DISCRIMINANT AND CLIFFORD ALGEBRAS

ANNE QUÉGUINER-MATHIEU AND JEAN-PIERRE TIGNOL

Abstract. The centralizer of a square-central skew-symmetric unit in a central simple algebra with orthogonal involution carries a unitary involution. The discriminant algebra of this unitary involution is shown to be an orthogonal summand in one of the components of the Clifford algebra of the orthogonal involution. As an application, structure theorems for orthogonal involutions on central simple algebras of degree 8 are obtained.

Throughout this paper, F denotes a field of characteristic different from 2. Let A be a central simple F -algebra of degree n = 4m, for some integer m, endowed with an involution σ of orthogonal type. In the first two sections, we assume that the algebra A contains an element θ such that σ(θ) = −θ and θ^{2} = a ∈ F^{⋆}. We denote by Ã the centralizer of θ in A. Since θ is skew-symmetric, σ induces an involution σ̃ on Ã, and (Ã, σ̃) is a central simple algebra with unitary involution of degree 2m over the étale quadratic extension F (θ) of F.

With this data, we may associate two different central simple algebras with involution, namely the Clifford algebra of (A, σ), and the discriminant algebra of (Ã, σ̃), both endowed with their canonical involution (see [14, §8, §10]). Our aim in this paper is to relate these two algebras with involution. Reversing the viewpoint, we show in section 3 that every central simple algebra of even degree 2m with unitary involution and exponent 2 over a quadratic extension of F can be embedded in a central simple F -algebra of degree 4m with orthogonal involution as the centralizer of some skew-symmetric, square-central element. Therefore, our results yield information on discriminant algebras of any central simple algebra of even degree with unitary involution and exponent 2.

The algebra with involution (A, σ) is said to be decomposable if A contains a non trivial σ-stable central simple subalgebra A_{1}. Indeed, if this holds, then A is the tensor product of A_{1} and its centralizer A_{2}, and σ is the tensor product of its restrictions to A_{1} and A_{2}. If A has degree 4, criteria of decomposability have been proven for the different types of involutions. Namely, (A, σ) is always decomposable if σ is of symplectic type; it is decomposable if and only if the discriminant of σ is trivial (resp. the discriminant algebra of (A, σ) is split) if σ is of orthogonal type (resp. of unitary type) (see [14, (16.16)], [12, (5.2)] and [13]). For higher degrees, it is still true that any decomposable orthogonal involution on a 2-power degree algebra has trivial discriminant, but this condition is not sufficient anymore (see [6],

[17] and [18] for examples of indecomposable involutions with trivial discriminant). The main result of section 4 is Theorem 4.3, which gives a necessary and sufficient condition for a central simple algebra of degree 8 with orthogonal involution to be decomposable: one of the components of the Clifford algebra must contain a squarecentral symmetric unit whose centralizer is split. In particular, we recover the fact that one of the components of the Clifford algebra of a degree 8 decomposable algebra with orthogonal involution is of index at most 2 proven in [17, 5.17]. We also give in Proposition 4.12 a necessary and sufficient condition for a degree 4 algebra B with a K/F -unitary involution τ to decompose as (B, τ) = (B_{0}, τ_{0}) ⊗_{F} (K, ) where (B_{0}, τ_{0}) is a central simple algebra with involution of the first kind and denotes the canonical involution of the quadratic extension K/F (compare with [14, (2.22)]).

Date: December 13, 2000.

The authors gratefully acknowledge support from the FNRS–CNRS cooperation agreement and from the TMR network “K-theory and linear algebraic groups” (contract ERB FMRX CT 97-0107). The second author is partially supported by the National Fund for Scientific Research (Belgium).

Returning to the general situation described at the beginning, we now sketch the relations between the Clifford algebra C(A, σ) and the discriminant algebra D(Ã, σ̃) proven in sections 1 and 2. The discriminant of the orthogonal involution σ is easily determined: since σ(θ) = −θ, we have

(1) disc(σ) = Nrd_{A}(θ) = N_{F (θ)/F} (NrdÃ(θ)) = N_{F (θ)/F} (θ^{2m}) = a^{2m} ∈ F^{⋆2}.

Therefore, by [14, (8.10)], the center Z(A, σ) of the Clifford algebra C(A, σ) is isomorphic to F × F, and C(A, σ) is a product of two central simple F -algebras of degree 2^{2m−1}, which we denote by C_{1} and C_{2}. Jacobson proved that C_{1} ⊗_{F} C_{2} is Brauer-equivalent to A (see [14, §9.C]), and it can be derived from Proposition 3.2.3 in Bayer-Fluckiger–Parimala [2] that one of C_{1}, C_{2} is Brauer-equivalent to the discriminant algebra D(Ã, σ̃). However, we aim to obtain more precise information, taking into account the canonical involutions on the Clifford and discriminant algebra.

To give precise statements, we note that it is possible to use θ to discriminate between the components C_{1}, C_{2} of C(A, σ), as follows: recall from [14, §8.D] that there is a canonical homogeneous polynomial map of degree 2m defined on the space Skew(A, σ) of skew-symmetric elements

π: Skew(A, σ) → Z(A, σ)

such that T_{Z(A,σ)/F} (π(s)) = 0 and π(s)^{2} = Nrd_{A}(s) for all s ∈ Skew(A, σ). (In the split case, π is essentially the pfaffian map, see [14, (8.26)].) From (1), it follows that π(θ)^{2} = a^{2m}; therefore, the elements

z^{+} = \frac{1}{2}(1 + a^{−m}π(θ))  and  z^{−} = \frac{1}{2}(1 − a^{−m}π(θ))

are orthogonal idempotents in Z(A, σ), and we have C(A, σ) = C^{+}(A, σ)×C^{−}(A, σ) with

C^{+}(A, σ) = C(A, σ)z^{+}  and  C^{−}(A, σ) = C(A, σ)z^{−}.

Note that, although this is not apparent from the notation, the determination of which component of C(A, σ) is C^{+}(A, σ) and which is C^{−}(A, σ) really depends on the choice of θ: a different choice of θ may interchange the components C^{+}(A, σ) and C^{−}(A, σ), see Proposition 4.13 and Corollary 4.14.

The canonical involution \underline{σ} on C(A, σ) restricts to involutions of the first kind on C^{+}(A, σ) and C^{−}(A, σ), which we denote again by \underline{σ}. These involutions are both orthogonal if m is even and both symplectic if m is odd. In section 2, we consider the case where a ∉ F^{⋆2}. We construct a quaternion F -algebra Q (whose definition depends on θ) and canonical representations

Θ^{+} : C^{+}(A, σ) →^{∼} End_{Q} E^{+}, Θ^{−}: C^{−}(A, σ) →^{∼} End_{A⊗Q} E^{−}

(see Propositions 2.3 and 2.8). Moreover, we define (−1)^{m+1}-hermitian forms h^{+} on E^{+} and h^{−} on E^{−} (with respect to the conjugation involution on Q and the involution σ ⊗ on A ⊗ Q, respectively) whose adjoint involutions correspond to the canonical involutions on C^{+}(A, σ) and C^{−}(A, σ) under Θ^{+} and Θ^{−}.

We also construct canonical orthogonal decompositions

(2) (E^{+}, h^{+}) = \begin{cases}(⨁_{0≤r<m/2}^{⊥}(Ê_{r}^{+}, h_{r}^{+})) ⊕^{⊥} (E_{m/2}^{+}, h_{m/2}^{+}) & if m is even, \\ ⨁_{0≤r≤(m−1)/2}^{⊥}(Ê_{r}^{+}, h_{r}^{+}) & if m is odd, \end{cases}

(3) (E^{−}, h^{−}) =

⨁_{0≤r<m/2}^{⊥}(Ê_{r}^{−}, h_{r}^{−})  if m is even,
(⨁_{0≤r<(m−1)/2}^{⊥}(Ê_{r}^{−}, h_{r}^{−})) ⊕^{⊥} (E_{(m−1)/2}^{−}, h_{(m−1)/2}^{−}) if m is odd,

see Propositions 2.5 and 2.9. The relation with the discriminant algebra D(Ã, σ̃) is the following: there is a canonical isomorphism

Ψ_{m}: D(Ã, σ̃) →^{∼} \begin{cases}End_{Q} E_{m/2}^{+} & if m is even, \\ End_{A⊗Q} E_{(m−1)/2}^{−} & if m is odd, \end{cases}

under which the canonical involution on D(Ã, σ̃) corresponds to the adjoint involution with respect to h_{m/2}^{+} or h_{(m−1)/2}^{−} respectively (see Corollary 2.13). Thus, the algebra with involution (D(Ã, σ̃), \underline{σ̃}) is an “orthogonal direct summand” (in the sense of [4]) of (C^{+}(A, σ), \underline{σ}) if m is even, of (C^{−}(A, σ), \underline{σ}) if m is odd.

The other terms in the decompositions (2) and (3) above can also be related to the centralizer Ã: we show in Propositions 2.5 and 2.10 that (Ê_{r}^{+}, h_{r}^{+}) for r < m/2 and (Ê_{r}^{−}, h_{r}^{−}) for r < (m − 1)/2 are in a certain sense “extended” from (−1)^{m}- hermitian spaces (E_{r}^{+}, ℓ_{r}^{+}) and (E_{r}^{−}, ℓ_{r}^{−}) over F (θ) and A ⊗ F (θ) respectively, and that there are canonical isomorphisms

Ψ_{2r}: λ^{2r}Ã →^{∼} End_{F (θ)} E_{r}^{+}, Ψ_{2r+1} : λ^{2r+1}Ã →^{∼} End_{A⊗F (θ)} E_{r}^{−}

under which the involutions σ̃^{∧2r}, σ̃^{∧2r+1} correspond to the adjoint involutions with respect to ℓ_{r}^{+} and ℓ_{r}^{−} respectively (see Proposition 2.12).

If a ∈ F^{⋆2}, then substituting \sqrt{a}^{−1}θ for θ we may assume θ^{2} = 1. The element e = \frac{1}{2}(1 + θ) ∈ A is an idempotent such that σ(e) = 1 − e, hence σ is hyperbolic by [3]. In this case, F (θ) ≃ F × F, so Ã ≃ B^{op} × B for some central simple F -algebra B of degree 2m and σ̃ is the exchange involution on B^{op} × B. The results above then take the following simple form: there are canonical representations

Θ^{+} : (C^{+}(A, σ), \underline{σ}) →^{∼} (End_{F} E^{+}, ad_{f^{+}}),
Θ^{−}: (C^{−}(A, σ), \underline{σ})) →^{∼} (End_{A} E^{−}, ad_{f^{−}})

for some canonical (−1)^{m}-symmetric form f^{+} and some (−1)^{m}-hermitian form f^{−} with respect to σ (see Propositions 1.1 and 1.5). Moreover, there is a canonical decomposition

E^{+} = ⨁_{r=0}^{m} E_{r}^{+}

such that f^{+}(ξ, η) = 0 for ξ ∈ E_{r}^{+} and η ∈ E_{s}^{+}, unless r + s = m (see Proposition 1.1). Therefore, for r ≤ (m − 1)/2 the space Ê_{r} = E_{r}^{+} ⊕ E_{m−r}^{+} is a hyperbolic subspace of E^{+}, and E^{+} is Witt-equivalent to E_{m/2}^{+} if m is even (and it is hyperbolic if m is odd, but this is clear a priori since f^{+} is skew-symmetric in this case).

Similarly, there is a decomposition

E^{−} = ⨁_{r=0}^{m−1} E_{r}^{−}

such that f^{−}(ξ, η) = 0 for ξ ∈ E_{r}^{−} and η ∈ E_{s}^{−}, unless r + s = m − 1 (see Proposition 1.5). Therefore, the space Ê_{r}^{−} = E_{r}^{−} ⊕ E_{m−1−r}^{−} is hyperbolic, and E^{−} is hyperbolic if m is even, and Witt-equivalent to E_{(m−1)/2}^{−} if m is odd.

The relation with the centralizer Ã = B^{op} × B is as follows: there are canonical isomorphisms

Ψ_{2r}: λ^{2r}B →^{∼} End_{F} E_{r}^{+}  for r = 0, …, m,
Ψ_{2r+1} : λ^{2r+1}B →^{∼} End_{A} E_{r}^{−}  for r = 0, …, m − 1,

which induce an isomorphism of algebras with involution

Ψ_{m}: (D(Ã, σ̃), \underline{σ̃}) = (λ^{m}B, γ) →^{∼} \begin{cases}(End_{F} E_{m/2}^{+}, ad_{f_{m/2}^{+}}) & if m is even, \\ (End_{A} E_{(m−1)/2}^{−}, ad_{f_{(m−1)/2}^{−}}) & if m is odd, \end{cases}

where γ is the canonical involution on λ^{m}B ([14, §10.B]), see Propositions 1.9 and 1.11.

We thus recover (by substantially different methods) the results obtained by Garibaldi on the Clifford algebras of hyperbolic involutions [7], which provided the main source of inspiration for our investigations. We first discuss this particular case, to which the general case is reduced by scalar extension to F (\sqrt{a}).

1. The hyperbolic case

In this section, we consider the case where a ∈ F^{⋆2}. After scaling, we may assume a = 1. The elements

e = \frac{1}{2}(1 + θ)  and  e′ = \frac{1}{2}(1 − θ)

are idempotents and σ(e) = e′ = 1 − e, hence σ is hyperbolic by [3].

1.1. Representation of the Clifford algebra. Let c: A → C(A, σ) be the canonical map (see [14, (8.13)]). Consider

ρ(e) = c(eAe′)^{m} ⊂ C(A, σ)

and

E^{+} = C(A, σ)ρ(e) ⊂ C(A, σ).

It is known (see [9, §4] or [14, (8.29)]) that ρ(e) is a 1-dimensional subspace of C(A, σ) and dim E^{+} = 2^{2m−1} = deg C^{±}(A, σ). Moreover, Lemma 4.2 of [9] shows that π(θ)u = u for all u ∈ ρ(e), hence z^{+}ξ = ξ and z^{−}ξ = 0 for all ξ ∈ E^{+}. Therefore, left multiplication induces an F -algebra homomorphism

Θ^{+} : C^{+}(A, σ) → End_{F} E^{+}.

This homomorphism is injective since C^{+}(A, σ) is simple, hence it is surjective by dimension count.

Consider the map f^{+} : C(A, σ) × C(A, σ) → C(A, σ) defined by

f^{+}(ξ, η) = \underline{σ}(ξ)η for ξ, η ∈ C(A, σ), (4)

where \underline{σ} is the canonical involution on C(A, σ).

Proposition 1.1. For all ξ and η ∈ E^{+}, we have f^{+}(ξ, η) ∈ ρ(e). The map f^{+} is a regular bilinear form on E^{+}. It is symmetric if m is even and skew-symmetric if m is odd. Letting E_{r}^{+} = c(e′Ae)^{r}ρ(e) for r = 0, …, m, we have dim E_{r}^{+} = \binom{2m}{2r} and

E^{+} = ⨁_{r=0}^{m} E_{r}^{+}.

Moreover, for ξ ∈ E_{r}^{+} and η ∈ E_{s}^{+} we have f^{+}(ξ, η) = 0 unless r + s = m.

The left multiplication isomorphism Θ^{+} induces an isomorphism of algebras with involution

Θ^{+} : (C^{+}(A, σ), \underline{σ}) →^{∼} (End_{F} E^{+}, ad_{f^{+}})

where ad_{f^{+}} is the involution adjoint to f^{+}.

Proof. It suffices to check the assertions over a scalar extension of F. Extending scalars to a splitting field of A, we are reduced to the case where A is split. Assume thus A = End_{F} V and σ = ad_{q}, for some 4m-dimensional hyperbolic quadratic space (V, q) over F. The subspaces U = im e and W = ker e are supplementary totally isotropic subspaces of V. Let (u_{1}, …, u_{2m}) be a basis of U, and (w_{1}, …, w_{2m}) a basis of W such that q(u_{i} + w_{j}) = δ_{ij}. Under the canonical identification C(A, σ) ≃ C_{0}(V, q) ([14, (8.8)]), the vector space c(eAe′) (resp. c(e′Ae)) is the span of the products u_{i}u_{j} (resp. w_{i}w_{j}). Therefore, ρ(e) = u_{1} … u_{2m}F and E_{r}^{+} is the span of the products w_{i_{1}} … w_{i_{2r}}u_{1} … u_{2m}. It is canonically isomorphic to ⋀^{2r} W under the map which carries x_{1} ∧⋯∧x_{2r} to x_{1} … x_{2r}u_{1} … u_{2m}, for x_{1}, …, x_{2r} ∈ W, hence dim E_{r}^{+} = \binom{2m}{2r}. The equality E^{+} = ⨁_{r=0}^{m} E_{r}^{+} is easily verified.

If ξ = w_{i_{1}} … w_{i_{2r}}u_{1} … u_{2m} and η = w_{j_{1}} … w_{j_{2s}}u_{1} … u_{2m}, computation shows that f^{+}(ξ, η) ≠ 0 if and only if each w_{i}, 1 ≤ i ≤ 2m appears exactly once among w_{i_{1}}, …, w_{i_{2r}}, w_{j_{1}}, …, w_{j_{2s}}. If this condition holds, then r + s = m and f^{+}(ξ, η) = ±u_{1} … u_{2m}. This shows that f^{+} is regular and that f^{+}(ξ, η) ∈ ρ(e) for all ξ, η ∈ E^{+}.

For ξ, η, ζ ∈ C(A, σ), we have

f^{+}(ξη, ζ) = \underline{σ}(η)\underline{σ}(ξ)ζ = f^{+}(η, \underline{σ}(ξ)ζ),

hence \underline{σ} corresponds under Θ^{+} to the adjoint involution with respect to f^{+}.

Remark. The idea to consider the spaces E_{r}^{+} is inspired by Garibaldi [7, §4].

Corollary 1.2. Let Ê_{r}^{+} = E_{r}^{+} ⊕ E_{m−r}^{+} for r < m/2. The restriction of f^{+} to Ê_{r}^{+} is a hyperbolic form. If m is even, the restriction f_{m/2}^{+} of f^{+} to E_{m/2}^{+} is a regular symmetric bilinear form, and (E^{+}, f^{+}) is Witt-equivalent to (E_{m/2}^{+}, f_{m/2}^{+}).

Proof. The corollary readily follows from Proposition 1.1.

In order to obtain a representation of C^{−}(A, σ) similar to the representation Θ^{+} of C^{+}(A, σ), we use the Clifford bimodule B(A, σ) defined in [14, §9]. Recall from [14, (9.7)] that B(A, σ) is a left A-module as well as a C(A, σ)-bimodule, and that for A = End_{F} V, σ = ad_{q}, we have a canonical isomorphism

B(A, σ) = V ⊗ C_{1}(V, q)

where A acts on V and C(A, σ) = C_{0}(V, q) acts by multiplication. For a ∈ End_{F} V, c_{0} ∈ C_{0}(V, q), v ∈ V and c_{1} ∈ C_{1}(V, q) we thus set

a · (v ⊗ c_{1}) = a(v) ⊗ c_{1}, c_{0} ∗ (v ⊗ c_{1}) = v ⊗ (c_{0}c_{1}), (v ⊗ c_{1}) · c_{0} = v ⊗ (c_{1}c_{0}).

There is a canonical A-module homomorphism b: A → B(A, σ) which is given in the case where A = End_{F} V by the canonical map

End_{F} V →^{∼} V ⊗ V ↪ V ⊗ C_{1}(V, q).

Using the right action of C(A, σ) on B(A, σ), define

E^{−} = B(A, σ) · ρ(e) ⊂ B(A, σ).

The left actions of A and C(A, σ) on B(A, σ) endow E^{−} with a structure of left Aand C(A, σ)-module. For r = 0, …, m − 1, define

E_{r}^{−} = (Ae)^{b} · E_{r}^{+} ⊂ E^{−}

(where (Ae)^{b} is the image of Ae under b). This is an A-submodule of E^{−}.

Lemma 1.3. The dimensions of the F -vector spaces defined above are

dim E^{−} = 4m2^{2m−1} = deg A deg C^{±}(A, σ), dim E_{r}^{−} = 4m\binom{2m}{2r + 1},

and

E^{−} = ⨁_{r=0}^{m−1} E_{r}^{−}.

Moreover, z^{+} ∗ ξ = 0 and z^{−} ∗ ξ = ξ for all ξ ∈ E^{−}.

Proof. As in Proposition 1.1, it suffices to consider the case where A is split. Using the same notation as in the proof of that proposition, we have

E^{−} = V ⊗ C_{1}(V, q)u_{1} … u_{2m}

and E_{r}^{−} is spanned by the products v ⊗ (w_{i_{1}} … w_{i_{2r+1}}u_{1} … u_{2m}). Therefore, there is a canonical isomorphism of F -vector spaces E_{r}^{−} = V ⊗ ⋀^{2r+1} W which maps v⊗(x_{1}∧⋯∧x_{2r+1}) to v⊗(x_{1} … x_{2r+1}u_{1} … u_{2m}) for v ∈ V and x_{1}, …, x_{2r+1} ∈ W, and we have E^{−} = ⨁_{r=0}^{m−1} E_{r}^{−}. The dimensions of E^{−} and E_{r}^{−} are then easily computed.

Since π(θ) is a trace zero element in the center of C_{0}(V, q), it anticommutes with every element in C_{1}(V, q), hence

π(θ) ∗ (v ⊗ cu_{1} … u_{2m}) = −v ⊗ cπ(θ)u_{1} … u_{2m} = −v ⊗ cu_{1} … u_{2m}

for v ∈ V and c ∈ C_{1}(V, q). Therefore, z^{+} ∗ ξ = 0 and z^{−} ∗ ξ = ξ for all ξ ∈ E^{−}.

From the lemma, it follows that the left action of C(A, σ) on B(A, σ) induces an F -algebra homomorphism

Θ^{−}: C^{−}(A, σ) → End_{A} E^{−}.

This homomorphism is injective since C^{−}(A, σ) is simple, hence it is surjective by dimension count.

To describe a hermitian form on E^{−} whose adjoint involution corresponds to \underline{σ} under Θ^{−}, we use the following result:

Lemma 1.4. There exists a C(A, σ)-bimodule isomorphism

μ: B(A, σ) ⊗_{C(A,σ)} B(A, σ) → A ⊗_{F} C(A, σ)

which, in the case where A = End_{F} V and σ = ad_{q}, is given by

μ((v_{1} ⊗ c_{1}) ⊗ (v_{2} ⊗ c_{2})) = (v_{1} ⊗ v_{2}) ⊗ c_{1}c_{2}

under the canonical isomorphisms B(A, σ) = V ⊗ C_{1}(V, q), C(A, σ) = C_{0}(V, q) and A = V ⊗ V. This map satisfies

μ((aξ) ⊗ η) = a ⊗ 1 · μ(ξ ⊗ η)  and  μ(ξ ⊗ (aη)) = μ(ξ ⊗ η) · σ(a) ⊗ 1

for a ∈ A and ξ, η ∈ B(A, σ).

Proof. Let T (\underline{A}) be the tensor algebra on the underlying vector space \underline{A} of A, and let T_{+}(\underline{A}) be the sum of the components of strictly positive degree in T (\underline{A}). Recall from [14, (9.4)] the canonical representation ρ_{r}: 𝔖_{2r} → GL(\underline{A}^{⊗r}) of the symmetric group 𝔖_{2r} which for \underline{A} = V ⊗ V is given by

ρ_{r}(p)(v_{1} ⊗ ⋯ ⊗ v_{2r}) = v_{p^{−1}(1)} ⊗ ⋯ ⊗ v_{p^{−1}(2r)}.

For all r ≥ 1, let γ_{r} be the image of the cycle (1, 2, …, 2r)^{−1} ∈ 𝔖_{2r} under ρ_{r}. We define a map μ̃: T_{+}(\underline{A}) × T_{+}(\underline{A}) → \underline{A} ⊗ T (\underline{A}) by

μ̃(u, v) = ρ_{i+j}((1, 2)) ∘ γ_{i+j}^{−1}(u ⊗ γ_{j}(v)) for u ∈ \underline{A}^{⊗i} and v ∈ \underline{A}^{⊗j}.

Straightforward verifications in the split case show that the map μ̃ induces an isomorphism μ with the required properties under the canonical epimorphisms T (\underline{A}) → C(A, σ) and T_{+}(\underline{A}) → B(A, σ).

Remark. This lemma is Exercise 6 (a) in [14, Chapter II].

Recall from [14, (9.10)] the linear involution ω: B(A, σ) → B(A, σ) which in the split case is given by

(x ⊗ (v_{1} … v_{2r+1}))^{ω} = x ⊗ (v_{2r+1} … v_{1}) for x, v_{1}, …, v_{2r+1} ∈ V.

For ξ, η ∈ B(A, σ) we set

f^{−}(ξ, η) = μ(ξ^{ω} ⊗ η) ∈ A ⊗ C(A, σ).

Proposition 1.5. For all ξ, η ∈ E^{−} we have f^{−}(ξ, η) ∈ A ⊗ ρ(e). The map f^{−} is a regular (−1)^{m}-hermitian form on the left A-module E^{−} with respect to σ, and f^{−}(ξ, η) = 0 for ξ ∈ E_{r}^{−} and η ∈ E_{s}^{−}, unless r + s = m − 1.

The homomorphism Θ^{−} is an isomorphism of algebras with involution

Θ^{−}: (C^{−}(A, σ), \underline{σ}) →^{∼} (End_{A} E^{−}, ad_{f^{−}}).

Proof. As in Proposition 1.1, it suffices to consider the split case. Using the same notation as in the proof of that proposition, we let A = End_{F} V and σ = ad_{q}. Let v_{1}, v_{2} ∈ V. For ξ = v_{1} ⊗ (w_{i_{1}} … w_{i_{2r+1}}u_{1} … u_{2m}) and η = v_{2} ⊗ (w_{j_{1}} … w_{j_{2s+1}}u_{1} … u_{2m}), we have

f^{−}(ξ, η) = (v_{1} ⊗ v_{2}) ⊗ u_{2m} … u_{1}w_{i_{2r+1}} … w_{i_{1}}w_{j_{1}} … w_{j_{2s+1}}u_{1} … u_{2m}.

Computation shows that f^{−}(ξ, η) ≠ 0 if and only if each w_{i} for i = 1, …, 2m appears exactly once among w_{i_{1}}, …, w_{i_{2r+1}}, w_{j_{1}}, …, w_{j_{2s+1}}. If this condition holds, then r + s = m − 1 and

f^{−}(ξ, η) = ±(v_{1} ⊗ v_{2}) ⊗ u_{1} … u_{2m}.

It follows that f^{−} is regular. The following equations follow from straightforward computations: for ξ, η ∈ E^{−} and a ∈ A,

f^{−}(η, ξ) = (−1)^{m}σ ⊗ Id(f^{−}(ξ, η)),
f^{−}(aξ, η) = af^{−}(ξ, η)  and  f^{−}(ξ, aη) = f^{−}(ξ, η)σ(a).

Therefore, f^{−} is a (−1)^{m}-hermitian form on the left A-module E^{−}.

Finally, for ξ, η ∈ B(A, σ) and c ∈ C(A, σ) we have

(c ∗ ξ)^{ω} ⊗ η = (ξ^{ω}\underline{σ}(c)) ⊗ η = ξ^{ω} ⊗ (\underline{σ}(c) ∗ η),

hence f^{−}(c∗ξ, η) = f^{−}(ξ, \underline{σ}(c)∗η). Therefore, \underline{σ} corresponds to ad_{f^{−}} under Θ^{−}.

Corollary 1.6. Let Ê_{r}^{−} = E_{r}^{−} ⊕ E_{m−1−r}^{−} for r < (m − 1)/2. The restriction of f^{−} to Ê_{r}^{−} is a hyperbolic form. If m is even, the hermitian space (E^{−}, f^{−}) is hyperbolic. If m is odd, (E^{−}, f^{−}) is Witt-equivalent to (E_{(m−1)/2}^{−}, f_{(m−1)/2}^{−}), where f_{(m−1)/2}^{−} is the restriction of f^{−} to E_{(m−1)/2}^{−}.

Remark. This result was first proved by Garibaldi [7, Proposition 4.8].

1.2. Representation of the discriminant algebra of the centralizer. Our next goal is to relate the constructions of the preceding section to the discriminant algebra D(Ã, σ̃). We shall establish isomorphisms D(Ã, σ̃) ≃ End_{F} E_{m/2}^{+} if m is even, D(Ã, σ̃) ≃ End_{A} E_{(m−1)/2}^{−} if m is odd, and show that the canonical involution on the discriminant algebra corresponds to the restriction of the adjoint involutions ad_{f^{+}} and ad_{f^{−}} respectively.

Lemma 1.7. Ã = eAe ⊕ e′Ae′.

Proof. Since e + e′ = 1, every element x ∈ A decomposes as

x = exe + exe′ + e′xe + e′xe′.

We have eθ = θe = e and e′θ = θe′ = −e′, hence x commutes with θ if and only if exe′ + e′xe = 0. Therefore,

x = exe + e′xe′ ∈ eAe ⊕ e′Ae′.

This proves Ã ⊂ eAe ⊕ e′Ae′, and the reverse inclusion is clear.

Let B = e′Ae′, a central simple F -algebra of degree 2m which is Brauerequivalent to A. For x ∈ B we have σ(x) ∈ eAe, hence the map (x_{1}^{op}, x_{2}) ↦ σ(x_{1}) + x_{2} defines an F -algebra isomorphism

B^{op} × B →^{∼} eAe ⊕ e′Ae′ = Ã.

Under this isomorphism, σ̃ corresponds to the exchange involution ε on B^{op} × B. Therefore, there is a canonical isomorphism

(D(Ã, σ̃), \underline{σ̃}) = (λ^{m}B, γ), (5)

where γ is the canonical involution on λ^{m}B, by [14, (10.31)].

Recall from [14, (10.4)] that for r = 0, …, 2m, the algebra λ^{r}B is defined as follows: let s_{r} ∈ B^{⊗r} be the element which in the split case B = End_{F} W satisfies

s_{r}(w_{1} ⊗ ⋯ ⊗ w_{r}) = ∑_{p∈𝔖_{r}} sgn(p)w_{p(1)} ⊗ ⋯ ⊗ w_{p(r)} for w_{1}, …, w_{r} ∈ W,

where the sum runs over all the permutations p of {1, …, r} and sgn(p) is the signature of p. Let S_{r} = B^{⊗r}s_{r}. Then λ^{r}B = End_{B^{⊗m}} S_{r}.

To obtain an alternative description of λ^{r}B, consider the idealizer Ŝ_{r} and the annihilator S_{r}^{0}, defined by

Ŝ_{r} = {x ∈ B^{⊗r} | S_{r}x ⊂ S_{r}}  and  S_{r}^{0} = {x ∈ B^{⊗r} | S_{r}x = 0}.

As explained in [14, p.9], right multiplication induces an isomorphism

Ŝ_{r}/S_{r}^{0} →^{∼} End_{B^{⊗r}} S_{r} = λ^{r}B.

It turns out that the even powers λ^{2r}B can be represented as End_{F} E_{r}^{+}, and the odd powers λ^{2r+1}B as End_{A} E_{r}^{−}, as we now show.

First, we consider the case of even powers. Let σ_{⋆}: B ⊗ B → End_{F} (e′Ae) be the algebra homomorphism defined by σ_{⋆}(x ⊗ y)(z) = xzσ(y). It is injective since B is simple, hence it is an isomorphism by dimension count. Therefore, we get an algebra isomorphism

σ_{⋆}^{⊗r}: B^{⊗2r} →^{∼} End_{F} ((e′Ae)^{⊗r}).

On the other hand, pick a nonzero element κ ∈ ρ(e) and denote by Φ_{r}^{+} the epimorphism

Φ_{r}^{+} : (e′Ae)^{⊗r} → c(e′Ae)^{r}ρ(e) = E_{r}^{+}

which maps z_{1} ⊗ ⋯ ⊗ z_{r} to c(z_{1}) ⋯ c(z_{r})κ, for z_{1}, …, z_{r} ∈ e′Ae.

Lemma 1.8. For any x ∈ Ŝ_{2r}, σ_{⋆}^{⊗r}(x)(ker Φ_{r}^{+}) ⊂ ker Φ_{r}^{+}. Moreover, if x ∈ S_{2r}^{0}, then im σ_{⋆}^{⊗r}(x) ⊂ ker Φ_{r}^{+}.

Let us assume this lemma for the moment. For any x ∈ Ŝ_{2r}, σ_{⋆}^{⊗r}(x) induces an endomorphism of E_{r}^{+}, which is trivial as soon as x ∈ S_{2r}^{0}. Hence, we get an algebra homomorphism

Ψ_{2r}: λ^{2r}B = Ŝ_{2r}/S_{2r}^{0} → End_{F} E_{r}^{+}.

This homomorphism does not depend on the choice of κ ∈ ρ(e).

Proposition 1.9. The homomorphism Ψ_{2r} is an isomorphism. Moreover, if m is even, Ψ_{m} is an isomorphism of algebras with involution

Ψ_{m}: (λ^{m}B, γ) →^{∼} (End_{F} E_{m/2}^{+}, ad_{f_{m/2}^{+}})

where f_{m/2}^{+} is the restriction of f^{+} to E_{m/2}^{+}.

Proofs of Lemma 1.8 and Proposition 1.9. As in Proposition 1.1, we may assume that A is split. Using the same notation as in the proof of Proposition 1.1, we have B = End_{F} W,

e′Ae = Hom(U, W) = W ⊗ U^{∗} ≃ W ⊗ W

(where the last isomorphism is induced by the polar form of q), and σ_{⋆}^{⊗r} is the natural isomorphism (End_{F} W)^{⊗2r} ≃ End_{F} ((W ⊗ W)^{⊗r}). As observed in the proof of Proposition 1.1, E_{r}^{+} ≃ ⋀^{2r} W, and Φ_{r}^{+} is the canonical epimorphism Φ_{r}^{+} : W^{⊗2r} → ⋀^{2r} W, whose kernel is ker s_{2r} (see [14, proof of (10.3)]). Lemma 1.8 then follows by explicit calculations. The homomorphism Ψ_{2r} is thus well-defined. It is injective since λ^{2r}B is simple, hence it is an isomorphism by dimension count.

Assume now that m is even. The canonical involution γ of λ^{m}B ≃ End_{F} (⋀^{m} W) is adjoint to the exterior power ∧: ⋀^{m} W × ⋀^{m} W → ⋀^{2m} W ≃ F (see [14, (10.11)(a)]), and one may check that it coincides with f_{m/2}^{+} if m/2 is even and with −f_{m/2}^{+} if m/2 is odd. Hence, in both cases, we get that γ corresponds to ad_{f_{m/2}^{+}}, and this finishes the proof of Proposition 1.9.

Now, we consider the odd powers λ^{2r+1}B. Letting A act on Ae ⊗ (e′Ae)^{⊗r} by left multiplication on the factor Ae, we get an algebra isomorphism

σ_{2r+1} : B^{⊗2r+1} → End_{A}(Ae ⊗_{F} (e′Ae)^{⊗r})

such that

σ_{2r+1}(x_{1} ⊗ ⋯ ⊗ x_{2r+1})(y ⊗ z_{1} ⊗ ⋯ ⊗ z_{r}) =

yσ(x_{1}) ⊗ x_{2}z_{1}σ(x_{3}) ⊗ ⋯ ⊗ x_{2r}z_{r}σ(x_{2r+1})

for x_{1}, …, x_{2r+1} ∈ B, y ∈ Ae, and z_{1}, …, z_{r} ∈ e′Ae. Let κ ∈ ρ(e), κ ≠ 0, and consider the A-module homomorphism

Φ_{r}^{−}: Ae ⊗_{F} (e′Ae)^{⊗r} → (Ae)^{b} · E_{r}^{+} = (Ae)^{b} · c(e′Ae)^{r} · ρ(e)

which maps y ⊗ z_{1} ⊗ … ⊗ z_{r} to y^{b} · c(z_{1}) … c(z_{r})κ.

Lemma 1.10. For any x ∈ Ŝ_{2r+1}, σ_{2r+1}(x)(ker Φ_{r}^{−}) ⊂ ker Φ_{r}^{−}. Moreover, if x ∈ S_{2r+1}^{0}, then im(σ_{2r+1}(x)) ⊂ ker Φ_{r}^{−}.

Assuming the lemma, we may use σ_{2r+1} to define a canonical algebra homomorphism

Ψ_{2r+1} : λ^{2r+1}B = Ŝ_{2r+1}/S_{2r+1}^{0} → End_{A} E_{r}^{−}.

Proposition 1.11. The homomorphism Ψ_{2r+1} is an isomorphism. Moreover, if m is odd, Ψ_{m} is an isomorphism of algebras with involution

Ψ_{m}: (λ^{m}B, γ) →^{∼} (End_{A} E_{(m−1)/2}^{−}, ad_{f_{(m−1)/2}^{−}})

where f_{(m−1)/2}^{−} is the restriction of f^{−} to E_{(m−1)/2}^{−}.

The proofs of Lemma 1.10 and Proposition 1.11 follow the same lines as those of Lemma 1.8 and Proposition 1.9. We leave the details to the reader.

Recall that central simple algebras with involution are called Witt-equivalent if they can be represented as endomorphism algebras of Witt-equivalent hermitian spaces, see [5]. (If (A_{1}, σ_{1}) and (A_{2}, σ_{2}) are Witt-equivalent and deg A_{1} ≥ deg A_{2}, Garibaldi [7] writes that (A_{1}, σ_{1}) is a hyperbolic extension of (A_{2}, σ_{2}).)

Corollary 1.12. The algebra (D(Ã, σ̃), \underline{σ̃}) is Witt-equivalent to (C^{+}(A, σ), \underline{σ}) if m is even, to (C^{−}(A, σ), \underline{σ}) if m is odd.

Proof. Combine the isomorphism (5) with Propositions 1.1 and 1.9 if m is even, with Propositions 1.5 and 1.11 if m is odd.

Remark. For even m, this result was proved by Garibaldi [7, Main Theorem 0.1]. For odd m, Garibaldi proves a different result (see [7, Proposition 3.4]), which turns out to be equivalent to the above by [8, Theorem 1.1].

2. The general case

In this section, we consider the general case described in the introduction: (A, σ) is a central simple F -algebra of degree 4m with orthogonal involution, containing an element θ such that σ(θ) = −θ and θ^{2} = a ∈ F^{⋆}. We assume a ∉ F^{⋆2} and let K = F (\sqrt{a}). We denote (A_{K}, σ_{K}) = (A ⊗_{F} K, σ ⊗ Id_{K}). Now, θ ∈ A_{K} satisfies σ_{K}(θ) = −θ and θ^{2} ∈ K^{⋆2}, hence (A_{K}, σ_{K}) is hyperbolic and we may apply the results of the preceding section.

2.1. Representation of the Clifford algebra. Let α ∈ K satisfy α^{2} = a. Let also

e = \frac{1}{2}(1 + θ ⊗ α^{−1}) ∈ A_{K}, e′ = \frac{1}{2}(1 − θ ⊗ α^{−1}) ∈ A_{K},
ρ(e) = c(eA_{K}e′)^{m} ⊂ C(A_{K}, σ_{K})  and  E^{+} = C(A_{K}, σ_{K})ρ(e) ⊂ C(A_{K}, σ_{K}).

2.1.1. Representation of C^{+}(A, σ). By Proposition 1.1, left multiplication defines an isomorphism

Θ^{+} : (C^{+}(A_{K}, σ_{K}), \underline{σ_{K}}) → (End_{K} E^{+}, ad_{f^{+}}). (6)

Let denote the non trivial automorphism of K/F. We also denote by the natural action of this automorphism on C(A_{K}, σ_{K}) = C(A, σ) ⊗ K. Our first goal is to determine the corresponding action on End_{K} E^{+} under Θ^{+}, as in [9, §4]. As we proceed to show, it is given by f ↦ ψ ∘ f ∘ ψ^{−1}, where ψ is defined in the next lemma. This will enable us to define on E^{+} a module structure over a certain quaternion F -algebra Q. This algebra will be defined as the subalgebra of End_{F} E^{+} generated by K and a map ψ as in the following lemma.

Lemma 2.1. There is an invertible K-semilinear map ψ: E^{+} → E^{+} such that

ψ(ξη) = \overline{ξ}ψ(η) for all ξ ∈ C(A_{K}, σ_{K}), η ∈ E^{+}. (7)

This map is unique up to a factor in K^{⋆}. It satisfies

ψ^{2} = λ Id_{E^{+}}

for some λ ∈ F^{⋆}.

Proof. The Skolem–Noether theorem (see for instance [16, Chapter 8, Theorem 4.2]) yields an element t ∈ A^{⋆} such that

tθt^{−1} = −θ,

hence et = te′ and σ(t)e′ = eσ(t). Let

g = ete′ + e′σ(t)^{−1}e.

Straightforward computations show that geg^{−1} = e′ = \overline{e} and σ_{K}(g)g = 1, hence g is in the orthogonal group O(A_{K}, σ_{K}). We claim that g is in the special orthogonal group O^{+}(A_{K}, σ_{K}). To prove this, observe that gθg^{−1} = −θ, hence

π(gθg^{−1}) = (−1)^{2m}π(θ) = π(θ).

Since the automorphism of C(A_{K}, σ_{K}) induced by the isometry g maps π(θ) to π(gθg^{−1}), it follows that this automorphism restricts to the identity on the center of C(A_{K}, σ_{K}). This proves the claim.

Consider then the Clifford group Γ(A_{K}, σ_{K}). Since the vector representation χ: Γ(A_{K}, σ_{K}) → O^{+}(A_{K}, σ_{K}) is onto (see [14, §13.B]), there exists υ ∈ Γ(A_{K}, σ_{K}) such that χ(υ) = g. Lemma 4.5 of [9] shows that

υρ(e)υ^{−1} = ρ(geg^{−1}) = ρ(\overline{e}).

For ξ ∈ E^{+}, we have \overline{ξ} ∈ \overline{E^{+}} = C(A_{K}, σ_{K})ρ(\overline{e}), hence the preceding equation shows that \overline{ξ}υ ∈ E^{+}. Define ψ: E^{+} → E^{+} by

ψ(ξ) = \overline{ξ}υ for ξ ∈ E^{+}.

The map ψ is clearly invertible and satisfies (7). If ψ′: E^{+} → E^{+} is another such map, then ψ′ ∘ ψ^{−1} ∈ End_{K} E^{+} commutes with left multiplication by elements in C(A_{K}, σ_{K}). It is therefore central, since the homomorphism Θ^{+} of (6) is onto, hence ψ′ ∘ ψ^{−1} ∈ K^{⋆}. Similarly, ψ^{2} ∈ End_{K} E^{+} is central, hence ψ^{2} = λ Id_{E^{+}} for some λ ∈ K^{⋆}. Since ψ^{2} commutes with ψ, we must have \overline{λ} = λ, hence λ ∈ F^{⋆}.

Since E^{+} is a K-vector space, there is a natural embedding K ↪ End_{F} E^{+}. Since the map ψ of Lemma 2.1 is uniquely determined up to a factor in K, the subalgebra Q ⊂ End_{F} E^{+} generated by K and ψ is a uniquely determined quaternion F -algebra. The centralizer of Q consists of the endomorphisms in End_{K} E^{+} which commute with ψ. In view of Lemma 2.1, left multiplication by ξ ∈ C(A_{K}, σ_{K}) commutes with ψ if and only if \overline{ξ} = ξ; therefore, the centralizer of Q is the image of C^{+}(A, σ) under Θ.

Since Q ⊂ End_{F} E^{+}, we may consider E^{+} as a (left) Q-module. Thus, the isomorphism (6) restricts to

Θ^{+} : C^{+}(A, σ) →^{∼} End_{Q} E^{+}.

To complete the description of (C^{+}(A, σ), \underline{σ}), we next determine a sesquilinear form on E^{+} with respect to the quaternion conjugation on Q whose adjoint involution corresponds to the canonical involution \underline{σ} under Θ^{+}.

Lemma 2.2. There exists κ ∈ ρ(e), κ ≠ 0, such that \underline{σ} ∘ ψ(\overline{\underline{σ} ∘ ψ(κ)}) = ψ^{2}(κ).

Proof. Lemma 2.1 (and its proof) shows that we may assume ψ(ξ) = \overline{ξ}υ for some υ ∈ Γ(A_{K}, σ_{K}) such that υρ(e)υ^{−1} = ρ(\overline{e}). Then

\underline{σ} ∘ ψ(\overline{\underline{σ} ∘ ψ(ξ)}) = \underline{σ}(υ)\overline{ξ}υ for all ξ ∈ E^{+}.

Since υ ∈ Γ(A_{K}, σ_{K}), we have \underline{σ}(υ)υ ∈ K^{⋆}, hence \underline{σ}(υ)\overline{ξ}υ ∈ υ^{−1}\overline{ξ}υ · K^{⋆} for all ξ ∈ E^{+}. It follows that \underline{σ} ∘ ψ(\overline{\underline{σ} ∘ ψ(ξ)}) ∈ ρ(e) for all ξ ∈ ρ(e). Pick an arbitrary element κ_{0} ∈ ρ(e), κ_{0} ≠ 0. Since dim ρ(e) = 1, we have

\underline{σ} ∘ ψ(\overline{\underline{σ} ∘ ψ(κ_{0})}) = \underline{σ}(υ)\overline{κ_{0}}υ = νκ_{0} for some ν ∈ K^{⋆}.

Applying and multiplying on the left by \underline{σ}(υ) and on the right by υ, we get

\underline{σ}(\overline{υ}υ)κ_{0}\overline{υ}υ = \overline{ν}\underline{σ}(υ)\overline{κ_{0}}υ = \overline{ν}νκ_{0}. (8)

On the other hand, we have ψ^{2} = λ Id_{E^{+}} for some λ ∈ F^{⋆}, hence

ψ^{2}(ξ) = ξ\overline{υ}υ = ξλ for all ξ ∈ E^{+}.

Applying \underline{σ}, we obtain \underline{σ}(\overline{υ}υ)\underline{σ}(ξ) = λ\underline{σ}(ξ) for all ξ ∈ E^{+}. Since \underline{σ}(ρ(e)) = ρ(e), it follows that \underline{σ}(\overline{υ}υ)κ_{0} = λκ_{0}, hence

\underline{σ}(\overline{υ}υ)κ_{0}\overline{υ}υ = λ^{2}κ_{0}. (9)

Comparing (8) and (9), we find λ^{2} = \overline{ν}ν hence, by Hilbert’s Theorem 90 (see for instance [14, (29.3)]), νλ^{−1} = μ\overline{μ}^{−1} for some μ ∈ K^{⋆}. Setting κ = μκ_{0}, we have

\underline{σ} ∘ ψ(\overline{\underline{σ} ∘ ψ(κ)}) = ν\overline{μ}κ_{0} = λμκ_{0} = ψ^{2}(κ).

Fix a map ψ as in Lemma 2.1 and an element κ ∈ ρ(e) as in Lemma 2.2. Then ρ(e) = κK and we may modify the bilinear form f^{+} of (4) to get a bilinear form with values in K: we let

f^{+}(ξ, η) = g^{+}(ξ, η)κ for ξ, η ∈ E^{+}.

We also define maps ℓ^{+} : E^{+} × E^{+} → K and h^{+} : E^{+} × E^{+} → Q by

ℓ^{+}(ξ, η) = g^{+}(ξ, ψ(η)) for ξ, η ∈ E^{+}

and

h^{+}(ξ, η) = α(ℓ^{+}(ξ, η) − g^{+}(ξ, η)ψ) ∈ K ⊕ Kψ = Q

where α ∈ K is such that α^{2} = a.

Proposition 2.3. For ξ, η ∈ E^{+},

g^{+}(ξ, ψ(η)) = \overline{g^{+}(ψ(ξ), η)}. (10)

The map g^{+} is a regular bilinear form on the K-vector space E^{+}. It is symmetric if m is even and skew-symmetric if m is odd. The map ℓ^{+} is a regular (−1)^{m}- hermitian^{1} form on the K-vector space E^{+}, and h^{+} is a regular (−1)^{m+1}-hermitian form on the left Q-module E^{+}, with respect to the conjugation involution on Q. Moreover, the isomorphism Θ^{+} of (6) is an isomorphism of algebras with involution

Θ^{+} : (C^{+}(A, σ), \underline{σ}) →^{∼} (End_{Q} E^{+}, ad_{h^{+}}).

Proof. Let λ ∈ F^{⋆} be such that ψ^{2} = λ Id_{E^{+}} and let ξ, ζ ∈ E^{+}. Using \underline{σ} ∘ ψ(\overline{\underline{σ} ∘ ψ(κ)}) = λκ, we derive from g^{+}(ξ, ζ)κ = \underline{σ}(ξ)ζ

λ\overline{g^{+}(ξ, ζ)}κ = \underline{σ} ∘ ψ(\overline{\underline{σ} ∘ ψ(\underline{σ}(ξ)ζ)}).

To compute the right side, write ξ = ξ_{0}κ and ζ = ζ_{0}κ for some ξ_{0}, ζ_{0} ∈ C(A_{K}, σ_{K}). Using (7), we get

\underline{σ} ∘ ψ(\overline{\underline{σ} ∘ ψ(\underline{σ}(ξ)ζ)}) = \underline{σ} ∘ ψ(κ)\underline{σ}(\overline{ξ_{0}})\overline{ζ_{0}}ψ(κ) = \underline{σ} ∘ ψ(ξ)ψ(ζ),

hence

g^{+}(ψ(ξ), ψ(ζ))κ = \underline{σ} ∘ ψ(ξ)ψ(ζ) = λ\overline{g^{+}(ξ, ζ)}κ.

Substituting η for ψ(ζ), we obtain (10).

1Note that ℓ^{+} is linear in its first slot and semilinear in its second slot.

Proposition 1.1 shows that f^{+} is a regular (−1)^{m}-symmetric bilinear form, hence g^{+} is also a regular (−1)^{m}-symmetric bilinear form, and ℓ^{+} is a regular (−1)^{m}- hermitian form since ψ is semilinear and satisfies (10). The fact that h^{+} is a regular (−1)^{m+1}-hermitian form follows from straightforward computations.

Finally, since ad_{f^{+}} = ad_{g^{+}} corresponds to \underline{σ_{K}} under Θ^{+}, by Proposition 1.1, it is easily checked that

h^{+}(ξη, ζ) = h^{+}(η, \underline{σ}(ξ)ζ) for ξ ∈ C(A, σ) and η, ζ ∈ E^{+},

proving the last part.

To shed some light on the structure of E^{+} as (−1)^{m+1}-hermitian space over Q, consider as in section 1 the following K-subspaces:

E_{r}^{+} = c(e′A_{K}e)^{r}ρ(e) ⊂ E^{+} for r = 0, …, m.

In particular, E_{m}^{+} = ρ(e′)ρ(e).

Lemma 2.4. The map ψ satisfies

ψ(ρ(e)) = ρ(\overline{e})ρ(e) = ρ(e′)ρ(e),

and ψ(E_{r}^{+}) = E_{m−r}^{+} for r = 0, …, m.

Proof. It suffices to prove these properties after scalar extension to a splitting field of A. Therefore, we may assume A is split and use the same notation as in Proposition 1.1. We thus let A = End_{F} V for some 4m-dimensional vector space V over F, and σ = ad_{q} for some quadratic form q on V.

Since σ(θ) = −θ, every v ∈ V is orthogonal to θ(v). If V_{0} is a maximal non-

singular subspace of V such that V_{0} and θ(V_{0}) are orthogonal, then V = V_{0} ⊕^{⊥} θ(V_{0}). On the other hand, we also have V_{K} = U ⊕ W where U = e(V_{K}) and W = e′(V_{K}) are totally isotropic subspaces of V_{K}, as in the proof of Proposition 1.1. Let v_{1}, …, v_{2m} be an orthogonal basis of V_{0} and let λ_{i} = q(v_{i}) for i = 1, …, 2m. In V_{K}, consider the vectors

u_{i} = \frac{1}{2}(v_{i} + θ(v_{i})α^{−1}),  w_{i} = \frac{1}{2}(v_{i} − θ(v_{i})α^{−1})λ_{i}^{−1}  for i = 1, …, 2m.

Then u_{1}, …, u_{2m} (resp. w_{1}, …, w_{2m}) is a basis of U (resp. W), and these elements satisfy q(u_{i} + w_{j}) = δ_{ij}. Moreover, their images in the Clifford algebra satisfy

u_{i}v_{i} = u_{i}w_{i}λ_{i} = v_{i}w_{i}λ_{i} for i = 1, …, 2m (11)

hence also, since \overline{u_{i}} = λ_{i}w_{i},

v_{i}u_{i} = λ_{i}w_{i}u_{i} = λ_{i}w_{i}v_{i} for i = 1, …, 2m. (12)

As observed in Proposition 1.1, we have

ρ(e) = u_{1} … u_{2m}K ⊂ C_{0}(V_{K}, q) = C(A_{K}, σ_{K}).

Using (11) and (12), it is easily seen that for υ = v_{1} … v_{2m}, the equation ψ_{0}(ξ) = \overline{ξ}υ defines a K-semilinear map ψ_{0} : E^{+} → E^{+} satisfying condition (7) of Lemma 2.1, hence ψ = μψ_{0} for some μ ∈ K^{⋆}. Moreover, we have

\overline{u_{1} … u_{2m}}υ = υu_{1} … u_{2m} = λ_{1} … λ_{2m}w_{1} … w_{2m}u_{1} … u_{2m}.

Since ρ(e′)ρ(e) = w_{1} … w_{2m}u_{1} … u_{2m}K, it follows that ψ(ρ(e)) = ψ_{0}(ρ(e)) = ρ(e′)ρ(e), proving the first part of the lemma.

For the second part, observe that c(e′A_{K}e) is spanned by the products w_{i}w_{j}, hence E_{r}^{+} is the span of the products w_{i_{1}} … w_{i_{2r}}u_{1} … u_{2m}, where we may assume 1 ≤ i_{1} < ⋯ < i_{2r} ≤ 2m. It follows that ψ(E_{r}^{+}) = ψ_{0}(E_{r}^{+}) is spanned by the elements of the form \overline{w_{i_{1}} … w_{i_{2r}}u_{1} … u_{2m}}v_{1} … v_{2m}, hence also, since \overline{w_{i}} = λ_{i}^{−1}u_{i} and \overline{u_{i}} = λ_{i}w_{i}, by the products u_{i_{1}} … u_{i_{2r}}w_{1} … w_{2m}v_{1} … v_{2m}. By (11) and (12), we have for 1 ≤ i_{1} < ⋯ < i_{2r} ≤ 2m,

u_{i_{1}} … u_{i_{2r}}w_{1} … w_{2m}v_{1} … v_{2m} = u_{i_{1}} … u_{i_{2r}}w_{1} … w_{2m}u_{1} … u_{2m}
= ±w_{j_{1}} … w_{j_{2s}}u_{1} … u_{2m},

where the indices j_{1}, …, j_{2s} form the complementary subset of {i_{1}, …, i_{2r}} in {1, …, 2m}. Therefore, ψ(E_{r}^{+}) = E_{m−r}^{+}, and the proof is complete.

It follows from the lemma that for r < m/2 the sum

Ê_{r}^{+} = E_{r}^{+} ⊕ E_{m−r}^{+}

is a Q-submodule of E^{+}, and that E_{m/2}^{+} is a Q-submodule of E^{+} if m is even. Their dimensions over Q are respectively equal to \binom{2m}{2r} and \frac{1}{2} \binom{2m}{m}.

For ε = ±1, let W^{ε}(K, ) (resp. W^{ε}(Q, )) denote the Witt group of ε-hermitian spaces over K (resp. Q) with respect to . For any ε-hermitian space (E, ℓ) over K, a (−ε)-hermitian space R(E, ℓ) over Q is defined by R(E, ℓ) = (E ⊕ ψE, h) where

h(x_{1} + ψx_{2}, y_{1} + ψy_{2}) = α(ℓ(x_{1}, y_{1}) + \overline{ℓ(x_{2}, y_{2})}λ) − α(ℓ(x_{1}, y_{2}) + \overline{ℓ(x_{2}, y_{1})})ψ

for x_{1}, x_{2}, y_{1}, y_{2} ∈ E. As observed in [16, p. 359]^{2} the map R induces a group homomorphism

R: W^{ε}(K, ) → W^{−ε}(Q, ).

Proposition 2.5. If ξ ∈ E_{r}^{+} and η ∈ E_{s}^{+}, then

g^{+}(ξ, η) = 0 unless r + s = m and ℓ^{+}(ξ, η) = 0 unless r = s.

Therefore, letting h_{r}^{+} (resp. h_{m/2}^{+}) denote the restriction of h^{+} to Ê_{r}^{+} (resp. E_{m/2}^{+}, if m is even), we have

(E^{+}, h^{+}) = \begin{cases}⨁_{0≤r≤(m−1)/2}^{⊥}(Ê_{r}^{+}, h_{r}^{+}) & if m is odd, \\ (⨁_{0≤r<m/2}^{⊥}(Ê_{r}^{+}, h_{r}^{+})) ⊕^{⊥} (E_{m/2}^{+}, h_{m/2}^{+}) & if m is even. \end{cases}

Moreover, for r < m/2 we have (Ê_{r}^{+}, h_{r}^{+}) = R(E_{r}^{+}, ℓ_{r}^{+}), where ℓ_{r}^{+} is the restriction of ℓ^{+} to E_{r}^{+}.

Proof. The first part readily follows from Proposition 1.1 and Lemma 2.4. The equality (Ê_{r}^{+}, h_{r}^{+}) = R(E_{r}^{+}, ℓ_{r}^{+}) for r < m/2 follows from a straightforward computation.

Recall from [16, p. 359] the “extension of scalars” map S: W^{−1}(Q, ) → W K (denoted by π_{2} in [16]) which fits in an exact sequence

W^{1}(K, ) ⟶^{R} W^{−1}(Q, ) ⟶^{S} W K. (13)

Corollary 2.6. If m is even, S(Ê_{r}^{+}, h_{r}^{+}) = 0 for all r < m/2.

The result holds trivially if m is odd, since the corresponding map S carries hermitian forms over Q to skew-symmetric spaces over K.

2Scharlau denotes by ρ the map R. When comparing with [16], one has to keep in mind that Scharlau considers sesquilinear forms which are semilinear in the first slot and linear in the second slot, whereas the opposite convention is used here. Moreover, there is a misprint in Scharlau’s definition of the form h, on p. 359 of [16]: it should involve the term f(x_{2}, y_{2})^{∗} instead of f(x_{2}, y_{2}).

2.1.2. Representation of C^{−}(A, σ). In order to obtain an analogue of Proposition 2.3 for C^{−}(A, σ), we consider

E^{−} = B(A_{K}, σ_{K}) · ρ(e) ⊂ B(A_{K}, σ_{K})

and the left multiplication isomorphism

Θ^{−}: (C^{−}(A_{K}, σ_{K}), \underline{σ_{K}}) →^{∼} (End_{A_{K}} E^{−}, ad_{f^{−}})

of Proposition 1.5.

Lemma 2.7. There is a unique action of ψ ∈ Q on E^{−} such that

ψ · (ξ · η) = \overline{ξ} · ψ(η) for ξ ∈ B(A_{K}, σ_{K}) and η ∈ E^{+}.

This action is A-linear.

Proof. Let κ_{0} ∈ ρ(e), κ_{0} ≠ 0. The proof of Lemma 2.4 shows that ψ(κ_{0}) = \overline{κ_{0}}υ for some υ ∈ Γ(A_{K}, σ_{K}). Since

E^{−} = B(A_{K}, σ_{K}) · E^{+} = B(A_{K}, σ_{K}) · κ_{0},

it follows that the action, if it exists, is necessarily defined by

ψ(x) = \overline{x}υ. (14)

This proves uniqueness of the action. On the other hand, one may easily check that equation (14) defines an action of ψ on E^{−} having the required properties.

Since K acts naturally on E^{−}, the lemma yields a left A ⊗_{F} Q-module structure on E^{−}, and Θ^{−} restricts to an F -algebra isomorphism

Θ^{−}: C^{−}(A, σ) →^{∼} End_{A⊗Q} E^{−}. (15)

To define a sesquilinear form on E^{−} whose adjoint involution corresponds to \underline{σ} under this isomorphism, fix an element κ ∈ ρ(e) as in Lemma 2.2. Then A_{K} ⊗_{K} ρ(e) = A_{K} ⊗ κ and we may define a map g^{−}: E^{−} × E^{−} → A_{K} by the equation

f^{−}(ξ, η) = g^{−}(ξ, η) ⊗ κ for ξ, η ∈ E^{−}.

We also define maps ℓ^{−}: E^{−} × E^{−} → A_{K} and h^{−}: E^{−} × E^{−} → A ⊗_{F} Q by

ℓ^{−}(ξ, η) = g^{−}(ξ, ψ · η)

and

h^{−}(ξ, η) = α(ℓ^{−}(ξ, η) − g^{−}(ξ, η)ψ) ∈ A_{K} ⊕ A_{K}ψ = A ⊗_{F} Q.

Proposition 2.8. For ξ, η ∈ E^{−},

g^{−}(ξ, ψ · η) = \overline{g^{−}(ψ · ξ, η)}. (16)

The map g^{−} is a regular (−1)^{m}-hermitian form on the A_{K}-module E^{−} with respect to the involution σ_{K}, the map ℓ^{−} is a regular (−1)^{m}-hermitian form on the A_{K}-module E^{−} with respect to the involution σ ⊗ , and h^{−} is a regular (−1)^{m+1}- hermitian form on the A ⊗ Q-module E^{−} with respect to the involution σ ⊗ . Moreover, the isomorphism Θ^{−} of (15) is an isomorphism of algebras with involution

Θ^{−}: (C^{−}(A, σ), \underline{σ}) →^{∼} (End_{A⊗Q} E^{−}, ad_{h^{−}}).

Proof. Let ξ, ζ ∈ E^{−}. We compute in two different ways

τ ⊗ (\underline{σ_{K}} ∘ ψ)(\overline{τ ⊗ (\underline{σ_{K}} ∘ ψ)(f^{−}(ξ, ζ))}) (17)

where τ: A_{K} → A_{K} is the involution defined by τ(x) = σ_{K}(\overline{x}) for x ∈ A_{K}.

First, since f^{−}(ξ, ζ) = g^{−}(ξ, ζ) ⊗ κ and κ satisfies the condition in Lemma 2.2, the expression in (17) is equal to \overline{g^{−}(ξ, ζ)} ⊗ ψ^{2}(κ).

On the other hand, letting ξ = ξ_{0} · κ and ζ = ζ_{0} · κ for some ξ_{0}, ζ_{0} ∈ B(A_{K}, σ_{K}), we have ξ^{ω} = \underline{σ_{K}}(κ) ∗ ξ_{0}^{ω}, hence

f^{−}(ξ, ζ) = μ(\underline{σ_{K}}(κ) ∗ ξ_{0}^{ω} ⊗ ζ_{0} · κ) = 1 ⊗ \underline{σ_{K}}(κ) · μ(ξ_{0}^{ω} ⊗ ζ_{0}) · 1 ⊗ κ.

Since for x ∈ A_{K} and c ∈ C(A_{K}, σ_{K}),

τ ⊗ (\underline{σ_{K}} ∘ ψ)(x ⊗ c · 1 ⊗ κ) = 1 ⊗ (\underline{σ_{K}} ∘ ψ)(κ) · σ_{K} ⊗ \underline{σ_{K}}(\overline{x} ⊗ \overline{c}),

it follows that

τ ⊗ (\underline{σ_{K}} ∘ ψ)(f^{−}(ξ, ζ)) = 1 ⊗ (\underline{σ_{K}} ∘ ψ)(κ) · σ_{K} ⊗ \underline{σ_{K}}(μ(\overline{ξ_{0}}^{ω} ⊗ \overline{ζ_{0}})) · 1 ⊗ \overline{κ}

hence the expression in (17) is equal to

1 ⊗ (\underline{σ_{K}} ∘ ψ)(κ) · μ(\overline{ξ_{0}}^{ω} ⊗ \overline{ζ_{0}}) · 1 ⊗ ψ(κ) = μ((ψ · ξ)^{ω} ⊗ (ψ · ζ)).

Comparing the results, we obtain

λ\overline{g^{−}(ξ, ζ)} = g^{−}(ψ · ξ, ψ · ζ)

where λ ∈ F^{⋆} is such that ψ^{2} = λ Id_{E^{+}}. Substituting η for ψ · ζ, we obtain (16).

Since f^{−} is a regular (−1)^{m}-hermitian form with respect to σ_{K}, it is clear that g^{−} is also a regular (−1)^{m}-hermitian form with respect to σ_{K}. Using (16), we obtain

ℓ^{−}(η, ξ) = \overline{g^{−}(ψ · η, ξ)} = (−1)^{m}\overline{σ_{K}(g^{−}(ξ, ψ · η))} = (−1)^{m}\overline{σ_{K}(ℓ^{−}(ξ, η))}

for ξ, η ∈ E^{−}. It follows that ℓ^{−} is a regular (−1)^{m}-hermitian form with respect to the involution σ ⊗ (= τ) on A_{K}. Straightforward computations show that h^{−} is a (−1)^{m+1}-hermitian form with respect to the involution σ ⊗ on A ⊗_{F} Q. Finally, since ad_{f^{−}} = ad_{g^{−}} corresponds to \underline{σ_{K}} under Θ^{−}, by Proposition 1.5, it is easily checked that

h^{−}(ξ · η, ζ) = h^{−}(η, \underline{σ}(ξ) · ζ) for ξ ∈ C(A, σ) and η, ζ ∈ E^{−},

completing the proof.

As in section 1, we define A_{K}-submodules of E^{−} by

E_{r}^{−} = (A_{K}e)^{b} · E_{r}^{+} ⊂ E^{−},

for r = 0, …, m − 1. We also set

Ê_{r}^{−} = E_{r}^{−} ⊕ E_{m−r−1}^{−} for r < (m − 1)/2.

Proposition 2.9. For r = 0, …, m − 1, we have ψ · E_{r}^{−} = E_{m−r−1}^{−}, hence Ê_{r}^{−} is an A ⊗ Q-submodule of E^{−} for r < m/2, and if m is odd E_{(m−1)/2}^{−} is an A ⊗ Qsubmodule of E^{−}.

If ξ ∈ E_{r}^{−} and η ∈ E_{s}^{−}, then

g^{−}(ξ, η) = 0 unless r + s = m − 1 and ℓ^{−}(ξ, η) = 0 unless r = s.

Therefore, letting h_{r}^{−} (resp. h_{(m−1)/2}^{−}) denote the restriction of h^{−} to Ê_{r}^{−} (resp. E_{(m−1)/2}^{−}, if m is odd), we have

(E^{−}, h^{−}) = \begin{cases}(⨁_{0≤r<(m−1)/2}^{⊥}(Ê_{r}^{−}, h_{r}^{−})) ⊕^{⊥} (E_{(m−1)/2}^{−}, h_{(m−1)/2}^{−}) & if m is odd, \\ ⨁_{0≤r<m/2}^{⊥}(Ê_{r}^{−}, h_{r}^{−}) & if m is even. \end{cases}

Proof. Since E_{r}^{−} = (A_{K}e)^{b} · E_{r}^{+}, we have by Lemmas 2.4 and 2.7

ψ · E_{r}^{−} = \overline{(A_{K}e)^{b}} · ψ(E_{r}^{+}) = (A_{K}e′)^{b} · E_{m−r}^{+}.

Extending scalars to a splitting field of A_{K}, we may verify that

(A_{K}e′)^{b} · E_{m−r}^{+} = (A_{K}e)^{b} · E_{m−r−1}^{+},

hence ψ · E_{r}^{−} = E_{m−r−1}^{−}. The other assertions easily follow, by Proposition 1.5.

Parimala, Sridharan and Suresh [2, Appendix 2, Theorem 2] have defined an exact sequence

W^{ε}(A ⊗ K, σ ⊗ ) ⟶^{R} W^{−ε}(A ⊗ Q, σ ⊗ ) ⟶^{S} W^{ε}(A ⊗ K, σ ⊗ Id_{K})

which is an analogue of (13).

Proposition 2.10. For r < (m − 1)/2, we have R(E_{r}^{−}, ℓ_{r}^{−}) = (Ê_{r}^{−}, h_{r}^{−}), hence S(Ê_{r}^{−}, h_{r}^{−}) = 0. If m is even, S(E^{−}, h^{−}) = 0.

Proof. Straightforward computation.

2.2. Representation of the discriminant algebra of the centralizer. Our next goal is to establish an isomorphism

(D(Ã, σ̃), \underline{σ̃}) →^{∼} \begin{cases}(End_{Q} E_{m/2}^{+}, ad_{h_{m/2}^{+}}) & if m is even, \\ (End_{A⊗Q} E_{(m−1)/2}^{−}, ad_{h_{(m−1)/2}^{−}}) & if m is odd. \end{cases}

We first define a canonical isomorphism Ã = e′A_{K}e′, then use the isomorphism Ψ_{m} of Proposition 1.9 or 1.11.

For simplicity of notation, we let B = e′A_{K}e′. Since \overline{e′} = e = σ_{K}(e′), we have \overline{σ_{K}(x)} ∈ B for x ∈ B. Clearly, the map τ: B → B defined by

τ(x) = \overline{σ_{K}(x)} for x ∈ B

is a unitary involution.

Lemma 2.11. For x ∈ B, we have x + \overline{x} ∈ Ã. The map x ↦ x + \overline{x} is an isomorphism of F -algebras

T: (B, τ) →^{∼} (Ã, σ̃)

such that T (α) = −θ.

Proof. It is clear that x + \overline{x} ∈ A. Since θe′ = −αe′ = e′θ, we have θx = −αx = xθ hence also θ\overline{x} = α\overline{x} = \overline{x}θ for all x ∈ B, hence x + \overline{x} ∈ Ã. Moreover, for x, x′ ∈ B we have \overline{x}, \overline{x′} ∈ eA_{K}e, hence x\overline{x′} = \overline{x}x′ = 0 and therefore

(x + \overline{x})(x′ + \overline{x′}) = xx′ + \overline{xx′}.

It follows that T is an F -algebra homomorphism. It is injective since x + \overline{x} = 0 implies x ∈ (e′A_{K}e′) ∩ (eA_{K}e), hence also surjective by dimension count. To see that τ corresponds to σ̃ under T, observe that for x ∈ B

σ̃(x + \overline{x}) = σ_{K}(x) + σ_{K}(\overline{x}) = T (τ(x)).

The isomorphism T induces a canonical isomorphism

(D(Ã, σ̃), \underline{σ̃}) ≃ (D(B, τ), \underline{τ}). (18)

We may (and shall) therefore work with (B, τ) instead of (Ã, σ̃). Recall from Propositions 1.9 and 1.11 that there are canonical isomorphisms

Ψ_{2r}: λ^{2r}B →^{∼} End_{K} E_{r}^{+}  and  Ψ_{2r+1} : λ^{2r+1}B →^{∼} End_{A_{K}} E_{r}^{−}.

Proposition 2.12. The involution τ^{∧2r} on λ^{2r}B induced by τ corresponds under Ψ_{2r} to the adjoint involution with respect to ℓ_{r}^{+}, the restriction of ℓ^{+} to E_{r}^{+}. Similarly, the involution τ^{∧(2r+1)} corresponds under Ψ_{2r+1} to the adjoint involution with respect to ℓ_{r}^{−}, the restriction of ℓ^{−} to E_{r}^{−}.

Proof. Extending scalars to a splitting field, we are reduced to proving the proposition in the case where A is split. We may thus assume A = End_{F} V for some 4m-dimensional vector space V, and σ = ad_{q} for some quadratic form q on V, and use the notation of Lemma 2.4. In particular, we define ψ: E^{+} → E^{+} by ψ(ξ) = \overline{ξ}υ, where υ = v_{1} … v_{2m}. Computation shows that the element κ = u_{1} … u_{2m} satisfies the condition of Lemma 2.2, and

ψ(κ) = υκ = λw_{1} … w_{2m}κ.

A basis of E_{r}^{+} is given by the products (w_{i_{1}} … w_{i_{2r}}κ)_{1≤i_{1}<···<i_{2r}≤2m}. To compute the Gram matrix of ℓ_{r}^{+} with respect to this basis, observe that

(19) f^{+}(w_{i_{1}} … w_{i_{2r}}κ, ψ(w_{j_{1}} … w_{j_{2r}}κ)) =

\underline{σ_{K}}(κ)w_{i_{2r}} … w_{i_{1}} \overline{w_{j_{1}}} … \overline{w_{j_{2r}}}ψ(κ) =

λλ_{j_{1}}^{−1} … λ_{j_{2r}}^{−1} \underline{σ_{K}}(κ)w_{i_{2r}} … w_{i_{1}}u_{j_{1}} … u_{j_{2r}}w_{1} … w_{2m}κ.

If {i_{1}, …, i_{2r}} ≠ {j_{1}, …, j_{2r}}, say i_{1} ∉ {j_{1}, …, j_{2r}}, then w_{i_{1}} commutes with u_{j_{1}} … u_{j_{2r}}, and (19) vanishes since w_{i_{1}}w_{1} … w_{2m} = 0. Assume now {i_{1}, …, i_{2r}} = {j_{1}, …, j_{2r}}. Since q(u_{i} + w_{i}) = 1, we have u_{i}w_{i} = 1 − w_{i}u_{i} for all i = 1, …, 2m. Substituting 1−u_{i_{1}}w_{i_{1}} for w_{i_{1}}u_{i_{1}} in the rightmost side of (19), the term containing u_{i_{1}}w_{i_{1}} vanishes by the same argument as above. Therefore, we obtain

f^{+}(w_{i_{1}} … w_{i_{2r}}κ, ψ(w_{i_{1}} … w_{i_{2r}})κ) =

λλ_{i_{1}}^{−1} … λ_{i_{2r}}^{−1}\underline{σ_{K}}(κ)w_{1} … w_{2m}κ = λλ_{i_{1}}^{−1} … λ_{i_{2r}}^{−1}κ.

It follows that (w_{i_{1}} … w_{i_{2r}}κ)_{1≤i_{1}<···<i_{2r}≤2m} is an orthogonal basis for ℓ_{r}^{+}, and

ℓ_{r}^{+}(w_{i_{1}} … w_{i_{2r}}κ, w_{i_{1}} … w_{i_{2r}}κ) = λλ_{i_{1}}^{−1} … λ_{i_{2r}}^{−1}. (20)

On the other hand, as in the proof of Lemma 1.8, we have B = End_{K} W, and it is easily checked that the involution τ is adjoint to the hermitian form h_{q} defined by

h_{q}(x, y) = b_{q}(\overline{x}, y) = q(\overline{x} + y) − q(\overline{x}) − q(y) for x, y ∈ W,

where b_{q} is the polar form of q. Thus,

h_{q}(w_{i}, w_{j}) = b_{q}(λ_{i}^{−1}u_{i}, w_{j}) = λ_{i}^{−1}δ_{ij} for i, j = 1, …, 2m.

The involution τ^{∧2r} on λ^{2r}B = End_{K}(⋀^{2r} W) is then adjoint to the hermitian form h_{q}^{∧2r} such that

h_{q}^{∧2r}(x_{1} ∧ ⋯ ∧ x_{2r}, y_{1} ∧ ⋯ ∧ y_{2r}) = det(h_{q}(x_{i}, y_{j}))_{1≤i,j≤2r}.

For 1 ≤ i_{1} < ⋯ < i_{2r} ≤ 2m and 1 ≤ j_{1} < ⋯ < j_{2r} ≤ 2m, we have

h_{q}^{∧2r}(w_{i_{1}} ∧ ⋯ ∧ w_{i_{2r}}, w_{j_{1}} ∧ ⋯ ∧ w_{j_{2r}}) =

\begin{cases}0 & if \{i_{1}, …, i_{2r}\} ≠ \{j_{1}, …, j_{2r}\}, \\ λ_{i_{1}}^{−1} … λ_{i_{2r}}^{−1} & if \{i_{1}, …, i_{2r}\} = \{j_{1}, …, j_{2r}\}. \end{cases}

Comparing with (20), we see that the map ⋀^{2r} W → E_{r}^{+} which carries x_{1} ∧⋯∧x_{2r} to x_{1} … x_{2r}κ is a similitude of (⋀^{2r} W, h_{q}^{∧2r}) with (E_{r}^{+}, ℓ_{r}^{+}). Therefore, the adjoint involutions τ^{∧2r} and ad_{ℓ_{r}^{+}} correspond to each other.

The same computations as above show that for v, v′ ∈ V and 1 ≤ i_{1} < ⋯ < i_{2r+1} ≤ 2m, 1 ≤ j_{1} < ⋯ < j_{2r+1} ≤ 2m,

ℓ_{r}^{−}(v ⊗ w_{i_{1}} … w_{i_{2r+1}}κ, v′ ⊗ w_{j_{1}} … w_{j_{2r+1}}κ) =

\begin{cases}0 & if \{i_{1}, …, i_{2r+1}\} ≠ \{j_{1}, …, j_{2r+1}\}, \\ λλ_{i_{1}}^{−1} … λ_{i_{2r+1}}^{−1}v ⊗ v′ & if \{i_{1}, …, i_{2r+1}\} = \{j_{1}, …, j_{2r+1}\}. \end{cases}

Therefore, the map V ⊗ ⋀^{2r+1} W → E_{r}^{−} which carries v ⊗ (x_{1} ∧ ⋯ ∧ x_{2r+1}) to v ⊗ x_{1} … x_{2r+1}κ is a similitude of the form ℓ_{r}^{−} and the form on V ⊗ ⋀^{2r+1} W with values in V ⊗ V which maps (v ⊗ ξ, v′ ⊗ ξ′) to v ⊗ v′h_{q}^{∧2r+1}(ξ, ξ′) for v, v′ ∈ V and ξ, ξ′ ∈ ⋀^{2r+1} W. Therefore, the adjoint involutions ad_{l_{r}^{−}} and τ^{∧2r+1} correspond to each other.

Corollary 2.13. If m is even, the map Ψ_{m} induces an isomorphism of algebras with involution

Ψ_{m}: (D(B, τ), \underline{τ}) →^{∼} (End_{Q} E_{m/2}^{+}, ad_{h_{m/2}^{+}}).

Similarly, if m is odd, the map Ψ_{m} induces an isomorphism of algebras with involution

Ψ_{m}: (D(B, τ), \underline{τ}) →^{∼} (End_{A⊗Q} E_{(m−1)/2}^{−}, ad_{h_{(m−1)/2}^{−}}).

Proof. Recall from [14, §10.E] that D(B, τ) is the F -subalgebra of λ^{m}B fixed under the automorphism τ^{∧m} ∘ γ.

Suppose first that m is even. Proposition 1.9 shows that γ corresponds to ad_{f_{m/2}^{+}} = ad_{g_{m/2}^{+}} under Ψ_{m}, and Proposition 2.12 shows that τ^{∧m} corresponds

to ad_{ℓ_{m/2}^{+}}. Therefore, Ψ_{m} maps D(B, τ) to the F -subalgebra of End_{K} E_{m/2}^{+} fixed

under ad_{ℓ_{m/2}^{+}} ∘ ad_{g_{m/2}^{+}}. We claim that for f ∈ End_{K} E_{m/2}^{+},

ad_{ℓ_{m/2}^{+}} ∘ ad_{g_{m/2}^{+}}(f) = ψ^{−1} ∘ f ∘ ψ

or, equivalently,

ad_{ℓ_{m/2}^{+}}(f) = ψ^{−1} ∘ ad_{g_{m/2}^{+}}(f) ∘ ψ.

To check the latter equality, observe that for ξ, η ∈ E_{m/2}^{+},

ℓ^{+}(ξ, f(η)) = \overline{g^{+}(ψ(ξ), f(η))} =

\overline{g^{+}(ad_{g^{+}}(f) ∘ ψ(ξ), η)} = ℓ^{+}(ψ^{−1} ∘ ad_{g^{+}}(f) ∘ ψ(ξ), η).

Therefore, the subalgebra of End_{K} E_{m/2}^{+} fixed under ad_{ℓ_{m/2}^{+}} ∘ ad_{g_{m/2}^{+}} is the central-

izer of ψ, which is End_{Q} E_{m/2}^{+}. Clearly, ad_{ℓ_{m/2}^{+}} and ad_{g_{m/2}^{+}} both restrict to ad_{h_{m/2}^{+}}

on End_{Q} E_{m/2}^{+}.

If m is odd, it follows from Propositions 1.11 and 2.12 that Ψ_{m} maps D(B, τ) to the F -subalgebra of End_{A_{K}} E_{(m−1)/2}^{−} fixed under ad_{ℓ_{(m−1)/2}^{−}} ∘ ad_{g_{(m−1)/2}^{−}}. The same

computation as above (substituting ℓ_{(m−1)/2}^{−} for ℓ_{m/2}^{+} and g_{(m−1)/2}^{−} for g_{m/2}^{+}) shows that this subalgebra is the centralizer of ψ, which is End_{A⊗Q} E_{(m−1)/2}^{−}.

3. Algebras of exponent 2 with unitary involution

In this section, we depart from the viewpoint laid out in the introduction, and take as starting point a central simple algebra B of degree 2m over a quadratic extension K of the base field F. We assume that B has exponent 2 and carries a unitary involution τ which is the identity on F. Our goal is to construct a central simple F -algebra B̂ of degree 4m which contains B, and an orthogonal involution τ̂ on B̂ which extends τ. Then B is the centralizer of K in B̂, so the results of section 2 apply to relate the discriminant algebra D(B, τ) to the Clifford algebra C(B̂, τ̂).

Since B has exponent 2, we may find on B an orthogonal involution ν, by [14, (3.1), (2.8)].

Lemma 3.1. There exists u ∈ B^{×} such that ν(u) = τ(u) = u and (τ ∘ ν)^{2}(x) = uxu^{−1} for all x ∈ B.

Proof. The composite τ ∘ ν ∘ τ is an involution on B of the same type as ν, since Sym(B, τ ∘ν ∘τ) = τ(Sym(B, ν)) has the same dimension as Sym(B, ν). Therefore, we may find u_{0} ∈ B^{×} such that ν(u_{0}) = u_{0} and

τ ∘ ν ∘ τ = Int(u_{0}) ∘ ν

(where Int(u_{0}) is the inner automorphism which maps x ∈ B to u_{0}xu_{0}^{−1}), hence (τ ∘ν)^{2} = Int(u_{0}). Then Int(u_{0}) commutes with τ ∘ν, hence τ ∘ν(u_{0}) ≡ u_{0} mod K^{×}. Since ν(u_{0}) = u_{0}, it follows that τ(u_{0}) = λu_{0} for some λ ∈ K^{×}. Then N_{K/F} (λ) = 1, and Hilbert’s Theorem 90 (see for instance [14, (29.3)]) yields λ_{0} ∈ K^{×} such that λ = λ_{0}\overline{λ_{0}}^{−1}. The element u = λ_{0}u_{0} meets the requirements.

Let u ∈ B^{×} be as in Lemma 3.1. Define an F -algebra

B̂ = B ⊕ Bz

by the following multiplication rules:

z^{2} = u  and  zb = τ ∘ ν(b)z for b ∈ B.

Define also τ̂: B̂ → B̂ by

τ̂(b_{1} + b_{2}z) = τ(b_{1}) + zτ(b_{2}).

Proposition 3.2. The F -algebra B̂ is central simple of degree 4m, and τ̂ is an orthogonal involution on B̂.

Proof. To prove the first part, observe that the algebra B̂ is constructed from B by a “generalized crossed product” process, see Albert [1, Chapter XI, Theorem 10]. It is readily verified that τ̂ is an involution on B̂. Since ν is orthogonal and

Sym(B̂, τ̂) = Sym(B, τ) ⊕ z Sym(B, ν),

it follows that

dim Sym(B̂, τ̂) = (2m)^{2} + 2m(2m + 1) = \frac{4m(4m + 1)}{2},

hence τ̂ is orthogonal.

Remark. The element u is determined by the conditions in Lemma 3.1 up to a factor in F^{×}. The additivity property for crossed products (see [11, (1.15)]) shows that substituting u′ = λu for u in the definition of B̂ yields an algebra B̂′ which is Brauerequivalent to B̂ ⊗_{F} (a, λ)_{F}, if K = F (\sqrt{a}). On the other hand, suppose C is an arbitrary central simple F -algebra of degree 4m containing B. Left multiplication defines an F -algebra homomorphism C → End_{B} C, where C is viewed as a right B-module, hence C ⊗_{F} K ≃ End_{B} C ≃ M_{2}(B). Therefore, C ⊗_{F} K ≃ B̂ ⊗_{F} K, and it follows that C is Brauer-equivalent to B̂ ⊗_{F} (a, λ)_{F} for some λ ∈ F^{×}. Thus, any central simple F -algebra of degree 4m containing B can be obtained by the construction above.

4. Algebras of degree 8

In this section, we state the main results before giving the proofs, which are based on the results of section 2 and on the strong “triality” relationship between a degree 8 central simple algebra with orthogonal involution and its Clifford algebra described in [14, §42].

Let A be a central simple algebra endowed with an orthogonal involution σ. As mentioned in the introduction, if A contains a square-central skew-symmetric unit θ, the discriminant of σ is trivial. From Proposition 2.3, we get moreover that one of the components of C(A, σ), namely C^{+}(A, σ), has index at most 2. When A has degree 8, these conditions turn out to be sufficient for the existence of such an element θ, as the following theorem shows:

Theorem 4.1. Let (A, σ) be a central simple algebra of degree 8 with orthogonal involution. The following conditions are equivalent:

(1) A contains a square-central skew-symmetric unit;

(2) the discriminant of σ is trivial, so C(A, σ) = C_{1} × C_{2} for some central simple F -algebras C_{1}, C_{2} of degree 8, and at least one of C_{1}, C_{2} has index 1 or 2.

If these conditions hold, then A contains square-central units θ_{1}, θ_{2}, θ_{3} ∈ Skew(A, σ) such that F (θ_{1}, θ_{2}, θ_{3}) is an étale 8-dimensional commutative F -subalgebra of A.

Remark. Using this theorem, one may easily find degree 8 central simple algebras with orthogonal involution which do not contain any square-central skew-symmetric element. For split examples, use for instance [14, (42.11)].

The following result is proven in [14, (42.11)]:

Theorem 4.2 (Knus–Merkurjev–Rost–Tignol). Let (A, σ) be a central simple algebra of degree 8 with orthogonal involution. The following conditions are equivalent:

(1) (A, σ) = (A_{1}, σ_{1}) ⊗ (A_{2}, σ_{2}) ⊗ (A_{3}, σ_{3}) for some quaternion algebras with involution (A_{i}, σ_{i}), i = 1, 2, 3;

(2) the discriminant of σ is trivial, so C(A, σ) = C_{1} × C_{2} for some central simple F -algebras C_{1}, C_{2} of degree 8, and at least one of C_{1}, C_{2} splits.

If (A, σ) is decomposable, so that

(A, σ) = (H, σ_{H}) ⊗_{F} (A_{0}, σ_{0}) (21)

where H is a σ-stable quaternion subalgebra of A and A_{0} is of degree 4, it contains a square-central skew-symmetric unit, and hence satisfies the equivalent conditions of Theorem 4.1. On the other hand, Example 4.5 below shows that (A, σ) is not always isomorphic to a tensor product of three quaternion algebras with involution.

The next result gives a criterion of decomposability for (A, σ).

Theorem 4.3. Let (A, σ) be a central simple algebra of degree 8 with orthogonal involution. The following conditions are equivalent:

(1) (A, σ) is decomposable, i.e. (A, σ) = (H, σ_{H})⊗_{F} (A_{0}, σ_{0}) where H is a quaternion algebra and A_{0} is of degree 4;

(2) the discriminant of σ is trivial, so C(A, σ) = C_{1} × C_{2} for some central simple F -algebras C_{1}, C_{2} of degree 8, and at least one of C_{1}, C_{2} contains a squarecentral symmetric unit whose centralizer is split.

Remarks 4.4. 1. Let g ∈ C_{i} be a square-central symmetric unit as above. Its centralizer is Brauer-equivalent to C_{i} ⊗ F [g]. Hence, the splitting hypothesis implies that C_{i} has index at most 2. If C_{i} is represented as (End_{Q} V, ad_{h}) for some quaternion division algebra Q, the existence of such an element g is equivalent the existence of a certain type of diagonalisation for h (see the appendix). Note that if g ∈ F^{×2}, then F [g] is isomorphic to F ×F and by the hypothesis that the centralizer splits, we mean that C_{i} splits over F.

2. In the situation of Theorem 4.3, we may always find a decomposition of (A, σ) as (21) in which the involutions σ_{H} and σ_{0} are orthogonal. Indeed, if they are symplectic, then it follows from the second proof of [14, (16.1)] (see also (16.16) and exercise 2 of chapter II) that there is a decomposition

(A_{0}, σ_{0}) = (H′, σ′) ⊗_{F} (H′′, σ′′)

where H′, H′′ are quaternion algebras, and σ′ (resp. σ′′) is a symplectic (resp. orthogonal) involution. Substituting H′′ for H, we are reduced to the case where the restriction of σ to H is orthogonal.

The following examples show that the conditions of Theorems 4.1, 4.2 and 4.3 are not equivalent.

Example 4.5. Let ⟨1, u, v, w⟩ be a quadratic form over F with non-trivial discriminant. Consider the algebra A = M_{8}(K) over the field K = F (t), where t is an indeterminate over F, endowed with the adjoint involution with respect to the quadratic form q = ⟨1, t⟩ ⊗ ⟨1, u, v, w⟩. The algebra (A, σ) is decomposable, and to prove it does not decompose as a tensor product of three σ-invariant quaternion subalgebras, it suffices to prove, according to Theorem 4.2, that both components of C(A, σ) are non-split. To compute C(A, σ), we proceed as follows: consider an orthogonal basis (e_{1}, e_{2}, …, e_{8}) of q = ⟨1, u, v, w, t, tu, tv, tw⟩. One may check that the elements e_{1}e_{2}, e_{1}e_{3}, e_{1}e_{2}e_{3}e_{4}, e_{1}e_{2}e_{3}e_{5}, e_{1}e_{2}e_{3}e_{4}e_{5}e_{6} and e_{1}e_{2}e_{3}e_{4}e_{5}e_{7} generate a subalgebra of C(A, σ) = C_{0}(q) which is the tensor product of three quaternion algebras B = (u, v) ⊗ (uvw, uvt) ⊗ (vw, uw). Since q has trivial discriminant, the center of C_{0}(q) is isomorphic to F × F and both components of C(A, σ) are isomorphic to B, which is Brauer equivalent to (uvw, uvwt) and hence is not split.

Since (A, σ) is decomposable, the conditions of Theorem 4.3 hold. A squarecentral symmetric unit whose centralizer is split may here be explicitly found: the element e_{1}e_{2}e_{3}e_{4} has the required properties.

Example 4.6. Let A be the central simple algebra of degree 8 and exponent 2 constructed in [10, (5.7.37)]. As explained there, A contains a square-central element θ, but does not contain any quaternion subalgebra. By Theorem (4.14) of [14] we may endow A with an involution σ of orthogonal type which extends the canonical involution of the quadratic field extension F (θ)/F. Hence (A, σ) satisfies the equivalent conditions of Theorem 4.1, but not those of Theorem 4.3 since A itself is not decomposable.

Finally, we give a new version of Theorem 4.2 which, for a given choice of θ, yields the more precise decomposition described in (3), under the hypothesis that the split component of C(A, σ) is C(A, σ)^{+}.

Theorem 4.7. Let (A, σ) be a central simple F -algebra of degree 8 with orthogonal involution and let θ ∈ A be a square-central skew-symmetric unit. The following statements are equivalent:

(1) C^{+}(A, σ) is split;

(2) D(Ã, σ̃) is split;

(3) (A, σ) = (A_{1}, σ_{1}) ⊗ (A_{2}, σ_{2}) ⊗ (A_{3}, σ_{3}) for some quaternion algebras with involution (A_{i}, σ_{i}), i = 1, 2, 3, and θ ∈ A_{1}.

Moreover, if these conditions hold, we may find a decomposition of (A, σ) as above such that σ_{1} = Int(θ) ∘ γ_{1}, σ_{2} = γ_{2} and σ_{3} = γ_{3}, where γ_{i} denotes the canonical involution on A_{i}.

Remark. The choice of θ is essential in determining which of the two components of C(A, σ) is C^{+}(A, σ) (see Proposition 4.13).

Let us now prove those results. We start with the following:

Lemma 4.8. Let (A, σ) be a central simple F -algebra with orthogonal involution of even degree, and let e_{1}, …, e_{r} ∈ A be symmetric orthogonal idempotents such that e_{1} +⋯+e_{r} = 1. For i = 1, …, r, let A_{i} = e_{i}Ae_{i}, and let σ_{i} be the restriction of σ to A_{i}. Assume each A_{i} has even degree. We may then identify C(A_{i}, σ_{i}) to an F -subalgebra of C(A, σ). For s_{1} ∈ Skew(A_{1}, σ_{1}), …, s_{r} ∈ Skew(A_{r}, σ_{r}), we have s_{1} + ⋯ + s_{r} ∈ Skew(A, σ) and

π(s_{1} + ⋯ + s_{r}) = π_{1}(s_{1}) ⋯ π_{r}(s_{r}),

where π: Skew(A, σ) → Z(C(A, σ)), π_{1} : Skew(A_{1}, σ_{1}) → Z(C(A_{1}, σ_{1})), …, π_{r}: Skew(A_{r}, σ_{r}) → Z(C(A_{r}, σ_{r})) are the generalized pfaffian maps to the centers of the Clifford algebras. Moreover, the elements π_{1}(s_{1}), …, π_{r}(s_{r}) pairwise commute and F (π_{1}(s_{1}), …, π_{r}(s_{r})) is an étale F -algebra of dimension 2^{r} containing the center of C(A, σ). If the center is isomorphic to F × F, and if f is an idempotent ≠ 0, 1 in the center of C(A, σ), then F (π_{1}(s_{1}), …, π_{r}(s_{r})) · f = F (π_{1}(s_{1}), …, π_{r−1}(s_{r−1})) · f is an F -algebra of dimension 2^{r−1}.

Proof. The inclusion A_{i} ↪ A induces an algebra homomorphism on the tensor algebras of the underlying vector spaces T (A_{i}) → T (A), which induces an injective algebra homomorphism C(A_{i}, σ_{i}) → C(A, σ), see Dejaiffe [4]. (It suffices to check this last part in the split case. If A = End_{F} V, then A_{i} = End_{F} V_{i} for some evendimensional regular subspace V_{i} ⊂ V, and the map C(A_{i}, σ_{i}) → C(A, σ) reduces to the canonical map C_{0}(V_{i}) ↪ C_{0}(V).)

For s_{1} ∈ Skew(A_{1}, σ_{1}), …, s_{r} ∈ Skew(A_{r}, σ_{r}), it is clear that s_{1} + ⋯ + s_{r} ∈ Skew(A, σ). To prove that π_{1}(s_{1}), …, π_{r}(s_{r}) pairwise commute and that π(s_{1} + ⋯ + s_{r}) = π_{1}(s_{1}) ⋯ π_{r}(s_{r}), we may extend scalars to a splitting field of A. The previous equality implies in particular that π_{1}(s_{1}) ⋯ π_{r}(s_{r})f ∈ F^{⋆}f so that F (π_{1}(s_{1}), …, π_{r}(s_{r})) · f = F (π_{1}(s_{1}), …, π_{r−1}(s_{r−1})) · f.

Proof of Theorem 4.1. We only have to prove (2) ⇒ (1). Suppose C_{1} has index 1 or 2, and let σ_{1} be the restriction to C_{1} of the canonical involution of C(A, σ). We may then represent C_{1} = End_{Q} V, σ_{1} = ad_{h} for some quaternion F -algebra Q and some skew-hermitian Q-space (V, h) of rank 4, and we have

C(End_{Q} V, ad_{h}) ≃ A × C_{2}

by triality [14, (42.3)]. Consider an orthogonal decomposition

V = V_{1} ⊕^{⊥} V_{2} ⊕^{⊥} V_{3} ⊕^{⊥} V_{4}

where V_{1}, …, V_{4} are Q-subspaces of rank 1. For i = 1, …, 4, let e_{i} ∈ C_{1} be the orthogonal projection V → V_{i} ⊂ V. The elements e_{1}, …, e_{4} are symmetric orthogonal idempotents such that e_{1} + ⋯ + e_{4} = 1, and e_{i}C_{1}e_{i} is a quaternion F -algebra isomorphic to Q for i = 1, …, 4.

Let s_{i} ∈ e_{i}C_{1}e_{i} be an invertible element such that σ(s_{i}) = −s_{i} for i = 1, …, 4 and consider the generalized pfaffian

π_{i}(s_{i}) ∈ Z(C(e_{i}C_{1}e_{i}, σ_{1}|_{e_{i}C_{1}e_{i}})) ⊂ C(C_{1}, σ_{1}) = A × C_{2}.

Let

π_{i}(s_{i}) = (θ_{i}, θ′_{i}) ∈ A × C_{2}.

Since π_{i}(s_{i}) is skew-symmetric for the canonical involution of C(C_{1}, σ_{1}), we have σ(θ_{i}) = −θ_{i} for i = 1, …, 4. Moreover,

π_{i}(s_{i})^{2} = Nrd_{e_{i}C_{1}e_{i}}(s_{i}) ∈ F^{⋆},

hence θ_{1}^{2}, …, θ_{4}^{2} ∈ F^{⋆}.

The last statement is a consequence of lemma 4.8.

Remark 4.9. The idea to use an orthogonal basis of V to produce skew-symmetric units in the Clifford algebra of End_{Q} V is borrowed from [15].

Proof of Theorem 4.3: “if” part. Suppose C_{1} contains a unit g such that g^{2} ∈ F^{⋆}, σ_{1}(g) = g, where σ_{1} is the canonical involution on C_{1}, and the centralizer of g in C_{1} is split.

If C_{1} is split (see Remark 4.4), then the existence of a stable decomposition of A into a tensor product of quaternion subalgebras follows from Theorem 4.2. We may thus assume C_{1} is not split for the rest of the proof of the “if” part.

Then F [g] is a quadratic field extension of F and the index of C_{1} is 2. As in the proof of Theorem 4.1, we may choose a representation C_{1} = End_{Q}(V), σ_{1} = ad_{h} for some quaternion division F -algebra Q split by F [g] and some 4-dimensional skew-hermitian Q-space (V, h). Let j ∈ Q be a pure quaternion such that j^{2} = g^{2}.

We claim that there exists an anisotropic vector v ∈ V such that g(v) = vj. Assuming this result, we may complete the proof by considering s ∈ End_{Q} V defined by

s(x) = vh(v, x) for x ∈ V.

For x, y ∈ V we have

h(s(x), y) = \overline{h(v, x)}h(v, y) = −h(x, v)h(v, y) = −h(x, s(y)),

so s is skew-symmetric. Moreover, since σ_{1}(g) = g we have

s(g(x)) = vh(v, g(x)) = −vjh(v, x),

whereas

g(s(x)) = g(v)h(v, x) = vjh(v, x),

so s ∘ g = −g ∘ s.

Consider now the Clifford algebra

C(C_{1}, σ_{1}) = A × C_{2},

and let c(s) = (θ, θ′) be the image of s in A × C_{2} under the canonical map c: C_{1} → C(C_{1}, σ_{1}). If we call e the orthogonal projection on vF, s belongs to eC_{1}e which is a quaternion algebra. Hence, the restriction of c to eC_{1}e coincides with the generalized pfaffian π: Skew(eC_{1}e, σ_{1}|_{eC_{1}e}) → Z(C(eC_{1}e, σ_{1}|_{eC_{1}e}). Hence, as in the proof of Theorem 4.1, the element θ ∈ A is a square-central skew-symmetric unit. The inner automorphism Int(g) commutes with σ_{1} since σ_{1}(g)g ∈ F^{⋆}. Therefore, it induces an automorphism C(Int(g)) of the Clifford algebra C(C_{1}, σ_{1}). This automorphism is the identity on the center F ×F, since Nrd_{C_{1}}(g) = g^{8} = (σ_{1}(g)g)^{4}. Its restriction to A is therefore an inner automorphism Int(ρ) for some ρ ∈ A. Since g^{2} ∈ F^{⋆}, we have Int(g)^{2} = Id_{C_{1}}, hence also Int(ρ)^{2} = Id_{A}. Thus, ρ^{2} ∈ F^{⋆}. Moreover, C(Int(g)) commutes with the canonical involution of C(C_{1}, σ_{1}), hence Int(ρ) commutes with σ, and therefore σ(ρ)ρ ∈ F^{⋆}. Finally, since

C(Int(g))(c(s)) = c(gsg^{−1}) = −c(s),

we have ρθ = −θρ. It follows that the subalgebra H of A generated by ρ and θ is a quaternion algebra stable under σ. The proof of the “if” part is thus complete if we prove the claim on the existence of v ∈ V. This claim follows from the next lemma.

Lemma 4.10. Let (V, h) be a skew-hermitian space over some quaternion division algebra Q. Assume g ∈ End_{Q} V is symmetric for the adjoint involution ad_{h} and satisfies g^{2} = j^{2} for some (non-zero) pure quaternion j ∈ Q. Then there exists an orthogonal basis (v_{1}, …, v_{n}) of V such that g(v_{ℓ}) = v_{ℓ}j for ℓ = 1, …, n.

Proof. Let L = F (j) ⊂ Q and let i ∈ Q be a non-zero pure quaternion which anticommutes with j, so that Q = L ⊕ iL. We may then decompose the skewhermitian form h by letting

h(x, y) = h_{0}(x, y) + ih_{1}(x, y)

with h_{0}(x, y), h_{1}(x, y) ∈ L for x, y ∈ V. The form h_{0} : V × V → L is skewhermitian, and h_{1} is symmetric bilinear (compare [16, p. 359]). Consider now ψ ∈ End_{L} V defined by

ψ(x) = \frac{1}{2}(x − g(x)j^{−1}) for x ∈ V.

Since g^{2} = j^{2}, we have ψ^{2} = ψ, hence the L-vector space V decomposes as

V = ker ψ ⊕ im ψ.

For x ∈ V, we have

ψ(xi) = \frac{1}{2}(x + g(x)j^{−1})i = (x − ψ(x))i,

hence (ker ψ)i = im ψ. Therefore,

dim_{L} ker ψ = dim_{L} im ψ = dim_{Q} V,

and every L-basis of ker ψ is a Q-basis of V. For x, y ∈ ker ψ, we have g(x) = xj, g(y) = yj, hence the equation h(g(x), y) = h(x, g(y)) yields

−jh(x, y) = h(x, y)j

and it follows that h_{0}(x, y) = 0. Therefore, every h_{1}-orthogonal basis (v_{1}, …, v_{n}) of ker ψ is an h-orthogonal basis of V such that g(v_{ℓ}) = v_{ℓ}j for ℓ = 1, …, n.

Remark 4.11. The same arguments can be used in the case where g is skew-sym-metric. Then h_{1}(x, y) = 0 for all x, y ∈ ker ψ, and every h_{0}-orthogonal basis (v_{1}, …, v_{n}) of ker ψ is an h-orthogonal basis of V such that g(v_{ℓ}) = v_{ℓ}j for ℓ = 1, …, n.

To prove the “only if” part of Theorem 4.3, we shall need the following result of independent interest:

Proposition 4.12. Let B be a central simple algebra of degree 4 over a quadratic field extension K of F, and let τ be a unitary involution on B whose restriction to K is the non-trivial automorphism of K/F. There exists an F -subalgebra B_{0} ⊂ B stable under τ such that B = B_{0} ⊗_{F} K if and only if the discriminant algebra D(B, τ) contains a square-central unit g of reduced trace zero which is symmetric under the canonical involution \underline{τ}. Moreover, if this condition holds, then D(B, τ) is Brauer-equivalent to the quaternion algebra (a, λ)_{F}, where K = F (\sqrt{a}) and λ = g^{2}.

Proof. If it exists, the F -subalgebra B_{0} is the algebra of fixed points of a semi-linear automorphism φ of B such that φ ∘ τ = τ ∘ φ and φ^{2} = Id_{B}. By [14, (15.26)](see also (15.24)), we have

Aut_{F} (B, τ) = Aut_{F} (D(B, τ), \underline{τ}),

and the semi-linear automorphisms of (B, τ) correspond to inner automorphisms of D(B, τ) induced by improper similitudes. Therefore, an automorphism φ as above exists if and only if there is in D(B, τ) an element y such that

\underline{τ}(y)y ∈ F^{⋆}, y^{2} ∈ F^{⋆} and Nrd_{D(B,τ)}(y) = −(\underline{τ}(y)y)^{3}. (22)

It remains to prove that such an element y exists if and only if there is in D(B, τ) a square-central symmetric unit of reduced trace zero.

Suppose first g ∈ D(B, τ) is such that g^{2} ∈ F^{⋆}, \underline{τ}(g) = g and Trd_{D(B,τ)}(g) = 0. If g^{2} = λ, the reduced characteristic polynomial of g has the form (X^{2} − λ)^{3}, hence

Nrd_{D(B,τ)}(g) = −λ^{3} = −(\underline{τ}(g)g)^{3}

and we may take y = g.

Conversely, if there is in D(B, τ) an element y satisfying (22), let \underline{τ}(y)y = μ and y^{2} = λ. If λ ∉ F^{⋆2}, then the reduced characteristic polynomial of y is (X^{2} − λ)^{3}, hence Trd_{D(B,τ)}(y) = 0, Nrd_{D(B,τ)}(y) = −λ^{3} and (22) implies μ^{3} = λ^{3}. For ω = μλ^{−1}, we thus have ω^{3} = 1 and \underline{τ}(y)y = ωy^{2}, hence \underline{τ}(y) = ωy. This last equation implies ω^{2} = 1, hence ω = 1. It follows that y is a square-central symmetric unit of reduced trace zero.

If λ ∈ F^{⋆2}, let λ = λ_{0}^{2} with λ_{0} ∈ F^{⋆}. The reduced characteristic polynomial of y then has the form (X − λ_{0})^{m_{1}}(X + λ_{0})^{m_{2}} with m_{1} + m_{2} = 6, hence Nrd_{D(B,τ)}(y) = (−1)^{m_{1}}λ^{3}. Conditions (22) yield −μ^{3} = (−1)^{m_{1}}λ^{3}. Arguing as in the preceding case, we obtain \underline{τ}(y) = (−1)^{m_{1}+1}y. If m_{1} is even, we have \underline{τ}(y) = −y, hence −y has the same reduced characteristic polynomial as y. This is a contradiction since the reduced characteristic polynomial of −y is (X − λ_{0})^{m_{2}}(X + λ_{0})^{m_{1}} and m_{1} even implies m_{1} ≠ m_{2}.

If m_{1} = m_{2} = 3, then y is a square-central symmetric unit of trace zero. Finally, if m_{1} = 1 (resp. 5), then the right ideal generated by y − λ_{0} (resp. y + λ_{0}) has dimension 6 (note that y is diagonalisable), hence the algebra D(B, τ) is split, and we may represent D(B, τ) as the endomorphism algebra of some 6-dimensional vector space, and \underline{τ} as the adjoint involution with respect to some symmetric bilinear form. The orthogonal reflection with respect to any non-singular 3-dimensional subspace is a square-central symmetric unit of trace zero.

To prove the last statement, observe that g is an improper similitude of D(B, τ) with multiplier λ, hence Theorem (13.38) of [14] shows that D(B, τ) is Brauerequivalent to the quaternion algebra (disc \underline{τ}, λ)_{F}. By the equivalence A_{3} ≡ D_{3} (see [14, (15.24)]), the algebra B is isomorphic to the Clifford algebra of (D(B, τ), \underline{τ}), hence disc \underline{τ} = a · F^{⋆2} and the proof is complete.

Proof of Theorem 4.3: “only if” part. As observed in Remark 4.4, we may assume

(A, σ) = (H, σ_{H}) ⊗_{F} (A_{0}, σ_{0})

where σ_{H} is an orthogonal involution on the quaternion algebra H. Let θ ∈ H be a skew-symmetric unit. Since H is a quaternion algebra, θ is square-central.

Let a = θ^{2} ∈ F^{⋆}. If a ∈ F^{⋆2}, then the discriminant of σ_{H} is trivial, hence σ_{H} is hyperbolic by [14, (7.4)]. It follows that σ is also hyperbolic, hence one of C_{1}, C_{2} is split, by [14, (8.31)]. It is easy to find in the split component a squarecentral symmetric unit of trace zero (for instance an orthogonal reflection), hence the theorem follows.

For the rest of the proof, we may thus assume a ∉ F^{⋆2}, and apply the results of section 2. These results yield a quaternion F -algebra Q and a representation

(C^{+}(A, σ), \underline{σ}) →^{∼} (End_{Q} E^{+}, ad_{h^{+}})

for some skew-hermitian form h^{+} (see Proposition 2.3). Proposition 2.5 yields an orthogonal decomposition

E^{+} = Ê_{0}^{+} ⊕^{⊥} E_{1}^{+},

where dim_{Q} Ê_{0}^{+} = 1 and dim_{Q} E_{1}^{+} = 3, and Corollary 2.13 yields a representation

(D(Ã, σ̃), ˜ \underline{σ}) →^{∼} (End_{Q} E_{1}^{+}, ad_{h_{1}^{+}}),

where h_{1}^{+} is the restriction of h^{+} to E_{1}^{+}. Since A_{0} is a σ̃-stable F -subalgebra of Ã such that Ã = F (θ) ⊗_{F} A_{0}, Proposition 4.12 shows that D(Ã, σ̃) contains a squarecentral symmetric unit of reduced trace zero. We may thus find g_{1} ∈ End_{Q} E_{1}^{+} such that ad_{h_{1}^{+}}(g_{1}) = g_{1} and g_{1}^{2} ∈ F^{⋆}. Let λ = g_{1}^{2}. Proposition 4.12 also shows that D(Ã, σ̃) is Brauer-equivalent to the quaternion algebra (a, λ)_{F}, hence Q = (a, λ)_{F}.

Since disc σ = 1, we also have disc \underline{σ} = 1 by triality [14, (42.3)]. On the other hand, disc ˜ \underline{σ} = a · F^{⋆2} because Ã is the Clifford algebra of (D(Ã, σ̃), ˜ \underline{σ}) by [14, (15.24)]. Therefore, the restriction of ad_{h^{+}} to End_{Q} Ê_{0}^{+} has discriminant a · F^{⋆2}. Since dim_{Q} Ê_{0}^{+} = 1, we have End_{Q} Ê_{0}^{+} ≃ Q, and there is an isomorphism under which the restriction of ad_{h^{+}} to End_{Q} Ê_{0}^{+} corresponds to the involution of Q = (a, λ)_{F} which maps the elements i, j of the standard quaternion basis to −i, j respectively, since this orthogonal involution also has discriminant a · F^{⋆2} (see [14, (7.4)]). The element g_{0} ∈ End_{Q} Ê_{0}^{+} corresponding to j is such that ad_{h^{+}}(g_{0}) = g_{0}

and g_{0}^{2} = λ. Now, the endomorphism g = g_{0} ⊕ g_{1} of E^{+} = Ê_{0}^{+} ⊕^{⊥} E_{1}^{+} satisfies ad_{h^{+}}(g) = g and g^{2} = λ ∈ F^{⋆}, and its centralizer is split.

Proof of Theorem 4.7. The equivalence of (1) and (2) follows from the Brauerequivalence of C^{+}(A, σ) and D(Ã, σ̃), see Proposition 2.3 and Corollary 2.13.

(3) ⇒ (2) If (3) holds, we have Ã = F (θ) ⊗_{F} Q_{2} ⊗_{F} Q_{3} and σ̃ = ⊗ σ_{2} ⊗ σ_{3}, hence D(Ã, σ̃) is split, by [14, (10.33)].

(2) ⇒ (3) By Karpenko-Quéguiner [13], (2) implies that

(Ã, σ̃) = (F (θ) ⊗ Q_{2} ⊗ Q_{3}, ⊗ γ_{2} ⊗ γ_{3})

for some quaternion F -algebras Q_{2}, Q_{3} with conjugation involutions γ_{2}, γ_{3}. The centralizer of Q_{2} ⊗ Q_{3} in A is a quaternion algebra Q_{1} containing θ.

Finally, since γ_{2} ⊗ γ_{3} is orthogonal, the restriction σ_{1} of σ to Q_{1} must be orthogonal, otherwise σ would be symplectic. Since it maps θ to −θ, it coincides with Int(θ) ∘ γ_{1}

As explained in the introduction, the choice of which of the two components of C(A, σ) is C^{+}(A, σ) depends on the choice of θ. The following proposition makes this fact clear and emphasizes the difference between Theorems 4.7 and 4.2.

Proposition 4.13. Let Q be a quaternion division algebra over an arbitrary field F, and let A = M_{4}(Q). Let also σ be an orthogonal hyperbolic involution on A and let C(A, σ) = C_{1} × C_{2} for some central simple F -algebras C_{1}, C_{2}. Then one of the components (C_{1}, say) is split, and the other one is Brauer-equivalent to Q. There exist θ_{1}, θ_{2} ∈ Skew(A, σ) such that θ_{1}^{2} = θ_{2}^{2} ∈ F^{⋆} and, letting a = θ_{1}^{2} = θ_{2}^{2},

\frac{1}{2}(1 + a^{−2}π(θ_{1})) · C(A, σ) = C_{1}, \frac{1}{2}(1 + a^{−2}π(θ_{2})) · C(A, σ) = C_{2}.

Thus, the equivalent conditions of Theorem 4.7 hold for θ_{1} but not for θ_{2}.

Proof of Proposition 4.13. Note that since hyperbolic involutions on A are isomorphic, it is enough to prove the result for a particular one.

Let A_{0} = M_{2}(F) ⊗_{F} Q and let σ_{0} be the tensor product of the symplectic involutions on M_{2}(F) and Q. Since the symplectic involution on M_{2}(F) is hyperbolic, the involution σ_{0} is hyperbolic. Letting M_{2}(F)^{0}, Q^{0} denote the vector spaces of (reduced) trace zero elements in M_{2}(F) and Q respectively, we have

Skew(A_{0}, σ_{0}) = (M_{2}(F)^{0} ⊗ 1) ⊕ (1 ⊗ Q^{0}).

Moreover, C(A_{0}, σ_{0}) = M_{2}(F)×Q, by [14, (8.19)], and the generalized pfaffian map

π_{0} : Skew(A_{0}, σ_{0}) → Z(C(A_{0}, σ_{0})) = F × F

satisfies

(23)  π_{0}(u ⊗ 1) = (u^{2}, −u^{2})  for u ∈ M_{2}(F)^{0},
(24)  π_{0}(1 ⊗ q) = (−q^{2}, q^{2})  for q ∈ Q^{0},

see [14, (16.32)]. Let

A = M_{2}(F) ⊗_{F} A_{0} (≃ M_{4}(Q))

and let σ = t ⊗ σ_{0}, the tensor product of the transpose involution on M_{2}(F) and σ_{0}. Since σ_{0} is hyperbolic, the involution σ is hyperbolic. The elements

e_{1} = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ⊗ 1 ∈ A, e_{2} = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) ⊗ 1 ∈ A

are symmetric orthogonal idempotents such that e_{1} + e_{2} = 1. Using the same notation as in Lemma 4.8, we have canonical isomorphisms (A_{1}, σ_{1}) = (A_{0}, σ_{0}) = (A_{2}, σ_{2}).

Let q ∈ Q^{0} be an invertible pure quaternion and let a = q^{2} ∈ F^{⋆}. We may find a matrix u ∈ M_{2}(F)^{0} such that u^{2} = a. In A = M_{2}(F) ⊗ M_{2}(F) ⊗ Q, consider the elements

θ_{1} = 1 ⊗ 1 ⊗ q

and

θ_{2} = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ⊗ 1 ⊗ q + ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) ⊗ u ⊗ 1.

We have θ_{1}, θ_{2} ∈ Skew(A, σ) and θ_{1}^{2} = θ_{2}^{2} = a. Lemma 4.8 yields

π(θ_{1}) = π_{1}(1 ⊗ q)π_{2}(1 ⊗ q), π(θ_{2}) = π_{1}(1 ⊗ q)π_{2}(u ⊗ 1).

By (23) and (24), we have π_{2}(1⊗q) = −π_{2}(u⊗1), hence π(θ_{1}) = −π(θ_{2}). Therefore, letting

C_{1} = \frac{1}{2}(1 + a^{−2}π(θ_{1})) · C(A, σ) and C_{2} = \frac{1}{2}(1 − a^{−2}π(θ_{1})) · C(A, σ),

we have C(A, σ) = C_{1} × C_{2} and \frac{1}{2}(1 + a^{−2}π(θ_{2})) · C(A, σ) = C_{2}.

Since condition (3) of Theorem 4.7 holds for θ_{1}, it follows that C_{1} is split. Since C_{1} ⊗_{F} C_{2} is Brauer-equivalent to A by a theorem of Jacobson (see [14, (9.14)]), C_{2} is Brauer-equivalent to A, hence also to Q.

The index of the algebra A in Proposition 4.13 is 2. This turns out to be a general property for algebras exhibiting the behaviour of Proposition 4.13, as the following corollary shows:

Corollary 4.14. Let (A, σ) be a central simple algebra of degree 8 with orthogonal involution of trivial discriminant. Suppose the index of A is not 2 and one of the components of C(A, σ) is split. Then for every square-central unit θ ∈ Skew(A, σ), there is a decomposition of A into a tensor product of σ-invariant quaternion F - subalgebras A = Q_{1} ⊗_{F} Q_{2} ⊗_{F} Q_{3} such that θ ∈ Q_{1} and the restriction of σ to Q_{1} is an orthogonal involution.

Proof. Let C(A, σ) = C_{1} × C_{2} and suppose C_{1} is split. By Jacobson’s theorem (see [14, (9.14)]), it follows that C_{2} is Brauer-equivalent to A. On the other hand, Proposition 2.3 shows that for each square-central unit θ ∈ Skew(A, σ) the index of C^{+}(A, σ) is at most 2. Therefore, we must have C^{+}(A, σ) = C_{1} if the index of A is 4 or 8, and C^{+}(A, σ) ≃ C_{1} ≃ C_{2} if A is split. Thus in all cases C^{+}(A, σ) is split and Theorem 4.7 applies.

Appendix

In this appendix, we show that the existence of a square-central symmetric or skew-symmetric element in the endomorphism algebra of some skew-hermitian space over a quaternion division algebra can be accounted for by a certain type of diagonalisation of the corresponding skew-hermitian form.

Proposition. Let Q be a division quaternion algebra over F, and (V, h) a skewhermitian module over Q. The following conditions are equivalent:

(1) the algebra with involution (End_{Q}(V), ad_{h}) contains a square-central symmetric (resp. skew-symmetric) element g whose centralizer splits;

(2) the hermitian form h admits a diagonalisation of the form h = ⟨q_{1}, …, q_{n}⟩, where q_{1}, …, q_{n} are pure quaternions satisfying q_{ℓ}j = −jq_{ℓ} (resp. h = ⟨λ_{1}j, …, λ_{n}j⟩, where λ_{ℓ} ∈ F^{⋆}), for some pure quaternion j.

If these conditions hold, the element g and the quaternion j are related by j^{2} = g^{2}.

Proof. (1) ⇒ (2) Since the centralizer of g splits, the quaternion algebra Q contains a pure quaternion j satisfying j^{2} = g^{2}. If g is symmetric, we apply Lemma 4.10 to find an orthogonal basis (v_{1}, …, v_{n}) of V such that g(v_{ℓ}) = v_{ℓ}j for ℓ = 1, …, n. The equation h(g(v_{ℓ}), v_{ℓ}) = h(v_{ℓ}, g(v_{ℓ})) then yields q_{ℓ}j = −jq_{ℓ} for q_{ℓ} = h(v_{ℓ}, v_{ℓ}). The same arguments can be used in the case where g is skew-symmetric, see Remark 4.11.

(2) ⇒ (1) Let us consider an orthogonal basis (v_{1}, …, v_{n}) of V over Q in which the form h has a diagonalisation as in (2). The endomorphism g defined by g(v_{ℓ}) = v_{ℓ}j for ℓ = 1, …, n has the required properties.

Note that if g is skew-symmetric and V_{0} is the F -vector space spanned by the basis v_{1}, …, v_{n} constructed in the proof above, then End_{Q} V = Q ⊗_{F} End_{F} V_{0} and Q, End_{F} V_{0} are stable under ad_{h}.

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Anne Quéguiner-Mathieu  Jean-Pierre Tignol
Laboratoire Analyse, Géométrie  Institut de Mathématique Pure
et Applications, UMR 7539  et Appliquée
Institut Galilée  Université catholique de Louvain
Université Paris Nord  B-1348, Louvain-la-Neuve
F-93430 Villetaneuse  Belgium

France

e-mail:  e-mail:
queguin@math.univ-paris13.fr  tignol@agel.ucl.ac.be