HYPERBOLICITY CRITERIA FOR CERTAIN INVOLUTIONS

M. G. MAHMOUDI

Abstract. Using the ideas and techniques developed by Bayer-Fluckiger, Shapiro and Tignol about hyperbolic involutions of central simple algebras, criteria for the hyperbolicity of involutions of the form σ ⊗ τ and σ ⊗ ρ, where σ is an involution of a central simple algebra A, τ is the nontrivial automorphism of a quadratic extension of the center of A and ρ is an involution of a quaternion algebra are obtained.

2000 Mathematics Subject Classification: 16W10, 11E39

Key words: Hyperbolic involution, central simple algebra, hermitian form

1. Introduction

Given a ring R with unity, a map σ : R → R is called an involution if σ(x + y) = σ(x) + σ(y), σ(xy) = σ(y)σ(x) and σ(σ(x)) = x for all x, y ∈ R. The pair (R, σ) (or simply σ) is called hyperbolic if there exists an idempotent e ∈ R such that σ(e) = 1 − e. This notion was introduced in [2]. An involution σ of R is said to be isotropic if there exists a nonzero element a ∈ R such that σ(a)a = 0; otherwise, σ is called anisotropic.

Various hyperbolicity criteria have been obtained by several authors. It is helpful to recall some of these basic criteria here.

Let K be a field of characteristic different from 2. Let A be a K-central simple algebra with an involution σ and let k be the fixed field of σ|_{K}. When σ is of the first kind, i.e., K = k, it was shown in [2, Thm. 3.3] that if L = K(\sqrt{d}) is a quadratic extension of K, then (A ⊗ L, σ ⊗ id_{L}) is hyperbolic if and only if there exists r ∈ A such that r^{2} = d and σ(r) = −r. This criterion excludes the exceptional case, where A is split, σ is of orthogonal type and the Witt index of the quadratic form, to which σ is adjoint, is odd. This result was generalized in [13] and [10], in various ways, to the case where σ is of the second kind.

When L/k is an extension of odd degree, it was shown in [1] that if σ ⊗ id_{L} is hyperbolic, then so is σ. This result was originally stated in [1] in terms of hermitian forms (see [5, pp. 79-80]).

If L is the function field of a quadratic form, defined over k, the behavior of σ ⊗ id_{L} (in other words, the behavior of σ under the field extension L/k) has been studied by various authors. If σ becomes hyperbolic over L = K[t]/(π(t)), where π(t) ∈ K[t] is a separable monic polynomial of degree 2n, it was shown in [13] that (A, σ) is homomorphic image of a certain universal free K-algebra H_{π} on 2n indeterminates with an involution σ_{π}. A quadratic form-theoretic formulation of this criterion was given in [4] by showing that H_{π} has a homomorphic image, which is the Clifford algebra of some quadratic form over a certain polynomial ring. A necessary and sufficient condition for a quaternion or a biquaternion algebra to become hyperbolic over a field extension was obtained in [3]. An interesting question about hyperbolicity of a central simple algebra with involution (A, σ) (of low dimension) over the function field of a quadratic form q and its relationship with the existence of some homomorphic images of the even Clifford algebra C_{0}(q) in A was studied in [7]. The hyperbolicity of (A, σ) (of low dimension) over the function field of three-dimensional forms was studied in [12].

Let A be a K-central simple algebra with an involution σ and let k be the fixed field of σ|_{K}. Let τ be the nontrivial automorphism of a quadratic extension L/k. Let Q be a quaternion algebra over K with an involution ρ such that σ|_{K} = ρ|_{K}. In this article we provide criteria for hyperbolicity of σ ⊗τ and σ ⊗ρ (see Theorem 3.2, Corollary 4.2 and Corollary 4.6). Our main motivation to search for such criteria arises from our previous work [11], which investigates the hyperbolicity of canonical involutions of Clifford algebra (or the even Clifford algebra) of a quadratic form q and its connection with the existence of particular subforms of q. In fact, the standard isomorphism theorems of Clifford algebras like C(q′⊥q) ≃ C(q′) ⊗ C(d · q) and C_{0}(q′′⊥q) ≃ C_{0}(q′′) ⊗ C(−d · q), where q′ is a form of even dimension, q′′ is a form of odd dimension and d is the discriminant of q (see [8, Ch. V, §2]), induce isomorphisms of algebras with involution (see [9]). If we choose, in the first (resp. second) isomorphism, q to be a two (resp. one) dimensional form, the canonical involutions of C(q′⊥q) (resp. C_{0}(q′′⊥q)) are decomposed as σ ⊗ ρ (resp. σ ⊗ τ). Finding criteria for the hyperbolicity of σ ⊗ τ and σ ⊗ ρ are therefore useful in this regard.

The criterion for hyperbolicity of σ ⊗ ρ is deduced from a general criterion for the hyperbolicity of a central simple algebra with involution in terms of certain subalgebra of codimension two. This criterion is given in Theorem 4.1 (see also Theorem 4.5).

As an application, a corollary about the hyperbolicity of the tensor product of two quaternion algebras with involution is drawn (c.f. Proposition 4.10, compare with [11, Prop. 3.9]).

2. Preliminaries

All fields considered in this paper are supposed to be of characteristic different from 2.

Let A be a K-central simple algebra with an involution σ. If σ|_{K} is the identity map, σ is called of the first kind. Otherwise σ|_{K} is a nontrivial automorphism of K. In this case, σ is called of the second kind. If k is the fixed field of σ|_{K}, in both cases, we say, in short, that σ is a K/k-involution. Let ε ∈ K be an element with σ(ε)ε = 1. An element a ∈ A which satisfies σ(a) = εa is called ε-hermitian. A (1)-hermitian element is usually called a symmetric element and a (−1)-hermitian element is usually called a skew-symmetric element.

We define

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

Note that A^{+} and A^{−} are k-vector spaces.

The involution σ is said to be of orthogonal type if dim A^{+} > dim A^{−}. It is said to be of symplectic type if dim A^{+} < dim A^{−}.

If σ is of the first kind, it is known that the square class of the reduced norm of any skew-symmetric invertible element a of A with respect to σ is independent of the choice of a (cf. [6]). The discriminant of σ is so defined in [5] as the square class of (−1)^{m} Nrd(a), where a is a skew-symmetric element of A and m = \frac{1}{2} deg(A). Here deg(A) is the degree of A (the degree of a central simple algebra is square root of its dimension, as a vector space, over its center).

Let (A, σ) be a K-central simple algebra with involution. Let ε ∈ K be an element with εσ(ε) = 1 and let V be a right A-module of finite rank. An εhermitian form over V with respect to σ, is a biadditive map h : V × V → A such that

1) h(xα, yβ) = σ(α)h(x, y)β for all x, y ∈ V and all α, β ∈ A,

2) εh(y, x) = σ(h(x, y)) for all x, y ∈ V.

If ε = 1, h is called a hermitian form and if ε = −1, h is called a skew-hermitian form.

For a non-degenerate ε-hermitian space (V, h) over (A, σ), the adjoint involution of End_{A}(V) with respect to (V, h), is the unique involution I_{h} of End_{A}(V) such that

1) I_{h}(α) = σ(α) for every α ∈ K,
2) h(x, f(y)) = h(I_{h}(f)(x), y) for all x, y ∈ V and all f ∈ End_{A}(V).

When the role of End_{A}(V) is clear in the context, we simply say that I_{h} is the involution, which is adjoint to h.

3. Quadratic extensions and hyperbolic involutions

Lemma 3.1. Let A be a K-central simple algebra with an anisotropic K/k-involution σ. For d ∈ k, let L = k(\sqrt{d}) be a quadratic extension of k with the nontrivial kautomorphism τ. If σ ⊗ τ is hyperbolic, then there exists an element r ∈ A such that r^{2} = d and σ(r) = r.

Proof. If σ ⊗ τ is hyperbolic, there exists e ∈ A ⊗_{k} L such that e^{2} = e and (σ ⊗ τ)(e) = 1 − e. We can write e = e_{1} ⊗ 1 + e_{2} ⊗ \sqrt{d}, where e_{1}, e_{2} ∈ A. The following systems of equations are obtained:

(1)  \{ \begin{matrix}e_{1}^{2} + de_{2}^{2} = e_{1} \\ e_{1}e_{2} + e_{2}e_{1} = e_{2}, \end{matrix} \{ \begin{matrix}σ(e_{1}) = 1 − e_{1} \\ σ(e_{2}) = e_{2}. \end{matrix}

We first show that e_{2} is invertible. As σ is anisotropic, the right ideal I = {x ∈ A : e_{2}x = 0} is generated by a symmetric idempotent f (cf. [2, Cor. 1.8]). The relation e_{2}f = 0 implies e_{1}e_{2}f = 0 and e_{2}^{2}f = 0. Using the previous system of equations and e_{2}^{2}f = 0 we obtain (e_{1} − e_{1}^{2})f = 0. We deduce that (fe_{1})σ(fe_{1}) = fe_{1}σ(e_{1})f = fe_{1}(1 − e_{1})f = f(e_{1} − e_{1}^{2})f = 0. As σ is anisotropic, we obtain fe_{1} = 0. A similar argument shows that σ(e_{1}f)e_{1}f = f(1 − e_{1})e_{1}f = f(e_{1} − e_{1}^{2})f = 0. Thus e_{1}f = 0.

We now have: 0 = σ(fe_{1}) = (1 − e_{1})f = f − e_{1}f = f. Therefore I = {0}. This implies that e_{2} is invertible. Now consider the element r = e_{1}e_{2}^{−1}. According to (1), we have e_{1} + e_{2}e_{1}e_{2}^{−1} = 1 thus e_{2}e_{1}e_{2}^{−1} = 1 − e_{1}.

On the other hand σ(r) = σ(e_{2})^{−1}σ(e_{1}) = e_{2}^{−1}(1 − e_{1}) = e_{1}e_{2}^{−1} = r. Similarly r^{2} = e_{1}e_{2}^{−1}e_{1}e_{2}^{−1} = e_{1}e_{2}^{−1}(1−e_{2}e_{1}e_{2}^{−1})e_{2}^{−1} = e_{1}e_{2}^{−2} −e_{1}^{2}e_{2}^{−2} = (e_{1} −e_{1}^{2})e_{2}^{−2} = de_{2}^{2}e_{2}^{−2} = d. This completes the proof. □

Theorem 3.2. Let A be a K-central simple algebra with a K/k-involution σ. We exclude the case where A is split, σ is symplectic and deg(A) = 2m, where m is an odd integer. Let L = k(\sqrt{d}) be a quadratic extension of k and let τ be the nontrivial automorphism of L/k. Then there exists an element r ∈ A with the properties r^{2} = d and σ(r) = r if and only if (A ⊗_{k} L, σ ⊗ τ) is hyperbolic.

Proof. First, suppose that the element r with the indicated properties exists. Take t = d^{−1}r ⊗ \sqrt{d} ∈ A ⊗_{k} L. We have: t^{2} = (d^{−1}r ⊗ \sqrt{d})^{2} = d^{−2}r^{2} ⊗ d = 1 and (σ ⊗ τ)(t) = d^{−1}r ⊗ (−\sqrt{d}) = −t. Take e = \frac{1}{2}(1 + t). We obtain e^{2} = e and σ(e) = 1 − e. Thus, (A ⊗_{k} L, σ ⊗ τ) is hyperbolic.

Conversely, suppose that (A⊗_{k} L, σ ⊗τ) is hyperbolic. Let A = End_{D}(V), where D is a division algebra which is Brauer-equivalent to A. Let σ′ be an involution of D of the same kind as σ. Finally let (V, h) be a ε-hermitian space over (D, σ′) such that σ is the adjoint involution with respect to (V, h) (we take ε = 1 when σ is of the second kind).

Consider the Witt decomposition (V, h) = (V_{0}, h_{0})⊥(V_{1}, h_{1}), where (V_{0}, h_{0}) is hyperbolic and (V_{1}, h_{1}) is anisotropic. Let σ_{0} and σ_{1} be the adjoint involutions of End_{D}(V_{0}) and End_{D}(V_{1}) with respect to h_{0} and h_{1}, respectively.

The hermitian space (V_{1}, h_{1}) becomes hyperbolic over (L, τ). The previous lemma implies the existence of an element r_{1} ∈ End_{D}(V_{1}) such that r_{1}^{2} = d and σ_{1}(r_{1}) = r_{1}.

Suppose that σ is of the second kind or of the first kind and of orthogonal type. According to [2, Thm. 2.2], End_{D}(V_{0}) contains a σ_{0}-invariant subalgebra M such that

(2)  (M_{2}(K), Θ_{1}) ≃ (M, σ_{0}|_{M}),

where Θ_{1} is the involution of M_{2}(K) defined by

Θ_{1} ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}σ_{0}(d) & σ_{0}(b) \\ σ_{0}(c) & σ_{0}(a) \end{matrix} ) .

Let r_{0} ∈ M ⊂ End_{D}(V_{0}) be the image of ( \begin{matrix}0 & d \\ 1 & 0 \end{matrix} ) under the isomorphism given

in (2). We have σ_{1}(r_{0}) = r_{0} and r_{0}^{2} = d. Let r ∈ A = End(V) be the element defined by

(3)  r(x, y) = (r_{0}(x), r_{1}(y)).

We obtain σ(r) = r and r^{2} = d.

If σ is of the first kind and of symplectic type and deg(End_{D}(V_{0})) is divisible by 4 (this is always the case except when D is split and deg(A) = 2m, where m is an odd integer), then again using [2, Thm. 2.2], there exists a σ_{0}-invariant subalgebra M such that

(4)  (M, σ_{0}|_{M}) ≃ (M_{2}(K), Θ_{1}).

Let r_{0} be the image of ( \begin{matrix}0 & d \\ 1 & 0 \end{matrix} ) under the isomorphism given in (4). We have

σ_{0}(r_{0}) = r_{0} and r_{0}^{2} = d. The element r defined in (3) satisfies σ(r) = r and r^{2} = d. □

4. Quadratic extensions of algebras and hyperbolic involutions

Theorem 4.1. Let (A, σ) be a K-central simple algebra with involution. Suppose that there exist λ, μ ∈ A^{×} such that λμ = −μλ, σ(λ) = −λ, σ(μ) = −μ and K(λ)/K is a quadratic extension. Let Ã = C_{A}(λ) be the centralizer of λ in A. Suppose that σ|Ã is anisotropic. Then σ is hyperbolic if and only if there exists r ∈ Ã such that σ(r)μ = μr and σ(r)r = μ^{2}.

Proof. As σ is hyperbolic, there exists an idempotent e ∈ A such that σ(e) = 1−e. We can write e = x + μy, where x, y ∈ Ã. We obtain the following systems of equations:

(5)  \{ \begin{matrix}x^{2} + μyμy = x \\ xμy + μyx = μy, \end{matrix} \{ \begin{matrix}σ(x) = 1 − x \\ σ(y) = μyμ^{−1}. \end{matrix}

We first show that y is invertible. We observe that the right ideal I = {z ∈ Ã : yz = 0} is generated by a symmetric idempotent f (cf. [2, Cor. 1.8]). The relation yf = 0 implies μyμyf = 0. Therefore (x − x^{2})f = 0. We now have: (fx)σ(fx) = fx(1 − x)f = f(x − x^{2})f = 0. In the same way σ(xf)(xf) = f(1 − x)xf = f(x − x^{2})f = 0. As σ|Ã is anisotropic we deduce that fx = xf = 0. On the other hand 0 = σ(fx) = (1 − x)f = f − xf = f. This implies I = {0} and thus y is invertible.

Now take r = xy^{−1}. Using (5), we obtain:

σ(r)r = σ(xy^{−1})xy^{−1} = σ(y^{−1})σ(x)xy^{−1} = μy^{−1}μ^{−1}(1 − x)xy^{−1}
= μy^{−1}μ^{−1}(x − x^{2})y^{−1} = μy^{−1}μ^{−1}(μyμy)y^{−1}
= μ^{2},

σ(r)μ = σ(xy^{−1})μ = σ(y)^{−1}σ(x)μ = μy^{−1}μ^{−1}(1 − x)μ

= μy^{−1} − μy^{−1}μ^{−1}xμ

= μy^{−1} − μy^{−1}μ^{−1}(μ − μyxy^{−1}) = μxy^{−1} = μr.

Conversely, suppose that there exists r ∈ Ã such that σ(r)μ = μr and σ(r)r = μ^{2}. Take t = rμ^{−1}. We have t^{2} = μ^{−1}σ(r)rμ^{−1} = 1 and σ(t) = −μ^{−1}σ(r) = −rμ^{−1} = −t. Now, if e = \frac{1}{2}(1 + t), then e^{2} = e and σ(e) = 1 − e. Therefore σ is hyperbolic. □

Corollary 4.2. Let A be a K-central simple algebra with an involution σ. Let Q be a quaternion algebra over K endowed with an involution ρ. Suppose that ρ and σ have the same restriction to K. Let λ, μ be elements in Q such that ρ(λ) = −λ, ρ(μ) = −μ, μ^{2} ∈ K, λμ = −μλ and let L = K(λ) be a quadratic extension of K. Suppose that σ ⊗ ρ|_{L} is anisotropic. Then the involution σ ⊗ ρ of A ⊗_{K} Q is hyperbolic if and only if there exist a, b ∈ A with σ(a) = a, σ(b) = b, ab = ba and σ(a)a − λ^{2}σ(b)b = μ^{2}.

Proof. First, suppose that σ ⊗ ρ is hyperbolic. According to Theorem 4.1, there exists an element r ∈ A⊗L such that (σ⊗ρ)(r)r = 1⊗μ^{2} and σ(r)(1⊗μ) = (1⊗μ)r. We can write r = a ⊗ 1 + b ⊗ λ for some a, b ∈ A. We have (σ ⊗ ρ)(r) = σ(a) ⊗ 1 − σ(b) ⊗ λ.

The condition (σ ⊗ ρ)(r)(1 ⊗ μ) = (1 ⊗ μ)r implies σ(a) ⊗ μ − σ(b) ⊗ λμ = a⊗μ+b⊗μλ. Therefore σ(a) = a and σ(b) = b. The condition (σ ⊗ρ)(r)r = 1⊗μ^{2} implies σ(a)a ⊗ 1 − σ(b)b ⊗ λ^{2} + σ(a)b ⊗ λ − σ(b)a ⊗ λ = 1 ⊗ μ^{2}. We deduce that σ(a)a − λ^{2}σ(b)b = μ^{2} and σ(a)b − σ(b)a = 0, which implies ab = ba.

Conversely, suppose that a and b with the indicated properties exist. Take r = a ⊗ 1 + b ⊗ λ. We have: (σ ⊗ ρ)(r)r = 1 ⊗ μ^{2} and (σ ⊗ ρ)(r)(1 ⊗ μ) = (1 ⊗ μ)r. From Theorem 4.1 we conclude that σ ⊗ ρ is hyperbolic. □

Remark 4.3. In Corollary 4.2, the condition concerning the anisotropy of σ ⊗ ρ|_{L} can also be stated as follows: If a, b ∈ A satisfy σ(a)a − λ^{2}σ(b)b = 0 and σ(a)b = σ(b)a, then a = b = 0.

Remark 4.4. If (A, σ) = (K, id), Corollary 4.2 states that the canonical involution of the quaternion algebra Q = (α, β)_{K} is hyperbolic if and only if the quadratic form ⟨1, −α, −β, αβ⟩ is isotropic over K if and only if Q is split. This is well known (see [4, Prop. 18]).

Theorem 4.1 can be established under more general hypotheses. In fact, we have:

Theorem 4.5. Let (A, σ) be a K-central simple algebra with involution. Suppose that there exist λ, μ ∈ A^{×} such that λμ = −μλ, σ(λ) = ε_{λ}λ, σ(μ) = ε_{μ}μ and K(λ)/K is a quadratic extension, where ε_{λ}, ε_{μ} ∈ K satisfy σ(ε_{λ})ε_{λ} = 1 and σ(ε_{μ})ε_{μ} = 1. Let Ã = C_{A}(λ) be the centralizer of λ in A. Suppose that σ|Ã is anisotropic. Then σ is hyperbolic if and only if there exists r ∈ Ã such that σ(r)μ = −ε_{μ}μr and σ(r)r = −ε_{μ}μ^{2}.

Proof. Using an argument similar to the one given in the proof of Theorem 4.1, we find the following systems of equations:

(6)  \{ \begin{matrix}x^{2} + μyμy = x \\ xμy + μyx = μy, \end{matrix} \{ \begin{matrix}σ(x) = 1 − x \\ σ(y) = −σ(ε_{μ})μyμ^{−1}. \end{matrix}

For r = xy^{−1} we have: σ(r)r = −ε_{μ}μ^{2} and σ(r)μ = −ε_{μ}μr.

Conversely, suppose that there exists r ∈ Ã such that σ(r)μ = −ε_{μ}μr and σ(r)r = −ε_{μ}μ^{2}. Take t = rμ^{−1}. We have: t^{2} = rμ^{−1} · rμ^{−1} = −σ(ε_{μ})μ^{−1}σ(r) · rμ^{−1} = σ(ε_{μ})ε_{μ} = 1 and σ(t) = σ(μ^{−1})σ(r) = (ε_{μ}μ)^{−1}(−ε_{μ}μrμ^{−1}) = −rμ^{−1} = −t. Now, if e = \frac{1}{2}(1 + t), then e^{2} = e and σ(e) = 1 − e. Therefore σ is hyperbolic. □

Corollary 4.6. Let (A, σ) be a K-central simple algebra with involution. Let (Q, ρ) be a K-quaternion algebra with involution. Suppose that ρ and σ have the same restriction to K. Let λ, μ be elements of Q such that ρ(λ) = ε_{λ}λ, ρ(μ) = ε_{μ}μ, μ^{2} ∈ K, λμ = −μλ and let L = K(λ) be a quadratic extension of K, where ε_{λ}, ε_{μ} ∈ K satisfy ρ(ε_{λ})ε_{λ} = 1 and ρ(ε_{μ})ε_{μ} = 1. Suppose that σ⊗ρ|_{L} is anisotropic. Then the involution σ ⊗ ρ of A ⊗ Q is hyperbolic if and only if there exist a, b ∈ A with σ(a) = −ε_{μ}a, σ(b) = ε_{λ}^{−1}ε_{μ}b, ab = ba and σ(a)a + ε_{λ}λ^{2}σ(b)b + ε_{μ}μ^{2} = 0.

Proof. According to Theorem 4.5, there exists an element r ∈ A ⊗ L such that (σ ⊗ ρ)(r)r = −ε_{μ}(1 ⊗ μ^{2}) and σ(r)(1 ⊗ μ) = −ε_{μ}(1 ⊗ μ)r. We can write r = a ⊗ 1 + b ⊗ λ for some a, b ∈ A. We have (σ ⊗ ρ)(r) = σ(a) ⊗ 1 + ε_{λ}σ(b) ⊗ λ.

The condition σ(r)(1 ⊗ μ) = −ε_{μ}(1 ⊗ μ)r implies that σ(a) ⊗ μ + ε_{λ}σ(b) ⊗ λμ = −ε_{μ}(a ⊗ μ − b ⊗ μλ). Therefore σ(a) = −ε_{μ}a and ε_{λ}σ(b) = ε_{μ}b. The condition (σ⊗ρ)(r)r = −ε_{μ}(1⊗μ^{2}) implies σ(a)a⊗1+ε_{λ}σ(b)b⊗λ^{2}+σ(a)b⊗λ+ε_{λ}σ(b)a⊗λ = −ε_{λ}(1 ⊗ μ^{2}). We deduce that σ(a)a + ε_{λ}λ^{2}σ(b)b = −ε_{μ}μ^{2} and σ(a)b + ε_{λ}σ(b)a = 0, which implies ab = ba. □

Remark 4.7. In Corollary 4.6, the condition concerning the anisotropy of σ ⊗ ρ|_{L} can also be stated as follows: If a, b ∈ A satisfy σ(a)a + ε_{λ}λ^{2}σ(b)b = 0 and σ(a)b + ε_{λ}σ(b)a, then a = b = 0.

Remark 4.8. Let (A, σ) = (K, id). Suppose that Q = (a, b)_{K} is the quaternion algebra with the orthogonal involution ρ defined by ρ(i) = i, ρ(j) = j. Then Corollary 4.6 implies that the involution ρ is hyperbolic if and only if −ab ∈ K^{×}^{2}. This is well known (see [3, Thm. 2.1]).

As an application of above results, we present an alternative proof for the following result stated in [3, Prop. 3.1]. As it is mentioned in [3], this result is an immediate consequence of [2, Cor. 2.5].

Corollary 4.9. Let A be a biquaternion algebra over a field K with an orthogonal involution σ such that disc(σ) = 1. We can write the following decomposition (A, σ) = (Q_{1}, σ_{1}) ⊗_{K} (Q_{2}, σ_{2}), where σ_{1} and σ_{2} are the symplectic involutions. Then σ is hyperbolic if and only if Q_{1} or Q_{2} is split.

Proof. If one of the algebras Q_{1} or Q_{2} is split, then one of the involutions σ_{1} or σ_{2} is hyperbolic. Therefore σ is hyperbolic.

Conversely, suppose that σ is hyperbolic. We can write Q_{2} = (c, d)_{K}, where c, d ∈ K^{×}. If Q_{2} is not split, we have in particular, c ∉ K^{×}^{2}. If the involution σ_{1} ⊗σ_{2}|_{K(\sqrt{c})} of Q_{1} ⊗K(\sqrt{c}) is isotropic, then [3, Lem. 2.3] implies that Q_{1} is split. If σ_{1} ⊗ σ_{2}|_{K(\sqrt{c})} is anisotropic, Corollary 4.6 implies the existence of x, y ∈ K such that x^{2} − cy^{2} − d = 0. Thus the quadratic form ⟨1, −c, −d⟩ is isotropic. Therefore Q_{2} is split. □

Proposition 4.10. Let Q_{1} = (a, b)_{K} and Q_{2} = (c, d)_{K} be two K-quaternion algebras. Let σ_{1} be the orthogonal involution of Q_{1} defined by σ_{1}(λ) = λ, σ_{1}(μ) = μ, where {λ, μ} is a standard basis of Q_{1} with λμ = −μλ, λ^{2} = a, μ^{2} = b and let σ_{2} be the canonical involution of Q_{2}. Then σ = σ_{1} ⊗ σ_{2} is hyperbolic if and only if Q_{2} is split or at least one of the quadratic forms ⟨a, b, −ac, −bc, −d⟩ or ⟨a, b, −c⟩ is isotropic over K.

Proof. Suppose that σ is hyperbolic. If c ∈ K^{×}^{2}, then Q_{2} is split. Now suppose that c ∉ K^{×}^{2}. Consider the quadratic extension L/K, where L = K(\sqrt{c}). The restriction σ_{2}|_{L} is the nontrivial automorphism of L/K.

If σ_{1} ⊗ σ_{2}|_{L} is isotropic, then Q_{1} ⊗ L is split. Therefore σ_{1} ⊗ σ_{2}|_{L} is adjoint to an isotropic hermitian form of dimension 2 over a field. In particular, σ_{1} ⊗ σ_{2}|_{L} is hyperbolic. According to Theorem 3.2, there exists r ∈ Q_{1} with the properties r^{2} = c and σ_{1}(r) = r. Thanks to these last properties one can write r = γ_{1}1 + γ_{2}λ + γ_{3}μ for some γ_{1}, γ_{2}, γ_{3} ∈ K. However, r^{2} = c implies that γ_{1}^{2} + aγ_{2}^{2} + bγ_{3}^{2} = c and γ_{1}γ_{2} = γ_{1}γ_{3} = 0. If γ_{1} = 0, we deduce that the quadratic form ⟨a, b, −c⟩ is isotropic. If γ_{1} ≠ 0, we deduce that γ_{2} = γ_{3} = 0. Therefore c is a square, which is a contradiction.

Now consider the case where σ_{1} ⊗ σ_{2}|_{L} is anisotropic. In this case, Corollary 4.6 implies the existence of x, y ∈ Q_{1} such that σ_{1}(x) = x, σ_{1}(y) = y, xy = yx and

(7)  σ_{1}(x)x − cσ_{1}(y)y − d = 0.

We can write x = γ_{1}1 + γ_{2}λ + γ_{3}μ and y = γ′_{1}1 + γ′_{2}λ + γ′_{3}μ for some γ_{1}, γ_{2}, γ_{3}, γ′_{1}, γ′_{2}, γ′_{3} ∈ K. The condition xy = yx implies that the elements w = γ_{2}λ + γ_{3}μ and w′ = γ′_{2}λ + γ′_{3}μ are linearly dependent over K. So there exists a nonzero element v ∈ Kλ ⊕ Kμ ⊂ Q_{1} such that w = θv and w′ = θ′v for some θ, θ′ ∈ K. Now (7) implies that λ_{1}^{2} − cλ_{1}^{′2} + q(v)θ^{2} − cq(v)θ^{′2} − d = 0, where q is the quadratic form ⟨a, b⟩. It follows that the quadratic form ⟨1, −c⟩⊗⟨a, b⟩⊥⟨−d⟩ ≃ ⟨a, b, −ac, −bc, −d⟩ is isotropic over K. Conversely, suppose that one of the following conditions holds:

Q_{2} is split or

⟨a, b, −c⟩ is isotropic or
⟨a, b, −ab, −ac, −d⟩ is isotropic.

In the first case, σ is hyperbolic because σ_{2} is hyperbolic too. In the second case, σ is hyperbolic because σ_{1} ⊗ σ_{2}|_{L} is hyperbolic. In the third case, we obtain a system of the form of (7). Therefore σ is hyperbolic. □

Acknowledgments. I thank Behrooz Mirzaii for careful reading of an earlier version of this manuscript and for his useful suggestions. I also thank Eva Bayer-Fluckiger and Karim Becher for their constructive comments on this work. I acknowledge the financial support provided by the Research Council of Sharif University of Technology.

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Department of Mathematical Sciences, Sharif University of Technology, P. O. Box: 11155-9415, Tehran, Iran. E-Mail address: mmahmoudi@sharif.ir