On decomposable biquaternion algebras with involution of orthogonal type

A.-H. Nokhodkar

August 9, 2015

Abstract We investigate the pfaffians of decomposable biquaternion algebras with involution of orthogonal type. In characteristic two, a classification of these algebras in terms of their pfaffians is studied. Also, in arbitrary characteristic, a criterion for an orthogonal involution on a biquaternion algebra to be metabolic is obtained. Mathematics Subject Classification: 11E88, 15A63, 16W10.

1 Introduction

A biquaternion algebra is a tensor product of two quaternion algebras. Every biquaternion algebra is a central simple algebra of degree 4 and exponent 2 or 1. A result proved by A. A. Albert shows that the converse is also true (see [9, (16.1)]). An Albert form of a biquaternion algebra A is a 6-dimensional quadratic form with trivial discriminant whose Clifford algebra is isomorphic to M_{2}(A). It is known that two biquaternion algebras over a field F are isomorphic as F -algebras if and only if their Albert forms are similar (see [9, (16.3)]).

The Albert form of a biquaternion algebra with involution arises naturally as the quadratic form induced by a pfaffian (see [13, (3.3)]). The classical pfaffian is a polynomial map Pf defined on alternating matrices under the transpose involution, which satisfies Pf(X)^{2} = det X for every alternating matrix X (see [2, (3.27)]). In [10], a pfaffian of certain modules over Azumaya algebras was defined and used to classify 6-dimensional quadratic spaces over commutative rings. This construction was used in [13] to find a criterion for involutions on an Azumaya algebra of rank 16, which contains 2 as a unit, to admit an invariant rank 4 Azumaya subalgebra. A similar decomposition criterion for involutions on a biquaternion algebra in arbitrary characteristic was also obtained in [11].

It is known that involutions of symplectic type on a biquaternion algebra can be classified, up to conjugation, by their pfaffian norms (see [9, (16.19)]). For orthogonal involutions the situation is a little more complicated. In characteristic ≠ 2, [13, (5.3)] yields a classification of decomposable orthogonal involutions on a biquaternion algebra A in terms of the pfaffian and the pfaffian adjoint (introduced in [10] and [13]). This classification was originally stated in [13] for the more general case where A is an Azumaya algebra which contains 2 as a unit.

In this work, the pfaffians of decomposable biquaternion algebras with orthogonal involution are investigated. In §3, we recall the notions of pfaffian and pfaffian adjoint of a biquaternion algebra with involution (A, σ). For a decomposable orthogonal involution σ, let q_{σ} be a pfaffian satisfying q_{σ}(x)^{2} = Nrd_{A}(x) for every alternating element x. Set Alt(A, σ)^{+} = {x + p_{σ}(x) | x ∈ Alt(A, σ)} and Alt(A, σ)^{−} = {x − p_{σ}(x) | x ∈ Alt(A, σ)}, where p_{σ} is the linear endomorphism of Alt(A, σ) satisfying xp_{σ}(x) = p_{σ}(x)x = q_{σ}(x) and p_{σ}^{2}(x) = x for x ∈ Alt(A, σ). We shall see in (3.9) that the union of the sets Alt(A, σ)^{+} and Alt(A, σ)^{−} coincides with the set of all square-central elements in Alt(A, σ). At the end of §3, we study in more details the classification of orthogonal involutions on biquaternion algebras in characteristic ≠ 2, obtained in [13]. Although this result is already presented in [13], it is useful to restate it to enable comparison with the corresponding result in characteristic 2 (see (3.17) and (4.12)).

In §4, we study the characterization of decomposable biquaternion algebras with involution in characteristic 2. We also investigate the relation between the restriction of the form q_{σ} to Alt(A, σ)^{+}, denoted by q_{σ}^{+} and the Pfister invariant 𝔓𝔣(A, σ) introduced in [4]. The key result is (4.11), which states that if σ and σ′ are two decomposable orthogonal involutions on A, then q_{σ}^{+} ≃ q_{σ′}^{+} if and only if 𝔓𝔣(A, σ) ≃ 𝔓𝔣(A, σ′). Using this and [15, (6.5)], it can be shown that σ and σ′ are conjugate if and only if q_{σ}^{+} ≃ q_{σ′}^{+} (see (4.12)). Finally, we obtain in (4.16) and (4.19) some criteria for an orthogonal involution on a biquaternion algebra in arbitrary characteristic to be metabolic.

2 Preliminaries

Let V be a finite dimensional vector space over a field F. A quadratic form over F is a map q : V → F such that (i) q(av) = a^{2}q(v) for every a ∈ F and v ∈ V; (ii) the map 𝔟_{q} : V × V → F defined by 𝔟_{q}(u, v) = q(u + v) − q(u) − q(v) is a bilinear form. Note that for every v ∈ V we have 𝔟_{q}(v, v) = 2q(v). In particular, if char F = 2, then 𝔟_{q}(v, v) = 0 for v ∈ V, i.e., 𝔟_{q} is an alternating form. The orthogonal complement of a subspace W ⊆ V is defined as W^{⊥} = {x ∈ V | b_{q}(x, y) = 0 for all y ∈ W }. If V = U ⊕ W is the direct sum of two subspaces U and W with W ⊆ U^{⊥}, we write (V, q) = (U ⊥ W, q|_{U} ⊥ q|_{W}).

A quadratic form q (resp. a bilinear form 𝔟) on V is called isotropic if there exists a nonzero vector v ∈ V such that q(v) = 0 (resp. 𝔟(v, v) = 0). For α ∈ F, we say that q (resp. 𝔟) represents α if there exists a nonzero vector v ∈ V such that q(v) = α (resp. 𝔟(v, v) = α). The sets of all elements of F represented by q and 𝔟 are denoted by D_{F} (q) and D_{F} (𝔟) respectively. For α ∈ F^{×}, the scaled quadratic form α · q is defined as α · q(v) = αq(v) for every v ∈ V.

If char F = 2, for a ∈ F, we denote by [a] (the isometry class of) the quadratic form q(x) = ax^{2}. If char F ≠ 2 and a_{1}, ⋯, a_{n} ∈ F, the quadratic form q(x_{1}, ⋯, x_{n}) = ∑_{i=1}^{n} a_{i}x_{i}^{2} is denoted by ⟨a_{1}, ⋯, a_{n}⟩_{q}. Also, in arbitrary characteristic, the bilinear form defined by 𝔟((x_{1}, ⋯, x_{n}), (y_{1}, ⋯, y_{n})) = ∑_{i=1}^{n} a_{i}x_{i}y_{i} is denoted by ⟨a_{1}, ⋯, a_{n}⟩. Finally, the bilinear form ⟨1, a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, a_{n}⟩ is called a bilinear n-fold Pfister form and is denoted by ⟨⟨a_{1}, ⋯, a_{n}⟩⟩.

An involution on a central simple F -algebra A is an antiautomorphism σ of A of order 2. We say that σ is of the first kind if σ|_{F} = id. An involution σ of the first kind is said to be of symplectic type (or symplectic) if over a splitting field of A, it becomes adjoint to an alternating bilinear form. Otherwise σ is said to be of orthogonal type (or orthogonal). The discriminant of an orthogonal involution σ is denoted by disc σ. The set of alternating elements of A is defined as

Alt(A, σ) = {a − σ(a) | a ∈ A}.

A quaternion algebra over a field F is a central simple algebra Q of degree 2. The canonical involution γ on Q is defined by γ(x) = Trd_{Q}(x) − x for x ∈ Q, where Trd_{Q}(x) is the reduced trace of x. It is known that the canonical involution on Q is the unique involution of symplectic type on Q and it satisfies γ(x)x ∈ F for every x ∈ Q (see [9, Ch. 2]). The map N_{Q} : Q → F defined by N_{Q}(x) = γ(x)x is called the norm form of Q. An element x ∈ Q is called a pure quaternion if Trd_{Q}(x) = 0. The set of all pure quaternions of Q is a 3-dimensional subspace of Q denoted by Q_{0}. Note that an element x ∈ Q lies in Q_{0} if and only if γ(x) = −x, or equivalently, N_{Q}(x) = −x^{2}.

A central simple algebra with involution (A, σ) over a field F is called totally decomposable if it decomposes as tensor products of σ-invariant quaternion F - algebras. If A is a biquaternion algebra, we will use the term decomposable instead of “totally decomposable”. It is known that a biquaternion algebra with orthogonal involution (A, σ) is decomposable if and only if disc σ is trivial (see [11, (3.7)]).

Let (A, σ) be an algebra with involution over a field F. An idempotent e ∈ A is called a metabolic (resp. hyperbolic) idempotent with respect to σ if σ(e)e = 0 and (1−e)(1−σ(e)) = 0 (resp. σ(e) = 1−e). The pair (A, σ) is called metabolic (resp. hyperbolic) if A contains a metabolic (resp. hyperbolic) idempotent with respect to σ. Every hyperbolic involution σ is metabolic but the converse is not always true. If σ is symplectic or char F ≠ 2, the involution σ is metabolic if and only if it is hyperbolic, (see [5, (4.10)] and [3, (A.3)]).

3 The pfaffian and the pfaffian adjoint

We begin our discussion by looking at special cases of [12, (2.1)] and [12, (3.1)]:

Theorem 3.1. ([12]) Let (A, σ) be a biquaternion algebra with orthogonal involution over a field F and let d_{σ} ∈ F^{×} be a representative of the class disc σ ∈ F^{×}/F^{×2}, i.e., d_{σ}F^{×2} = disc σ. There exists a map pf_{σ} : Alt(A, σ) → F such that pf_{σ}(x)^{2} = d_{σ}Nrd_{A}(x) for every x ∈ Alt(A, σ). The map pf_{σ} is uniquely determined up to a sign. Moreover, there exists an F -linear map π_{σ} : Alt(A, σ) → Alt(A, σ) such that xπ_{σ}(x) = π_{σ}(x)x = pf_{σ}(x) and π_{σ}^{2}(x) = d_{σ}x for every x ∈ Alt(A, σ).

Remark 3.2. The map π_{σ} in (3.1) is uniquely determined by pf_{σ}. In fact it is easily seen by scalar extension to a splitting field that Alt(A, σ) has a basis B consisting of invertible elements. For every x ∈ B, we must have π_{σ}(x) = x^{−1}pf_{σ}(x). As π_{σ} is F -linear, it is uniquely defined on Alt(A, σ).

Definition 3.3. The map pf_{σ} in (3.1) is called a pfaffian of (A, σ). We also call the map π_{σ}, the pfaffian adjoint of pf_{σ}.

Note that by [13, (3.3)], every pfaffian of (A, σ) is an Albert form of A.

Definition 3.4. Let F be a field. The pfaffian of an alternating matrix X = (x_{ij}) ∈ M_{4}(F) (under transpose involution) is defined as

Pf(X) = x_{12}x_{34} − x_{13}x_{24} + x_{14}x_{23}.

Notations 3.5. Let (A, σ) be a decomposable biquaternion algebra with orthogonal involution over a field F. Since disc σ is trivial, using (3.1) one can find a pfaffian pf_{σ} of Alt(A, σ) satisfying pf_{σ}(x)^{2} = Nrd_{A}(x) for every x ∈ Alt(A, σ). If pf′_{σ} is another pfaffian with this property, then pf′_{σ} = ±pf_{σ}. After scalar extension to an algebraic closure of F, exactly one of these pfaffians corresponds to the pfaffian Pf. We denote this pfaffian by q_{σ}. Moreover, we denote by p_{σ} the pfaffian adjoint of q_{σ}, hence

q_{σ}(x)^{2} = Nrd_{A}(x), xp_{σ}(x) = p_{σ}(x)x = q_{σ}(x) and p_{σ}^{2}(x) = x,

for every x ∈ Alt(A, σ). We also use the following notations:

Alt(A, σ)^{+} := {x + p_{σ}(x) | x ∈ Alt(A, σ)},
Alt(A, σ)^{−} := {x − p_{σ}(x) | x ∈ Alt(A, σ)}.

Note that if char F = 2, then Alt(A, σ)^{+} = Alt(A, σ)^{−}. As proved in [13, p. 597] and [11, (3.5)], Alt(A, σ)^{+} and Alt(A, σ)^{−} are 3-dimensional subspaces of Alt(A, σ). Since p_{σ}^{2} = id, we have p_{σ}(x) = x for every x ∈ Alt(A, σ)^{+} and p_{σ}(x) = −x for every x ∈ Alt(A, σ)^{−}. The converse is also true, i.e.,

Alt(A, σ)^{+} = {x ∈ Alt(A, σ) | p_{σ}(x) = x},  (1)
Alt(A, σ)^{−} = {x ∈ Alt(A, σ) | p_{σ}(x) = −x}.  (2)

In fact if char F ≠ 2, then for every x ∈ Alt(A, σ) with p_{σ}(x) = x we have x = \frac{1}{2}(x+p_{σ}(x)) ∈ Alt(A, σ)^{+}. Similarly if p_{σ}(x) = −x, then x = \frac{1}{2}(x−p_{σ}(x)) ∈ Alt(A, σ)^{−}. If char F = 2, then the relation (1) follows from the dimension formula for the image and the kernel of the linear map p_{σ} + id.

The following result is implicitly contained in [9, pp. 249-250] over a field of characteristic different form 2:

Lemma 3.6. ([9]) Let (A, σ) be a decomposable biquaternion algebra with orthogonal involution over a field F. Then p_{σ} is an isometry of (Alt(A, σ), q_{σ}). Furthermore 𝔟_{q_{σ}}(x, y) = xp_{σ}(y) + yp_{σ}(x), for x, y ∈ Alt(A, σ).

Proof. For every x ∈ Alt(A, σ) we have q_{σ}(p_{σ}(x)) = p_{σ}(p_{σ}(x))p_{σ}(x) = xp_{σ}(x) = q_{σ}(x). Thus, p_{σ} is an isometry. The second assertion is easily obtained from the relations q_{σ}(x) = xp_{σ}(x) and 𝔟_{q_{σ}}(x, y) = q_{σ}(x + y) − q_{σ}(x) − q_{σ}(y).

Lemma 3.7. Let (A, σ) be a decomposable biquaternion algebra with orthogonal involution over a field F. Then Alt(A, σ)^{+} = (Alt(A, σ)^{−})^{⊥} ⊆ C_{A}(Alt(A, σ)^{−}).

Proof. Let 𝔟 = 𝔟_{q_{σ}} and let x ∈ Alt(A, σ)^{+}. Since p_{σ} ∈ O(Alt(A, σ), q_{σ}), we have 𝔟(x, y) = 𝔟(p_{σ}(x), p_{σ}(y)) = 𝔟(x, p_{σ}(y)) for every y ∈ Alt(A, σ). Thus, 𝔟(x, y − p_{σ}(y)) = 0, i.e., Alt(A, σ)^{+} ⊆ (Alt(A, σ)^{−})^{⊥}. By dimension count we obtain Alt(A, σ)^{+} = (Alt(A, σ)^{−})^{⊥}. Now let z ∈ Alt(A, σ)^{−}. By (3.6) we have 0 = 𝔟(x, z) = −xz + zx. Thus, xz = zx, which implies that Alt(A, σ)^{+} commutes with Alt(A, σ)^{−}, i.e., Alt(A, σ)^{+} ⊆ C_{A}(Alt(A, σ)^{−}).

Lemma 3.8. Let (A, σ) be a decomposable biquaternion algebra with orthogonal involution over a field F and let x ∈ Alt(A, σ). If x^{2} ∈ F, then p_{σ}(x) = ±x.

Proof. Set α = x^{2} ∈ F and β = q_{σ}(x) ∈ F. Then β^{2} = q_{σ}(x)^{2} = Nrd_{A}(x) = ±α^{2}. Thus, β = λα for some λ ∈ F with λ^{4} = 1, i.e., q_{σ}(x) = λx^{2}. If α ≠ 0, then multiplying xp_{σ}(x) = q_{σ}(x) = λx^{2} on the left by x^{−1} we obtain p_{σ}(x) = λx. The relation p_{σ}^{2} = id then implies that λ = ±1 and we are done. So suppose that α = 0, i.e., x^{2} = 0. By (3.6) we have 𝔟_{q_{σ}}(p_{σ}(x), x) = p_{σ}(x)^{2} + x^{2} = p_{σ}(x)^{2}, hence p_{σ}(x)^{2} ∈ F. On the other hand, the relations xp_{σ}(x) = q_{σ}(x) = λx^{2} = 0 show that p_{σ}(x) is not invertible. Thus,

p_{σ}(x)^{2} = 0.  (3)

Suppose that p_{σ}(x) ≠ x, hence x ∉ Alt(A, σ)^{+}. In view of (3.7) one can find w ∈ Alt(A, σ)^{−} such that 𝔟_{q_{σ}}(x, w) = 1. By (3.6) we have

−xw + wp_{σ}(x) = 1.  (4)

Multiplying (4) on the left by x we get xwp_{σ}(x) = x. Using (4), it follows that (wp_{σ}(x) − 1)p_{σ}(x) = x, which yields p_{σ}(x) = −x by (3). This completes the proof (note that if char F = 2, this argument shows that the assumption p_{σ}(x) ≠ x leads to the contradiction p_{σ}(x) = −x, hence p_{σ}(x) = x).

Proposition 3.9. Let (A, σ) be a decomposable biquaternion algebra with orthogonal involution over a field F and let Alt(A, σ)^{0} = Alt(A, σ)^{+} ∪ Alt(A, σ)^{−}. Then Alt(A, σ)^{0} = {x ∈ Alt(A, σ) | p_{σ}(x) = ±x} = {x ∈ Alt(A, σ) | x^{2} ∈ F }.

Proof. The relations (1) and (2) below (3.5) yield the first equality. The second equality follows from (3.8).

Notation 3.10. For a decomposable biquaternion algebra with involution of orthogonal type (A, σ) over a field F, we use the notations Q(A, σ)^{+} = F + Alt(A, σ)^{+} and Q(A, σ)^{−} = F + Alt(A, σ)^{−}. We will simply denote Q(A, σ)^{+} by Q^{+} and Q(A, σ)^{−} by Q^{−}, if the pair (A, σ) is clear from the context.

Lemma 3.11. ([11]) Let (A, σ) be a decomposable biquaternion algebra with orthogonal involution over a field F.

(1) If char F ≠ 2, then Q^{+} and Q^{−} are two σ-invariant quaternion subalgebras of A with Q_{0}^{+} = Alt(A, σ)^{+} and Q_{0}^{−} = Alt(A, σ)^{−}. Furthermore we have (A, σ) ≃ (Q^{+}, σ|_{Q^{+}})⊗(Q^{−}, σ|_{Q^{−}}), where σ|_{Q^{+}} and σ|_{Q^{−}} are the canonical involutions of Q^{+} and Q^{−} respectively.

(2) If char F = 2, then Q^{+} = Q^{−} is a maximal commutative subalgebra of F satisfying x^{2} ∈ F for every x ∈ Q^{+}.

Proof. As observed in [11, (3.5)], Q^{+} is a σ-invariant quaternion subalgebra of A and σ|_{Q^{+}} is of symplectic type. By dimension count and (3.7) we obtain Q^{−} = C_{A}(Q^{+}), hence A ≃ Q^{+} ⊗_{F} Q^{−}. By [9, (2.23 (1))], σ|_{Q^{−}} is of symplectic type. Finally, since Trd_{Q^{+}}(x) = 0 for every x ∈ Alt(A, σ)^{+}, we have Q_{0}^{+} = Alt(A, σ)^{+}. Similarly Q_{0}^{−} = Alt(A, σ)^{−}. This proves the first part. The second part follows from [11, (3.6)].

Notation 3.12. Let (A, σ) be a decomposable biquaternion algebra with orthogonal involution over a field F. We denote by q_{σ}^{+} and q_{σ}^{−} the restrictions of q_{σ} to Alt(A, σ)^{+} and Alt(A, σ)^{−} respectively.

Lemma 3.13. Let (A, σ) be a decomposable biquaternion algebra with orthogonal involution over a field F.

(1) Every unit u ∈ Alt(A, σ)^{+} (resp. u ∈ Alt(A, σ)^{−}) can be extended to a basis {u, v, w} of Alt(A, σ)^{+} (resp. Alt(A, σ)^{−}) such that w = uv.

(2) Every basis {u, v, w} of Alt(A, σ)^{+} (resp. Alt(A, σ)^{−}) with w = uv is orthogonal with respect to q_{σ}^{+} (resp. q_{σ}^{−}).

(3) If char F ≠ 2, then N_{Q^{+}} ≃ ⟨1⟩_{q} ⊥ (−1) · q_{σ}^{+} and N_{Q^{−}} ≃ ⟨1⟩_{q} ⊥ q_{σ}^{−}.

(4) If char F = 2 and (A, σ) ≃ (Q_{1}, σ_{1})⊗(Q_{2}, σ_{2}) is a decomposition of (A, σ), then q_{σ}^{+} ≃ [α] ⊥ [β] ⊥ [αβ], where α ∈ F^{×} and β ∈ F^{×} are representatives of the classes disc σ_{1} ∈ F^{×}/F^{×2} and disc σ_{2} ∈ F^{×}/F^{×2} respectively.

Proof. We just prove the result for q_{σ}^{+}. The proof for q_{σ}^{−} is similar.

(1) Choose an element u′ ∈ Alt(A, σ)^{+} \F u and set α = u^{2} ∈ F^{×}. By (3.11), uu′ ∈ Q^{+} = F + Alt(A, σ)^{+}. Thus, there exist λ ∈ F and w ∈ Alt(A, σ)^{+} such that uu′ = λ+w. Set v = u′ −λα^{−1}u ∈ Alt(A, σ)^{+}. Then uv = w ∈ Alt(A, σ)^{+}. Thus, {u, v, w} is the desired basis.

(2) Let B = {u, v, w} be a basis of Alt(A, σ)^{+} with w = uv. Then vu = σ(uv) = −uv. Using (3.6) we obtain 𝔟(u, v) = uv + vu = 0. Similarly, 𝔟(u, w) = 𝔟(v, w) = 0.

(3) Let {u, v, w} be a basis of Alt(A, σ)^{+} with w = uv. By (2), q_{σ}^{+} ≃ ⟨α, β, −αβ⟩_{q}, where α = u^{2} ∈ F and β = v^{2} ∈ F. On the other hand since vu = −uv, {1, u, v, w} is a quaternion basis of Q^{+}. Thus, N_{Q^{+}} ≃ ⟨1, −α, −β, αβ⟩_{q} by [6, (9.6)].

(4) Let u ∈ Alt(Q_{1}, σ_{1}) and v ∈ Alt(Q_{2}, σ_{2}) be two units and set α = u^{2}, β = v^{2} and w = uv. Then disc σ_{1} = αF^{×2} ∈ F^{×}/F^{×2} and disc σ_{2} = βF^{×2} ∈ F^{×}/F^{×2}. Also, since w ∈ Alt(A, σ) and w^{2} ∈ F, by (3.9) we obtain w ∈ Alt(A, σ)^{+}. So {u, v, w} is a basis of Alt(A, σ)^{+} and q_{σ}^{+} ≃ [α] ⊥ [β] ⊥ [αβ].

Proposition 3.14. ([13, (5.3)]) Let (A, σ) and (A′, σ′) be two decomposable biquaternion algebras with orthogonal involution over a field F. If (A, σ) ≃ (A′, σ′), then there exists an isometry f : (Alt(A, σ), q_{σ}) → (Alt(A′, σ′), q_{σ′}) such that f(Alt(A, σ)^{+}) = Alt(A′, σ′)^{+}.

Proof. See the implication (1) ⇒ (2) in [13, (5.3)] (note that the proof given there also works in characteristic 2).

Lemma 3.15. ([13]) Let (A, σ) be a decomposable biquaternion algebra with orthogonal involution over a field F of characteristic different from 2. Then (Alt(A, σ), q_{σ}) ≃ (Alt(A, σ)^{+}, q_{σ}^{+}) ⊥ (Alt(A, σ)^{−}, q_{σ}^{−}).

Proof. See [13, p. 597].

Remark 3.16. Let (A, σ) be a decomposable biquaternion algebra with orthogonal involution over a field F of characteristic different from 2. By (3.15) every x ∈ Alt(A, σ) can be written uniquely as x = x^{+} + x^{−}, where x^{+} ∈ Alt(A, σ)^{+} = Q_{0}^{+} and x^{−} ∈ Alt(A, σ)^{−} = Q_{0}^{−}. In view of [9, (16.24)] and (3.11 (1)), the maps p_{σ} and q_{σ} can be defined explicitly as p_{σ}(x^{+} + x^{−}) = x^{+} − x^{−} and q_{σ}(x^{+} + x^{−}) = (x^{+})^{2} − (x^{−})^{2}. Using this fact, one can find a shorter proof of (3.8) in characteristic different from 2 (see also the decomposition of q_{σ}^{+} and q_{σ}^{−} in the proof of (3.13 (3))).

The next result complements [13, (5.3)] for biquaternion algebras:

Theorem 3.17. ([13]) Let (A, σ) and (A′, σ′) be two decomposable biquaternion algebras with orthogonal involution over a field F of characteristic different from 2. Let Q^{+} = Q(A, σ)^{+}, Q^{−} = Q(A, σ)^{−}, Q′^{+} = Q(A′, σ′)^{+} and Q′^{−} = Q(A′, σ′)^{−}. The following statements are equivalent:

(1) (A, σ) ≃ (A′, σ′).

(2) q_{σ} ≃ q_{σ′} and q_{σ}^{+} ≃ q_{σ′}^{+}.

(3) A ≃ A′ and q_{σ}^{+} ≃ q_{σ′}^{+}.

(4) A ≃ A′ and Q^{+} ≃ Q′^{+}.

Furthermore, the above statements are equivalent to those obtained by changing “+” to “−”.

Proof. The implication (1) ⇒ (2) follows from (3.14). Since q_{σ} and q_{σ′} are Albert forms of (A, σ) and (A′, σ′) respectively, q_{σ} ≃ q_{σ′} implies that A ≃ A′ by [9, (16.3)]. This proves (2) ⇒ (3). The implication (3) ⇒ (4) follows from (3.13 (3)) and [14, Ch. III, (2.5)]. To prove (4) ⇒ (1) observe that by (3.11 (1)) we have C_{A}(Q^{+}) = Q^{−} and C_{A′}(Q′^{+}) = Q′^{−}. Thus, the isomorphisms Q^{+} ≃_{F} Q′^{+} and A ≃_{F} A′ imply that Q^{−} ≃_{F} Q′^{−}. Since the restrictions of σ to Q^{+} and Q^{−} and the restrictions of σ′ to Q′^{+} and Q′^{−} are all symplectic, we obtain

(A, σ) ≃_{F} (Q^{+}, σ|_{Q^{+}}) ⊗_{F} (Q^{−}, σ|_{Q^{−}})
≃_{F} (Q′^{+}, σ′|_{Q′}^{+}) ⊗_{F} (Q′^{−}, σ′|_{Q′}^{−}) ≃_{F} (A′, σ′).

To prove the last statement of the result, observe that by (3.14), (3.15) and the Witt’s cancellation theorem [14, Ch. I, (4.2)], the statement (1) implies that q_{σ} ≃ q_{σ′} and q_{σ}^{−} ≃ q_{σ′}^{−}. Thus, the implication (1) ⇒ (2) again follows from (3.14). The rest implications can be verified easily.

4 Relation with the Pfister invariant in characteristic two

Throughout this section, unless stated otherwise, F is a field of characteristic 2.

Definition 4.1. Let A be a finite-dimensional associative F -algebra. The minimum number r such that A can be generated as an F -algebra by r elements is called the minimum rank of A and is denoted by r_{F} (A).

Theorem 4.2. ([15]) Let (A, σ) be a totally decomposable algebra with involution of orthogonal type over F. There exists a symmetric and self-centralizing subalgebra S ⊆ A such that x^{2} ∈ F for every x ∈ S and dim_{F} S = 2^{n}, where n = r_{F} (S). Furthermore, for every subalgebra S with these properties, we have S = F + S_{0}, where S_{0} = S ∩ Alt(A, σ). In particular, S ⊆ F + Alt(A, σ). Finally, the subalgebra S is uniquely determined up to isomorphism.

Proof. See [15, pp. 10-11].

Notation 4.3. The isomorphism class of S in (4.2) is denoted by Φ(A, σ).

The next result shows that for biquaternion algebras with orthogonal involution, the subalgebra Φ(A, σ) is unique as a set:

Corollary 4.4. Let (A, σ) be a decomposable biquaternion algebra with involution of orthogonal type over F. Then Φ(A, σ) = Q^{+}.

Proof. Write Φ(A, σ) = F + S_{0}, where S_{0} = Φ(A, σ) ∩ Alt(A, σ). Since every element of Φ(A, σ) is square-central, using (3.9) we have S_{0} ⊆ Alt(A, σ)^{+}. Then S_{0} = Alt(A, σ)^{+} by dimension count, hence Φ(A, σ) = F +Alt(A, σ)^{+} = Q^{+}.

The following result is implicitly contained in [15]:

Lemma 4.5. Let (A, σ) be a totally decomposable algebra of degree 2^{n} with involution of orthogonal type over F. Suppose that there exists a set {u_{1}, ⋯, u_{n}} ⊆ Alt(A, σ) ∩ A^{∗} consisting of pairwise commutative square-central elements such that u_{i_{1}} ⋯ u_{i_{l}} ∈ Alt(A, σ) for every 1 ≤ l ≤ n and 1 ≤ i_{1} < ⋯ < i_{l} ≤ n. Then Φ(A, σ) ≃ F [u_{1}, ⋯, u_{n}].

Proof. By [8, (2.2.3)], S := F [u_{1}, ⋯, u_{n}] is self-centralizing. The other required properties of S, stated in (4.2), are easily verified.

Definition 4.6. The set {u_{1}, ⋯, u_{n}} ⊆ Alt(A, σ) ∩ A^{∗} in (4.5) is called a set of alternating generators of Φ(A, σ).

We recall the following definition from [4]:

Definition 4.7. Let (A, σ) = (Q_{1}, σ_{1})⊗⋯⊗(Q_{n}, σ_{n}) be a totally decomposable algebra with orthogonal involution over F. Let α_{i} ∈ F^{×}, i = 1, ⋯, n, be a representative of the class disc σ_{i} ∈ F^{×}/F^{×2}. The bilinear n-fold Pfister form ⟨⟨α_{1}, ⋯, α_{n}⟩⟩ is called the Pfister invariant of (A, σ) and is denoted by 𝔓𝔣(A, σ).

Note that by [4, (7.5)], 𝔓𝔣(A, σ) is independent of the decomposition of (A, σ).

Theorem 4.8. ([15]) Let (A, σ) be a totally decomposable algebra of degree 2^{n} with involution of orthogonal type over F. Then Φ(A, σ) can be considered as an underlying vector space of the bilinear form 𝔓𝔣(A, σ) in such a way that 𝔓𝔣(A, σ)(x, x) = x^{2} for every x ∈ Φ(A, σ). Also, 𝔓𝔣(A, σ) ≃ ⟨⟨α_{1}, ⋯, α_{n}⟩⟩ if and only if there exists a set of alternating generators {u_{1}, ⋯, u_{n}} of Φ(A, σ) such that u_{i}^{2} = α_{i} ∈ F^{×}, i = 1, ⋯, n.

Proof. See [15, pp. 11-12].

Lemma 4.9. Let ⟨⟨α, β⟩⟩ be an isotropic bilinear Pfister form over F. If αβ ≠ 0, then ⟨⟨α, β⟩⟩ ≃ ⟨⟨α, β + α^{−1}λ^{2}⟩⟩ for every λ ∈ F.

Proof. Since ⟨⟨α, β⟩⟩ is isotropic, by [6, (4.14)] either α ∈ F^{×2} or β ∈ D_{F} ⟨1, α⟩. If α ∈ F^{×2}, using [6, (4.15 (2))] and [6, (4.15 (1))] we obtain

⟨⟨α, β⟩⟩ ≃ ⟨⟨β + α^{−1}λ^{2}, αβ⟩⟩ ≃ ⟨⟨β + α^{−1}λ^{2}, αβ(α^{−1}λ^{2} − (β + α^{−1}λ^{2}))⟩⟩
≃ ⟨⟨β + α^{−1}λ^{2}, αβ^{2}⟩⟩ ≃ ⟨⟨α, β + α^{−1}λ^{2}⟩⟩.

If β ∈ D_{F} ⟨1, α⟩, then there exist b, c ∈ F such that β = b^{2} + c^{2}α. Let s = α^{−1}β^{−1}λ ∈ F. Using [6, (14.15 (1))] we obtain

⟨⟨α, β⟩⟩ ≃ ⟨⟨α, β((1 + csα)^{2} − (bs)^{2}α)⟩⟩ ≃ ⟨⟨α, β(1 + c^{2}s^{2}α^{2} + b^{2}s^{2}α)⟩⟩
≃ ⟨⟨α, β + s^{2}αβ(c^{2}α + b^{2})⟩⟩ ≃ ⟨⟨α, β + s^{2}αβ^{2}⟩⟩ ≃ ⟨⟨α, β + α^{−1}λ^{2}⟩⟩.

Lemma 4.10. Let (A, σ) be a decomposable biquaternion algebra with involution of orthogonal type over F and let α, β ∈ F^{×}. Then 𝔓𝔣(A, σ) ≃ ⟨⟨α, β⟩⟩ if and only if q_{σ}^{+} ≃ [α] ⊥ [β] ⊥ [αβ].

Proof. If 𝔓𝔣(A, σ) ≃ ⟨⟨α, β⟩⟩, by (4.8) there exists a set of alternating generators {u, v} of Φ(A, σ) such that u^{2} = α and v^{2} = β. By (4.4) and (3.13 (2)), {u, v, uv} is an orthogonal basis of Alt(A, σ)^{+}, hence q_{σ}^{+} ≃ [α] ⊥ [β] ⊥ [αβ].

To prove the converse, choose a basis {x, y, z} of Alt(A, σ)^{+} with x^{2} = α, y^{2} = β and z^{2} = αβ. Consider the element xy ∈ Φ(A, σ). By (4.4), Φ(A, σ) = F + Alt(A, σ)^{+}. Thus, there exists a, b, c, d ∈ F such that

xy = a + bx + cy + dz.  (5)

If a = 0 then xy = bx + cy + dz ∈ Alt(A, σ)^{+}, which implies that {x, y} is a set of alternating generators of Φ(A, σ). As x^{2} = α and y^{2} = β we obtain 𝔓𝔣(A, σ) ≃ ⟨⟨α, β⟩⟩ by (4.8).

So suppose that a ≠ 0. By squaring both sides of (5), we obtain αβ = a^{2} + b^{2}α + c^{2}β + d^{2}αβ, which yields

1 + (ba^{−1})^{2}α + (ca^{−1})^{2}β + ((d + 1)a^{−1})^{2}αβ = 0.

Therefore, the form ⟨⟨α, β⟩⟩ is isotropic. Set y′ = y + α^{−1}ax ∈ Alt(A, σ)^{+}. By (5) we have xy′ = xy + a = bx + cy + dz ∈ Alt(A, σ)^{+}, hence {x, y′} is a set of alternating generators of Φ(A, σ). As x^{2} = α and y′^{2} = β + α^{−1}a^{2}, using (4.8) we obtain 𝔓𝔣(A, σ) ≃ ⟨⟨α, β + α^{−1}a^{2}⟩⟩. Thus, 𝔓𝔣(A, σ) ≃ ⟨⟨α, β⟩⟩ by (4.9).

Using (4.10) and (3.13 (4)), we obtain the following relation between the Pfister invariant and the quadratic form q_{σ}^{+}:

Proposition 4.11. Let (A, σ) and (A′, σ′) be two decomposable biquaternion algebras with orthogonal involution over F. Then q_{σ}^{+} ≃ q_{σ′}^{+} if and only if 𝔓𝔣(A, σ) ≃ 𝔓𝔣(A′, σ′).

The following result is analogous to (3.17):

Theorem 4.12. Let (A, σ) and (A′, σ′) be two decomposable biquaternion algebras with involution of orthogonal type over F. The following statements are equivalent:

(1) (A, σ) ≃ (A′, σ′).

(2) q_{σ} ≃ q_{σ′} and q_{σ}^{+} ≃ q_{σ′}^{+}.

(3) A ≃ A′ and q_{σ}^{+} ≃ q_{σ′}^{+}.

(4) A ≃ A′ and 𝔓𝔣(A, σ) ≃ 𝔓𝔣(A′, σ′).

Proof. The implication (1) ⇒ (2) follows from (3.14).

(2) ⇒ (3): Since q_{σ} and q_{σ′} are Albert forms of (A, σ) and (A′, σ′) respectively, q_{σ} ≃ q_{σ′} implies that A ≃ A′ by [9, (16.3)].

The implications (3) ⇒ (4) and (4) ⇒ (1) follow from (4.11) and [15, (6.5)] respectively.

Lemma 4.13. If ⟨⟨α, β⟩⟩ be an anisotropic bilinear Pfister form over F, then ⟨⟨α, β⟩⟩ ≄ ⟨⟨α + 1, β⟩⟩.

Proof. As proved in [1, p. 16], two bilinear Pfister forms are isometric if and only if their pure subforms are isometric. Thus, it is enough to show that the pure subform of ⟨⟨α, β⟩⟩ does not represents α + 1. If α + 1 ∈ D_{F} (⟨α, β, αβ⟩), then there exists a, b, c ∈ F such that a^{2}α + b^{2}β + c^{2}αβ = α + 1. Thus, 1 + (a + 1)^{2}α + b^{2}β + c^{2}αβ = 0, i.e., ⟨⟨α, β⟩⟩ is isotropic which contradicts the assumption.

Definition 4.14. For α ∈ F^{×}, define an involution T_{α} : M_{2}(F) → M_{2}(F) via

T_{α} ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}a & cα^{−1} \\ bα & d \end{matrix} ) .

Note that T_{α} is an involution of orthogonal type on M_{2}(F) and disc T_{α} = αF^{×2} ∈ F^{×}/F^{×2}.

The following example shows that if char F = 2, the conditions A ≃_{F} A′ and Q^{+} ≃_{F} Q′^{+} don’t necessarily imply that (A, σ) ≃ (A′, σ′) (compare (3.17)):

Example 4.15. Let ⟨⟨α, β⟩⟩ be an anisotropic Pfister form over a field F of characteristic 2 and let A = M_{4}(F). Consider the involutions σ = T_{α} ⊗ T_{β} and σ′ = T_{α+1} ⊗ T_{β} on A. Then 𝔓𝔣(A, σ) ≃ ⟨⟨α, β⟩⟩ and 𝔓𝔣(A, σ′) ≃ ⟨⟨α + 1, β⟩⟩, hence 𝔓𝔣(A, σ) ≄ 𝔓𝔣(A, σ′) by (4.13). Using (4.12), we obtain (A, σ) ≄ (A, σ′).

On the other hand by (4.8) there exists a set of alternating generators {u, v} (resp. {u′, v′}) of Φ(A, σ) (resp. Φ(A, σ′)) such that u^{2} = α and v^{2} = β (resp. u′^{2} = α + 1 and v′^{2} = β). Then Φ(A, σ) ≃ F [u, v] and Φ(A, σ′) ≃ F [u′, v′]. The linear map f : F [u, v] → F [u′, v′] induced by f(1) = 1, f(u) = u′ + 1, f(v) = v′ and f(uv) = (u′ + 1)v′ is an F -algebra isomorphism. Thus, Φ(A, σ) ≃ Φ(A, σ′), which implies that Q(A, σ)^{+} ≃ Q(A, σ′)^{+} by (4.4).

We conclude with some results on metabolic involutions. We recall that if char F ≠ 2, then an involution on a central simple F -algebra is metabolic if and only if it is hyperbolic (see [5, (4.10)]).

Proposition 4.16. Let (A, σ) be a decomposable biquaternion algebra with orthogonal involution over a field F of arbitrary characteristic. The following statements are equivalent:

(1) (A, σ) is metabolic.

(2) Q^{+} or Q^{−} is not a division ring.

(3) 1 ∈ D_{F} (q_{σ}^{+}) or −1 ∈ D_{F} (q_{σ}^{−}).

(4) q_{σ}^{+} or q_{σ}^{−} is isotropic.

Proof. If char F ≠ 2, by (3.11 (1)) we have (A, σ) ≃ (Q^{+}, σ|_{Q^{+}}) ⊗ (Q^{−}, σ|_{Q^{−}}), where σ|_{Q^{+}} and σ|_{Q^{−}} are the canonical involutions of Q^{+} and Q^{−} respectively. Thus, the equivalence (1) ⇔ (2) follows from [7, (3.1)]. The equivalences (2) ⇔ (3) and (2) ⇔ (4) both follow from (3.13 (3)) and [14, Ch. III, (2.7)].

Now, suppose that char F = 2. Then the equivalence (1) ⇔ (2) follows from [15, (6.6)].

(1) ⇒ (3): Let e be a metabolic idempotent with respect to σ and let x = e + σ(e). By (4.17), we have x^{2} = 1. Since x ∈ Alt(A, σ), (3.9) implies that x ∈ Alt(A, σ)^{+}, hence q_{σ}^{+}(x) = 1.

(3) ⇒ (4): Suppose that q_{σ}^{+}(u) = 1 for some u ∈ Alt(A, σ)^{+}. By (3.13 (1)) and (3.13 (2)), {u} extends to an orthogonal basis {u, v, w} of Alt(A, σ)^{+} with w = uv. Since Q^{+} is commutative (3.11 (2)), we obtain q_{σ}^{+}(v + w) = (v + w)^{2} = v^{2} + (uv)^{2} = 0, i.e., q_{σ}^{+} is isotropic.

(4) ⇒ (2): If q_{σ}^{+} is isotropic, then there exists a nonzero x ∈ Alt(A, σ)^{+} ⊆ Q^{+} such that x^{2} = 0. Thus, Q^{+} is not a division ring.

Lemma 4.17. Let (A, σ) be a central simple algebra with orthogonal involution over F and let e ∈ A be a metabolic idempotent. Then (e + σ(e))^{2} = 1.

Proof. As (1−e)(1−σ(e)) = 0, we obtain 1−σ(e)−e+eσ(e) = 0, which implies that e + σ(e) = 1 + eσ(e). Thus,

(e + σ(e))^{2} = (1 + eσ(e))^{2} = 1 + eσ(e)eσ(e) = 1.

Corollary 4.18. Let (A, σ) be a central simple algebra with involution over a field F of arbitrary characteristic. If σ is metabolic, then disc σ is trivial.

Proof. The result follows from (4.17) and [3, (2.3)].

Proposition 4.19. Let (A, σ) be a biquaternion algebra with involution of orthogonal type over a field F of arbitrary characteristic. Then σ is metabolic if and only if there exists u ∈ Alt(A, σ) such that u^{2} = 1.

Proof. If σ is metabolic, then by (4.18), disc σ is trivial. Thus, σ is decomposable and the result follows from (4.16). Conversely, suppose that there exists u ∈ Alt(A, σ) such that u^{2} = 1. Then disc σ = Nrd_{A}(u)F^{×2} is trivial, so (A, σ) is decomposable by [11, (3.7)]. Since u^{2} = 1 ∈ F and u ∈ Alt(A, σ), by (3.9) we have u ∈ Alt(A, σ)^{+} ∪ Alt(A, σ)^{−}. Therefore, either u ∈ Alt(A, σ)^{+} (i.e., q_{σ}^{+}(u) = 1) or u ∈ Alt(A, σ)^{−} (i.e., q_{σ}^{−}(u) = −1). By (4.16), σ is metabolic.

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A.-H. Nokhodkar, anokhodkar@yahoo.com

Department of Pure Mathematics, Faculty of Science, University of Kashan, P. O. Box 87317-51167, Kashan, Iran.