K. Let K = F (j) with j^{2} = j + λ. Let r(x, y) = R(x, y) − εR(y, x) be the polar of R.

Proposition 3.4. 1) q_{[P]}(x) = \overline{ε}j[r(x, T x)]

2) q_{[k]}(x) = \overline{ε}j[r(x, T x)] + ℓq_{[R]}(x)

3) The map T is a semilinear similitude of (q_{[R]}, V^{0}) with multiplier −μ.

Proof. It follows from the relations (2) that

\overline{P (x, x)} + εP (x, x) = R(x, T x) − εR(T x, x) = r(x, T x) (3)

and obviously this relation determines P (x, x) up to a function with values in Sym^{−ε}(K, σ). Since Sym^{−ε}(K, σ) = Alt^{+ε}(K, σ) by (2.5), [P] is determined by (3). Since \overline{r(x, T x)} = \overline{ε}r(x, T x) by (2), we have \overline{\overline{ε}jr(x, T x)} + ε(\overline{ε}jr(x, T x)) = r(x, T x) and 1) follows. The second claim follows from 1) and 3) is again a consequence of the identities (2).

Corollary 3.5. Any pair ([R], T ) with [R] ∈ Q_{1}^{ε}(U, K) and T a semilinear similitude with multiplier −μ ∈ F^{×} and such that T^{2} = μ, determines the structure of a (σ, ε)-quadratic space on U over D = [K, μ).

Proposition 3.6. The assignments h ↦ P and h ↦ R induce homomorphisms of groups π_{1} : W^{ε}(D, −) → W^{ε}(K, −) and π_{2} : W^{−ε}(D, −) → W^{ε}(K, Id).

Proof. The assignments are obviously compatible with orthogonal sums and Witt equivalence.

We recall that W^{ε}(K, −) can be identified with the corresponding Witt group of ε-hermitian forms (apply (2.5)). However, it is more convenient for the following computations to view ε-hermitian forms over K as (σ, ε)- quadratic forms. Let i ∈ K^{×} be such that σ(i) = −i (take i = 1 if Char F = 2). The map k ↦ ik induces an isomorphism s : W^{ε}(K, −) ⟶^{∼} W^{−ε}(K, −) (“scaling”). For any space U over K, let U_{D} = U ⊗_{K} D. We identify U_{D} with U ⊕ Uℓ through the map u ⊗ (x + ℓy) ↦ (ux, u\overline{y}l) and get a natural D-module structure on U_{D} = U ⊕ Uℓ. Any K-sesquilinear form k on U extends to a D-sesquilinear form k_{D} on U_{D} through the formula

k_{D}(x ⊗ a, y ⊗ b) = \overline{a}k(x, y)b

for x, y ∈ U and a, b ∈ D.

Lemma 3.7. The assignment k ↦ (ik)_{D} induces a homomorphism

β : W^{ε}(K, −) → W^{−ε}(D, −)

Proof. Let k̃ = (ik)_{D}. We have (k̃)^{∗} = −k̃^{∗}.

Theorem 3.8 (Lewis). With the notations above, the sequence

W^{ε}(D, −) ⟶^{π_{1}} W^{ε}(K, −) ⟶^{β} W^{−ε}(D, −) ⟶^{π_{2}} W^{ε}(K, Id)

is exact.

Proof. This is essentially the proof given in Appendix 2 of [2] with some changes due to the use of generalized quadratic forms, instead of hermitian forms. We first check that the sequence is a complex. Let [k] ∈ Q_{σ}^{ε}(V, D) and let V^{0} = U. We write elements of U_{D} = U ⊕Uℓ as pairs (x, yℓ) and decompose k_{D} = P +ℓR. By definition we have βπ_{1}([k]) = [β(P)] and

\begin{aligned}β(P)((x_{1}, y_{1}), (x_{2}, y_{2})) & = & i(P (x_{1}, x_{2}) + P (x_{1}, y_{2})ℓ + ℓP (y_{1}, x_{2}) \\ & & +ℓP (y_{1}, y_{2})ℓ). \end{aligned}

Let (xℓ, xℓ) ∈ U ⊕ Uℓ. We get β(P)((xℓ, xℓ), (xℓ, xℓ)) = 0 hence W = {(xℓ, xℓ)} ⊂ U ⊕ Uℓ is totally isotropic. It is easy to see that W ⊂ W^{⊥}, so that [β(P)] is hyperbolic and β ∘ π_{1} = 0. Let [g] ∈ Q_{σ}^{ε}(U, K). The subspace W = {(x, 0) ∈ U ⊕ Uℓ} is totally isotropic for π_{2}β([g]) and W ⊂ W^{⊥}. Hence π_{2}β([g]) = 0. We now prove exactness at W^{ε}(K, −). Since the claim is known if Char ≠ 2, we may assume that Char = 2 and ε = 1. Let [g] ∈ Q_{σ}^{ε}(U, K) be anisotropic such that β([g]) = 0 ∈ W^{−ε}(D, −). In particular β([g]) ∈ Q_{σ}^{−ε}(U_{D}, D) is isotropic. Hence the exist elements x_{1}, x_{2} ∈ U such that [g̃]((x_{1}, x_{2}ℓ), (x_{1}, x_{2}ℓ)) = 0. This implies (in Char 2) that

g(x_{1}, x_{1}) + μ\overline{g(x_{2}, x_{2})} ∈ F, g(x_{1}, x_{2})ℓ + ℓg(x_{2}, x_{1}) = 0. (4)

Let V_{1} be the K-subspace of V generated by x_{1} and x_{2}. Since [g] is anisotropic,

[g] = [g_{1}] ⊥ [g_{2}] with g_{1} = g|_{V_{1}}. We make V_{1} into a D-space by putting

(x_{1}a_{1} + x_{2}a_{2})ℓ = μx_{2}\overline{a}_{1} + x_{1}\overline{a}_{2}

To see that the action is well-defined, it suffices to show that dim_{K} V_{1} = 2. The elements x_{1} and x_{2} cannot be zero since [g] is anisotropic, so assume x_{2} = x_{1}c, c ∈ K^{×}. Then (4) implies g(x_{1}, x_{1}) + μc\overline{c}g(x_{1}, x_{1}) ∈ F, which contradicts the fact that g is anisotropic. Let g_{1}(x_{1}, x_{1}) + μ\overline{g_{1}(x_{2}, x_{2})} = z ∈ F. Let f ∈ Sesq_{σ}(V_{1}, K). Replacing g_{1} by g_{1} + f + f^{∗} defines the same class in Q_{σ}^{ε}(V_{1}, K) (recall that Char F = 2). Choosing f as

f(x_{1}, x_{1}) = jz, f(x_{2}, x_{2}) = 0, f(x_{1}, x_{2}) = f(x_{2}, x_{1}) = 0,

we may assume that

g_{1}(x_{1}, x_{1}) + μ\overline{g_{1}(x_{2}, x_{2})} = 0, g_{1}(x_{1}, x_{2})ℓ + ℓg_{1}(x_{2}, x_{1}) = 0. (5)

By (3.2) we may extend g_{1} to a sesquilinear form

g′(x, y) = g_{1}(x, y) + ℓμ^{−1}g_{1}(xℓ, y)

over D if g_{1} satisfies

g_{1}(xℓ, yℓ) = −μ\overline{g_{1}(x, y)}

This can easily be checked using (5) (and the definition of xℓ). Then g_{1} is in the image of π_{1}. Exactness at W^{ε}(K, −) now follows by induction on the dimension of U. We finally check exactness at W^{−ε}(D, −). Let [k] be anisotropic such that π_{2}([k]) = 0 in W^{−ε}(K, Id). In particular π_{2}([k]) is isotropic; let x ≠ 0 be such that π_{2}k(x, x) = 0 and let W be the D-subspace of V generated by x. Since [k] is anisotropic, [k′] = [k|_{W}] is nonsingular and [k] = [k′] ⊥ [k′′]. The condition π_{2}k(x, x) = 0 implies k(x, x) ∈ K. Let W_{1} be the K-subspace of W generated by x. Define g : W_{1} × W_{1} → K by g(xa, xb) = k(xa, xb)i^{−1} for a, b ∈ K. Then clearly [g] defines an element of W^{ε}(K, −) and β(g) = k′. Once again exactness follows by induction on the dimension of V.

4. Involutions on central simple algebras

Let D be a central division algebra over F, with involution σ and let b : V ×V → D be a nonsingular ε-hermitian form on a finite dimensional space over D. Let A = End_{D}(V). The map σ_{b} : A → A such that σ_{b}(λ) = σ(λ) for all λ ∈ F and

b(σ_{b}(f)(x), y) = b(x, f(y))

for all x, y ∈ V, is an involution of A, called the involution adjoint to b. We have σ_{b}(f) = b̂^{−1}f^{∗}b̂, where b̂ : V ⟶^{∼} V^{∗} is the adjoint of b. Conversely, any involution of A is adjoint to some nonsingular ε-hermitian form b and b is uniquely multiplicatively determined up to a σ-invariant element of F^{×}. Any automorphism ϕ of A compatible with σ_{b}, i.e., σ_{b}(ϕ(a))= ϕ(σ_{b}(a)), is of the form ϕ(a) = uau^{−1} with u : V ⟶^{∼} V a similitude of b. We say that an involution τ of A is a q-involution if τ is adjoint to the polar b_{θ} of a (σ, ε)-quadratic form θ. We write τ = σ_{θ}. Two algebras with q-involutions are isomorphic if the isomorphism is induced by a similitude of the corresponding quadratic forms. Over fields q-involutions differ from involutions only in characteristic 2 and for symplectic involutions. In view of possible generalizations (for example rings in which 2 ≠ 0 is not invertible) we keep to the general setting of (σ, ε)-quadratic forms. Let F_{0} be the subfield of F of σ-invariant elements and let T_{F/F_{0}} be the corresponding trace.

Lemma 4.1. The symmetric bilinear form on A given by Tr(x, y) = T_{F/F_{0}}(Trd_{A}(xy)) is nonsingular and Sym(A, τ)^{⊥} = Alt(A, τ).

Proof. If τ is of the first kind F_{0} = F and the claim is (2.3) of [6]. Assume that τ is of the second kind. Since the bilinear form (x, y) → Trd_{A}(xy) is nonsingular, Tr is also nonsingular and it is straightforward that Alt(A, τ) ⊂ Sym(A, τ)^{⊥}. Equality follows from the fact that dim_{F_{0}} Alt(A, τ) = dim_{F_{0}} Sym(A, τ) = dim_{F} A.

Proposition 4.2. Let (V, θ), θ = [k] be a (σ, ε)-quadratic space over D and let h = k̂ + εk̂^{∗} : V ⟶^{∼} V^{∗}. The F_{0}-linear form

f_{θ} : Sym(A, σ_{θ}) → F_{0}, f_{θ}(s) = Tr(h^{−1}k̂s), s ∈ Sym(A, σ_{θ})

depends only on the class θ and satisfies f_{θ}(x + σ_{θ}(x)) = Tr(x).

Proof. The first claim follows from (4.1) and the fact that if k ∈ Alt_{σ}^{ε}(V, D) then h^{−1}k̂ ∈ Alt_{σ_{θ}}^{1}(V, D). For the last claim we have:

f_{θ}(x + σ_{θ}(x)) = Tr(h^{−1}k̂(x + σ_{θ}(x))
= Tr(h^{−1}k̂x) + Tr(h^{−1}k̂h^{−1}x^{∗}h)
= Tr(h^{−1}k̂x) + Tr(k̂h^{−1}x^{∗})
= Tr(h^{−1}k̂x) + Tr(x(h^{−1})^{∗}k̂^{∗})
= Tr(h^{−1}k̂x) + Tr(h^{−1}εk̂^{∗}x) = Tr(x).

Lemma 4.3. Let τ be an involution of A = End_{D}(V) and let f be A F_{0}-linear form on Sym(A, τ) such that f(x + τ(x)) = Tr(x) for all x ∈ A. There exists an element u ∈ A such that f(s) = Tr(us) and u + τ(u) = 1. The element u is uniquely determined up to additivity by an element of Alt(A, τ). We take u = 1/2 if Char F ≠ 2.

Proof. The proof of (5.7) of [6] can easily be adapted.

Proposition 4.4. Let τ be an involution of A = End_{D}(V) and let f be A F_{0}-linear form on Sym(A, τ) such that f(x + τ(x)) = Tr(x) for all x ∈ A.

1) There exists a nonsingular (σ, ε)-quadratic form θ on V such that τ = σ_{θ} and f = f_{θ}.

2) (σ_{θ}, f_{θ}) = (σ_{θ′}, f_{θ′}) if and only if θ′ = λθ for λ ∈ F_{0}.

3) If τ = σ_{θ} and f = f_{θ} with f_{θ}(s) = Tr(us), the class of u in A/ Alt(A, σ_{θ}) is uniquely determined by θ.

Proof. Here the proof of (5.8) of [6] can adapted. We prove 1) for completeness. Let τ(x) = h^{−1}x^{∗}h, h = εh^{∗} : V ⟶^{∼} V^{∗}. Let f(s) = Tr(us) with u + τ(u) = 1 and let k ∈ Sesq_{σ}(V, D) be such that k̂ = hu : V → V^{∗}. We set θ = [k]. It is then straightforward to check that h = k + εk^{∗}.

Proposition 4.5. Let ϕ :( End_{D}(V), σ_{θ}) ⟶^{∼}( End_{D}(V′), σ_{θ′}) be an isomorphism of algebras with involution. Let f_{θ}(s) = Tr(us) and f_{θ′}(s′) = Tr(u′s′). The following conditions are equivalent:

1) ϕ is an isomorphism of algebras with q-involutions.

2) f_{θ′}(ϕ(s)) = f_{θ}(s) for all s ∈ Sym(End_{D}(V), σ_{θ}).

3) [ϕ(u)] = [u′] ∈ End_{D}(V′)/ Alt(End_{D}(V′), σ_{θ′}).

Proof. The implication 1) ⇒ 2) is clear. We check that 2) ⇒ 3). Let ϕ be induced by a similitude t : (V, b_{θ}) ⟶^{∼} (V′, b_{θ′}). Since f_{θ′}(ϕs) = f_{θ}(s), we have Tr(t^{−1}u′ts) = Tr(u′tst^{−1}) = Tr(us) for all s ∈ Sym( End_{D}(V), σ_{θ}), hence [ϕ(u)] = [u′]. The implication 3) ⇒ 1) follows from the fact that u can be chosen as h^{−1}k̂, h = k̂ + εk̂^{∗}.

Remark 4.6. We call the pair (σ_{θ}, f_{θ}) a (σ, ε)-quadratic pair or simply a quadratic pair. It determines θ up to the multiplication by a σ-invariant scalar λ ∈ F^{×}. In fact σ_{θ} determines the polar b_{θ} up to λ and f_{θ} determines u. We have θ = [b̂_{θ}u].

Example 4.7. Let q : V → F be a nonsingular quadratic form. The polar b_{q} induces an isomorphism ψ : V ⊗_{F} V ⟶^{∼} End_{F} (V) such that σ_{q}(ψ(x ⊗ y)) = ψ(y ⊗ x). Thus ψ(x ⊗ x) is symmetric and f_{q}(ψ(x ⊗ x)) = q(x) (see [6, (5.11)]. More generally, if V is a right vector space over D, we denote by ^{∗}V the space V viewed as a left D-space through the involution σ of D. The adjoint b̂_{θ} of a (σ, ε)-quadratic space (V, θ) induces an isomorphism ψ_{θ} : V ⊗_{D} ^{σ}V ⟶^{∼} End_{D}(V) and ψ_{θ}(xd ⊗ x) is a symmetric element of (End_{D}(V), σ_{θ}) for all x ∈ V and all ε-symmetric d ∈ D. One has f_{θ}(ψ(xd ⊗ x)) = [dk(x, x)], where θ = [k] (see [4, Theorem 7]).

5. Clifford algebras

Let σ be an involution of the first kind on D and let θ be a nonsingular (σ, ε)-quadratic form on V. Let σ_{θ} be the corresponding q-involution on A = End_{D}(V). We assume in this section that over a splitting A⊗_{F} F̃ ⟶^{∼} EndF̃(M) of A, θF̃ = θ ⊗ 1F̃ is a (Id, 1)-quadratic form q̃ over F̃, i.e. θF̃ is a (classical) quadratic form. In the terminology of [6] this means that σ_{θ} is orthogonal if Char ≠ 2 and symplectic if Char = 2. From now on we call such forms over D quadratic forms over D, resp. quadratic spaces over D if the forms are non-singular. Classical invariants of quadratic spaces (V, θ) are the dimension dim_{D} V and the discriminant disc(θ) and the Clifford invariant associated with the Clifford algebra. We refer to [6, §7] for the definition of the discriminant. We recall the definition of the Clifford algebra Cl(V, θ), following [10, 4.1]. Given (V, θ) as above, let θ = [k], k ∈ Sesq_{σ}(V, D), b_{θ} = k + εk^{∗} and h = b̂_{θ} ∈ Hom_{D}(V, V^{∗}). Let A = End_{D}(V), B = Sesq_{σ}(V, D) and B′ = V ⊗_{D} ^{σ}V. We identify A with V ⊗_{D} ^{σ}V^{∗} through the canonical isomorphism (x ⊗ ^{σ}f)(v) = xf(v) and B with V^{∗} ⊗_{D} ^{σ}V^{∗} through (f ⊗ ^{σ}g)(x, y) = \overline{g(x)}f(y). The isomorphism h can be used to define further isomorphisms:

φ_{θ} : B′ = V ⊗_{D} ^{σ}V ⟶^{∼} A = End_{D}(M), φ_{θ} : x ⊗ y ↦ x ⊗ h(y)

and the isomorphism ψ_{θ} already considered in (4.7):

ψ_{θ} : A ⟶^{∼} B, ψ_{θ} : x ⊗ ^{σ}f ↦ h(x) ⊗ ^{σ}f.

We use φ_{θ} and ψ_{θ} to define maps B′ × B → A, (b′, b) ↦ b′b and A × B′ → B′, (a, b′) ↦ ab′:

(x ⊗ ^{σ}y)(h(u) ⊗ g) = xb(y, u) ⊗ ^{σ}f and (x⊗,^{σ}f)(u⊗,^{σ}v) = xf(u) ⊗ ^{σ}h(v)

Furthermore, let τ_{θ} = φ_{θ}^{−1}σ_{θ}φ_{θ} : B′ → B′ be the transport of the involution σ_{θ} on A. We have τ_{θ}(x ⊗ ^{σ}y) = εy ⊗ ^{σ}x. Let S_{1} = {s_{1} ∈ B′ | τ_{θ}(s_{1}) = s_{1}}. We have S_{1} = (Alt^{ε}(V, D))^{⊥} for the pairing B′ × B → F, (b′, b) ↦ Trd_{A}(b′b). Let Sand be the bilinear map B′ ⊗B′ ×B → B′ defined by Sand(b′_{1} ⊗b′_{2}, b) = b′_{2}bb′_{1}. The Clifford algebra Cl(V, θ) of the quadratic space (V, θ) is the quotient of the tensor algebra of the F -module B′ by the ideal I generated by the sets

I_{1} = {s_{1} − Trd_{A}(s_{1}k)1, s_{1} ∈ S_{1}}
I_{2} = {c − Sand(c, k) | Sand(c, Alt^{ε}(V, D)) = 0}.

The Clifford algebra Cl(V, θ) has a canonical involution σ_{0} induced by the map τ. We have Cl(V, θ) ⊗_{F} F̃ = Cl(V ⊗_{F} F̃, θ ⊗ 1F̃) for any field extension F̃ of F and Cl(V, q) is the even Clifford algebra C_{0}(V, q) of (V, q) if D = F ([10, Théorème 2]). The reduction is through Morita theory for hermitian spaces (see for example [5, Chapter I, §9] for a description of Morita theory). In [6, §8] the Clifford algebra C(A, σ_{θ}, f_{θ}) of the triple (A, σ_{θ}, f_{θ}) is defined as the quotient of the tensor algebra T (A) of the F -space A by the ideal generated by the sets

J_{1} = {s − Trd_{A}(us), s ∈ Sym(A, σ_{θ})}
J_{2} = {c − Sand′(c, u), c ∈ A with Sand′(c, Alt(A, σ_{θ})) = 0}

where u = b̂_{θ}^{−1}k and Sand′ : (A⊗A, A) → A is defined as Sand′(a⊗b, x) = axb. The two definitions give in fact isomorphic algebras:

Proposition 5.1. The isomorphism φ_{θ} : V ⊗_{D} ^{σ}V ⟶^{∼} End_{D}(V) induces an isomorphism Cl(V, θ) ⟶^{∼} C(A, σ_{θ}, f_{θ}).

Proof. We only check that φ_{θ} maps I_{1} to J_{1}. By definition of τ and S_{1}, s = φ_{θ}(s_{1}) is a symmetric element of A. On the other hand we have by definition of the pairing B′ × B → A,

Trd_{A}(s_{1}k) = Trd_{A}(φ_{θ}(s_{1})ψ_{θ}^{−1}(k))
= Trd_{A}(sh^{−1}k̂) = Trd_{A}(su) = Trd_{A}(us),

hence the claim.

In particular we have C(End_{F} (V), σ_{q}, f_{q}) = C_{0}(V, q) for a quadratic space (V, q) over F. It is convenient to use both definitions of the Clifford algebra of a generalized quadratic space.

Let D = [K, μ) = K⊕ℓK be a quaternion algebra with conjugation σ. Let V be a D-module and let V^{0} be V as a right vector space over K (through restriction of scalars). Let T : V^{0} → V^{0}, T x = xℓ. We have End_{D}(V) ⊂ End_{K}(V^{0}) and

End_{D}(V) = {f ∈ End_{K}(V^{0}) | fT = T f}.

Let θ = [k] be a (σ, −1)-quadratic space and let k(x, y) = P (x, y) + ℓR(x, y) as in Section 3. It follows from (3.1) that R defines a quadratic space [R] on V^{0} over K.

Proposition 5.2. We have σ_{[R]}|_{End_{D}(V)} = σ_{θ} and f_{θ} = f_{[R]}|_{End_{D}(V)}.

Proof. We have an embedding D ↪ M_{2}(K), a + ℓb ↦ (\begin{matrix}a & μ\overline{b} \\ b & \overline{a} \end{matrix}) and conjuga-

tion given by x ↦ x^{∗} = c^{−1}x^{t}c, c = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}). The choice of a basis of V over

D identifies V with D^{n}, V^{0} with K^{2n}, End_{D}(V) with M_{n}(D) and End_{K}(V^{0}) with M_{2n}(K), where n = dim_{D} V. We further identify V and V^{∗} through the choice of the dual basis. We embed any element x = x_{1} + ℓx_{2} ∈ M_{k,l}(D),

x_{i} ∈ M_{k,l}(K) in M_{2k,2l}(K) through the map ι : x ↦ ξ = (\begin{matrix}x_{1} & μ\overline{x_{2}} \\ x_{2} & \overline{x_{1}} \end{matrix}). In

particular D^{n} is identified with a subspace of the space of (2n × 2)-matrices over K. Then D ⊂ M_{2}(K) operates on the right through (2 × 2)-matrices and M_{n}(D) ⊂ M_{2n}(K) operates on the left through (2n × 2n)-matrices. With the notations of Example (2.3) we have ι(x^{∗}) = Int(c^{−1})(x^{t}). Any D-sesquilinear form k on D^{n} can be written as k(x, y) = x^{∗}ay, where a ∈ M_{n}(D), as in (2.3). Let a = a_{1} + ℓa_{2}, a_{i} ∈ M_{n}(K) and let

α = ι(a) = (\begin{matrix}a_{1} & μ\overline{a_{2}} \\ a_{2} & \overline{a_{1}} \end{matrix}) .

Let η = ι(y), y = y_{1} + ℓy_{2}. We have

k(x, y) = x^{∗}ay = ξ^{∗}αη = (\begin{matrix}x_{1} & μ\overline{x_{2}} \\ x_{2} & \overline{x_{1}} \end{matrix})^{∗} (\begin{matrix}a_{1} & μ\overline{a_{2}} \\ a_{2} & \overline{a_{1}} \end{matrix}) (\begin{matrix}y_{1} & μ\overline{y_{2}} \\ y_{2} & \overline{y_{1}} \end{matrix}) .

On the other side it follows from h = P + ℓR that R(x, y) = ξ^{t}ρη with

ρ = (\begin{matrix}a_{2} & \overline{a_{1}} \\ −a_{1} & −μ\overline{a_{2}} \end{matrix}) .

Assume that θ = [k] , so that σ_{θ} corresponds to the involution Int(γ^{−1}) ∘ ∗, where γ = α − α^{∗}. Similarly σ_{[R]} corresponds to the involution Int(ρ^{−1}) ∘ t

where ρ̃ = ρ + ρ^{t}. We obviously have ρ = cα with c = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}), so that

ρ^{t} = α^{t}c^{t} = −α^{t}c = −ca^{∗} and ρ+ρ^{t} = c(α−α^{∗}) or cγ = ρ̃. Now^{∗} = Int(c^{−1})∘t implies σ_{[R]}|_{M_{n}(D)} = σ_{θ}. We finally check that f_{θ} = f_{[R]}|_{Sym(M_{n}(D),σ_{θ})}. We have f_{θ}(s) = Trd_{M_{n}(D)}(γ^{−1}αs) and f_{[R]}(s) = Trd_{M_{2n}(K)}(ρ^{−1}ρs), hence the claim, since ρ = cα and ρ̃ = cγ implies γ^{−1}α = ρ^{−1}ρ.

Corollary 5.3. The embedding End_{D}(V) ↪ End_{K}(V^{0}) induces

1) an isomorphism ( End_{D}(V), σ_{θ}, f_{θ}) ⊗ K ⟶^{∼}( End_{K}(V^{0}), σ_{[R]}, f_{[R]}),

2) an isomorphism C( End_{D}(V), σ_{θ}, f_{θ}) ⊗ K ⟶^{∼} C_{0}(V^{0}, [R]).

In view of (2) the semilinear automorphism T : V^{0} ⟶^{∼} V^{0}, T x = xℓ, is a semilinear similitude with multiplier −μ of the quadratic form [R], such that T^{2} = μ.

Lemma 5.4. The map T induces a semilinear automorphism C_{0}(T) of C_{0}(V^{0}, R) such that

C_{0}(T)(xy) = (−μ)^{−1}T (x)T (y) for x, y ∈ V^{0}

and C_{0}(T)^{2} = Id.

Proof. This follows (for example) as in [6, (13.1)]

Proposition 5.5.

C(End_{D}(V), σ_{θ}, f_{θ}) = {c ∈ C_{0}(V^{0}, R) | C_{0}(T)(c) = c}.

Proof. The claim follows from the defining relations of C(End_{D}(V), σ_{θ}, f_{θ}) and the fact that

End_{D}(V) = {f ∈ End_{K}(V^{0}) | T^{−1}fT = f}.

We call C(End_{D}(V), σ_{θ}, f_{θ}) or equivalently Cl(V, θ) the Clifford algebra of the quadratic quaternion space (V, θ).

Let t be a semilinear similitude of a quadratic space (U, q) of even dimension over K. Assume that disc(q) is trivial, so that C_{0}(U, q) decomposes as product of two K-algebras C^{+}(U, q) and C^{−}(U, q). We say that t is proper if C_{0}(t)(C^{±}(U, q)) ⊂ C^{±}(U, q) and we say that t is improper if C_{0}(t)(C^{±}(U, q)) ⊂ C^{∓}(U, q). In general we say that t is proper if t is proper over some field extension of F which trivializes disc(q). For any semilinear similitude t, let d(t) = 1 is t if proper and d(t) = −1 if t is improper.

Lemma 5.6. Let t_{i} be a semilinear similitude of (U_{i}, q_{i}), i = 1, 2. We have d(t_{1} ⊥ t_{2}) = d(t_{1})d(t_{2}).

Proof. We assume that disc(q_{i}), i = 1, 2, is trivial. Let e_{i} be an idempotent generating the center Z_{i} of C_{0}(q_{i}). We have t_{i}(e_{i}) = e_{i} if t_{i} is proper and t_{i}(e_{i}) = 1−e_{i} if t_{i} is improper. The idempotent e = e_{1}+e_{2}−2e_{1}e_{2} ∈ C_{0}(q_{1} ⊥ q_{2}) generates the center of C_{0}(q_{1} ⊥ q_{2}) (see for example [5, (2.3), Chap. IV] ) and the claim follows by case checking.

Lemma 5.7. Let V, θ, V^{0}, R and T be as above. Let dim_{K} V^{0} = 2m. Then T is proper if m is even and is improper if m is odd.

Proof. The quadratic space (V, θ) is the orthogonal sum of 1-dimensional spaces and we get a corresponding orthogonal decomposition of (V^{0}, [R]) into subspaces (U_{i}, q_{i}) of dimension 2. In view of (5.6) it suffices to check the case

m = 1. Let α = a = a_{1} + ℓa_{2} ∈ D and ρ = (\begin{matrix}a_{2} & \overline{a_{1}} \\ −a_{1} & −μ\overline{a_{2}} \end{matrix}) . We choose μ = 1,

a_{1} = j (j as in (2.4)), put i = 1 − 2j, so that \overline{i} = −i and choose a_{2} = 0. Let x = x_{1}e_{1} + x_{2}e_{2} ∈ V^{0}, so [R](x_{1}, x_{2}) = ix_{1}x_{2} and C([R]) is generated by e_{1}, e_{2} with the relations e_{1}^{2} = 0, e_{2}^{2} = 0, e_{1}e_{2} + e_{2}e_{1} = i. The element e = i^{−1}e_{1}e_{2} is an idempotent generating the center. Since T (x_{1}e_{1} + x_{2}e_{2}) = \overline{x}_{2}e_{1} + \overline{x}_{1}e_{2}, we have C_{0}(T)(e_{1}e_{2}) = −e_{2}e_{1} and C_{0}(T)(e) = 1 − e. Thus T is not proper.

Of special interest for the next section are quadratic quaternion forms [k] such that the induced quadratic forms π_{2}([k]) are Pfister forms. For convenience we call such forms Pfister quadratic quaternion forms. Hyperbolic spaces of dimension 2^{n} are Pfister forms, hence spaces of the form β([b]), b a hermitian form over K, are Pfister, in view of the exactness of the sequence of Lewis [7]. It is in fact easy to give explicit examples of Pfister forms using the following constructions:

Example 5.8 (Char F ≠ 2). Let q =< λ_{1}, …, λ_{n} > be a diagonal quadratic form on F^{n}, i.e., q(x) = ∑ λ_{i}x_{i}^{2}. Let [k] on D^{n} be given by the diagonal form ℓq. Then the corresponding quadratic form [R] on K^{2n} is given by the diagonal form < 1, −μ > ⊗q. In particular we get the 3-Pfister form << a, b, μ >> choosing for q the norm form of a quaternion algebra (a, b)_{F}.

Example 5.9 (Char F = 2). Let b =< λ_{1}, …, λ_{n} > be a bilinear diagonal form on F^{n}, i.e., b(x, y) = ∑ λ_{i}x_{i}y_{i}. Let k = (j + ℓ)b on D^{n}. Then the corresponding quadratic form [R] over K = R(j), j^{2} = j + λ, is given by the form [R] = b ⊗ [1, λ] where [ξ, η] = ξx_{1}^{2} + x_{1}x_{2} + ηx_{2}^{2}. In particular, for b =< 1, a, c, ac >, we get the 3-Pfister form << a, c, λ]] with the notations of [6], p. xxi.

6. Triality for semilinear similitudes

Let ℭ be a Cayley algebra over F with conjugation π : x ↦ \overline{x} and norm 𝔫: x ↦ x\overline{x}. The new multiplication x ⋆ y = \overline{x} \overline{y} satisfies

x ⋆ (y ⋆ x) = (x ⋆ y) ⋆ x = 𝔫(x)y (6)

for x, y ∈ ℭ. Further, the polar form b_{𝔫} is associative with respect to ⋆, in the sense that

b_{𝔫}(x ⋆ y, z) = b_{𝔫}(x, y ⋆ z).

Proposition 6.1. For x, y ∈ ℭ, let r_{x}(y) = y ⋆ x and ℓ_{x}(y) = x ⋆ y. The map ℭ → End_{F} (ℭ ⊕ ℭ) given by

x ↦ (\begin{matrix}0 & ℓ_{x} \\ r_{x} & 0 \end{matrix})

induces isomorphisms α: (C(ℭ, 𝔫), τ) ⟶^{∼} (End_{F} (ℭ ⊕ ℭ), σ_{𝔫⊥𝔫}) and

α_{0} : (C_{0}(ℭ, 𝔫), τ_{0}) ⟶^{∼} (End_{F} (ℭ), σ_{𝔫}) × (End_{F} (ℭ), σ_{𝔫}), (7)

of algebras with involution.

Proof. We have r_{x}(ℓ_{x}(y)) = ℓ_{x}(r_{x}(y)) = 𝔫(x) · y by (6). Thus the existence of the map α follows from the universal property of the Clifford algebra. The fact that α is compatible with involutions is equivalent to

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

for all x, y, z, u in ℭ. This formula follows from the associativity of b_{𝔫}. Since C(ℭ, 𝔫) is central simple, the map α is an isomorphism by a dimension count.

Assume from now on that ℭ is defined over a field K which is quadratic Galois over F. Any proper semilinear similitude t of 𝔫 induces a semilinear automorphism C(t) of the even Clifford algebra (C_{0}(ℭ, 𝔫), τ_{0}), which does not permute the two components of the center of C_{0}(ℭ, 𝔫). Thus α_{0} ∘ C_{0}(t) ∘ α_{0}^{−1} is a pair of semilinear automorphisms of (End_{K}(ℭ), σ_{𝔫}). It follows as in (4.5) that, for any quadratic space (V, q), semilinear automorphisms of (End_{K}(V), σ_{q}, f_{q}) are of the form Int(f), where f is a semilinear similitude of q. The following result is due to Wonenburger [12] in characteristic different from 2:

Proposition 6.2. For any proper semilinear similitude t_{1} of 𝔫 with multiplier μ_{1}, there exist proper semilinear similitudes t_{2}, t_{2} such that

α_{0} ∘ C_{0}(t_{1}) ∘ α_{0}^{−1} = (Int(t_{2}), Int(t_{3}))

and

\begin{aligned}μ_{3}^{−1}t_{3}(x ⋆ y) & = & t_{1}(x) ⋆ t_{2}(y), \\ μ_{1}^{−1}t_{1}(x ⋆ y) & = & t_{2}(x) ⋆ t_{3}(y), \\ μ_{2}^{−1}t_{2}(x ⋆ y) & = & t_{3}(x) ⋆ t_{1}(y). \end{aligned} (8)

Let t_{1} be an improper similitude with multiplier μ_{1}. There exist improper similitudes t_{2}, t_{3} such that

\begin{aligned}μ_{3}^{−1}t_{3}(x ⋆ y) & = & t_{1}(y) ⋆ t_{2}(x), \\ μ_{1}^{−1}t_{1}(x ⋆ y) & = & t_{2}(y) ⋆ t_{3}(x), \\ μ_{2}^{−1}t_{2}(x ⋆ y) & = & t_{3}(y) ⋆ t_{1}(x). \end{aligned}

The pair (t_{2}, t_{3}) is determined by t_{1} up to a factor (λ, λ^{−1}), λ ∈ K^{×}, and we have μ_{1}μ_{2}μ_{3} = 1. Furthermore, any of the formulas in (8) implies the two others.

Proof. The proof given in [6, (35.4)] for similitudes can also be used for semilinear similitudes.

Remark 6.3. The class of two of the t_{i}, i = 1, 2, 3, modulo K^{×} is uniquely determined by the class of the third t_{i}.

Corollary 6.4. Let T_{1} be a proper semilinear similitude of (ℭ, 𝔫) such that T_{1}^{2} = μ_{1}, μ_{1} ∈ K^{×} and with multiplier −μ_{1}. There exist elements a_{i} ∈ K^{×}, i = 1, 2, 3, and proper semilinear similitudes T_{i} of (ℭ, 𝔫), with T_{i}^{2} = μ_{i}, μ_{i} ∈ K^{×} and with multiplier −μ_{i}, i = 2, 3, such that a_{i}\overline{a_{i}}μ_{i} = μ_{i+1}μ_{i+2} and

\begin{aligned}a_{3}T_{3}(x ⋆ y) & = & T_{1}(x) ⋆ T_{2}(y) \\ a_{1}T_{1}(x ⋆ y) & = & T_{2}(x) ⋆ T_{3}(y) \\ a_{2}T_{2}(x ⋆ y) & = & T_{3}(x) ⋆ T_{1}(y \end{aligned}

The class of any T_{i} modulo K^{×} determines the two other classes and the μ_{i}’s are determined up to norms from K^{×}. Furthermore any of the three formulas determines the two others.

Proof. Counting indices modulo 3, we have relations

T_{i}(x) ⋆ T_{i+1}(y) = b_{i+2}T_{i+2}, b_{i} ∈ K^{×}

in view of (6.2). If we replace all T_{j} by T_{j} ∘ρ_{ν_{j}}, ν_{j} ∈ K^{×}, we get new constants a_{i}. The claim then follows from (3.3).

7. Triality for quadratic quaternion forms

Let D_{1} = K ⊕ℓ_{1}K = [K, μ_{1}) be a quaternion algebra over F and let (V_{1}, q_{θ_{1}}) be a quaternion quadratic space of dimension 4 over D_{1}. Let θ_{1} = [h_{1}], h_{1}(x, y) = P_{1}(x, y) + ℓR_{1}(x, y), so that [R_{1}] = π_{2}(θ_{1}) corresponds to a 8-dimensional (classical) quadratic form on V_{1}^{0} over K. The map T_{1} : V_{1}^{0} → V_{1}^{0}, T_{1}(x) = xℓ_{1}, is a semilinear similitude of (V_{1}^{0}, [R_{1}]) with multiplier −μ_{1} and such that T_{1}^{2} = μ_{1}. We recall that by (3.5) it is equivalent to have a quadratic quaternion space (V_{1}, q_{θ_{1}}) or a pair (V_{1}^{0}, [T_{1}]). We assume from now on that the quadratic form q_{[R_{1}]} is a 3-Pfister form, i.e.,the norm form 𝔫 of a Cayley algebra ℭ over K. In view of (6.4) T_{1} induces two semilinear similitudes T_{2}, resp. T_{3}, with multipliers μ_{2}, resp. μ_{3}, which in turn define a quaternion quadratic space (V_{2}, θ_{2}) of dimension 4 over D_{2} = [K, μ_{2}), resp. a quaternion quadratic space (V_{3}, θ_{3}) of dimension 4 over D_{3} = [K, μ_{3}). Let Br(F) be the Brauer group of F.

Proposition 7.1. 1) [D_{1}][D_{2}][D_{3}] = 1 ∈ Br(F),

2) The restriction of α : C_{0}(ℭ, 𝔫)) ⟶^{∼} End_{K}(ℭ) × End_{K}(ℭ) to C(V_{i}, D_{i}, θ_{i}) induces isomorphisms

α_{i} : (C(V_{i}, D_{i}, θ_{i}), τ) ⟶^{∼} (End_{D_{i+1}}(V_{i+1}), σ_{θ_{i+1}}) × (End_{D_{i+2}}(V_{i+2}), σ_{θ_{i+2}})

Proof. The first claim follows from the fact that μ_{1}μ_{2} = μ_{3} Nrd_{D_{3}}(a_{3}) and the second is a consequence of (5.5), (3.5) and the definition of α.

Example 7.2. Let ℭ_{0} be a Cayley algebra over F and let ℭ = ℭ_{0} ⊗_{F} K. For any c ∈ ℭ_{0} such that c^{2} = μ_{1} ∈ F^{×}, T_{1} : ℭ → ℭ given by T_{1}(k ⊗ x) = \overline{k} ⊗ xc is a semilinear similitude with multiplier −μ_{1} such that T_{1}^{2} = μ_{1}. The Moufang identity (cx)(yc) = c(xy)c in ℭ implies that

(xc) ⋆ (cy) = \overline{c}(x ⋆ y)\overline{c}.

Thus T_{2}(k ⊗ y) = \overline{k} ⊗ cy and T_{3}(k ⊗ z) = i\overline{k} ⊗ \overline{c}z\overline{c} (where i ∈ K^{×} is such that \overline{i} = −i) satisfy (6.4). The corresponding triple of quaternion algebras is ([K, μ_{1}), [K, μ_{1}), [K, i\overline{i}μ_{1}^{2})), the third algebra being split.

Example 7.3. Let D_{i}, i = 1, , 2, 3, be quaternion algebras over F such that [D_{1}][D_{2}][D_{3}] = 1 ∈ Br(F). We may assume that the D_{i} contain a common separable quadratic field K and that D_{i} = [K, μ_{i}), μ_{i} ∈ F^{×} such that μ_{1}μ_{2}μ_{3} ∈ F^{×}^{2}. In [6, (43.12)] similitudes S_{i} with multiplier μ_{i}, i = 1, 2, 3, of the split Cayley algebra ℭ_{s} over F are given, such that 1) μ_{3}^{−1}S_{3}(x ⋆ y) = S_{1}(x) ⋆ S_{2}(y) and 2) S_{i}^{2} = μ_{i}. Let ℭ = K ⊗ ℭ_{s}. Let u ∈ K^{×} be such that \overline{u} = −u. The semilinear similitudes T_{i}(k ⊗ x) = u\overline{k} ⊗ S_{i}(x), i = 1, 2, 3, satisfy

a_{3}T_{3}(x ⋆ y) = T_{1}(x) ⋆ T_{2}(y)

with a_{3} = uμ_{3}^{−1} (we use the same notation ⋆ in ℭ_{s} and in ℭ). Thus there exist a triple of quadratic quaternion forms (θ_{1}, θ_{2}, θ_{3}) corresponding to the three given quaternion algebras. We hope to describe the corresponding quadratic quaternion forms in a subsequent paper.

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Max-Albert Knus ETH Zentrum CH-8092-Zürich Switzerland knus@math.ethz.ch Oliver Villa ETH Zentrum CH-8092-Zürich Switzerland