HYPERBOLIC INVOLUTIONS AND QUADRATIC EXTENSIONS

M. G. MAHMOUDI

Abstract. This is a variation on a theme of Bayer-Fluckiger, Shapiro and Tignol related to hyperbolic involutions. More precisely criteria for the hyperbolicity of involutions of quadratic extensions of simple algebras and 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 Kcentral 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 [14] and [11], 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 [6, 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 [14] 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 [5] 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 [4]. 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 [8]. The hyperbolicity of (A, σ) (of low dimension) over the function field of three-dimensional forms was studied in [13].

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 3.3 and Corollary 4.2). The proofs follow the same line as the proof of Bayer-Fluckiger, Shapiro and Tignol (cf. [2, Thm 3.3]). Our main motivation to search for such criteria arises from our previous work [12], 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 [9, Ch. V, §2]), induce isomorphisms of algebras with involution (see [10]). 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 3.1 (see also Corollary 4.1). In other words, this theorem studies the hyperbolicity of involutions under quadratic extensions of algebras. The notion of quadratic extensions of algebras is also considered in the recent paper [3] in connection with orthogonal involutions of degree 12 algebras with trivial discriminant and Clifford invariant.

As an application, a corollary about the hyperbolicity of the tensor product of two quaternion algebras with involution is drawn (cf. Proposition 4.7, compare with [12, Prop. 3.9]). This application, which concerns degree 4 algebras, can also be obtained using the equivalence of categories described in §15 of the Book of Involutions [6] (see Remark 4.8).

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. [7]). The discriminant of σ is so defined in [6] 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 ε-hermitian 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

Theorem 3.1. Let A be a simple algebra that is a quadratic extension of a simple algebra B, in the sense that A = B ⊕ Bz for some element z such that zBz^{−1} = B and z^{2} = b ∈ B^{×}. Let σ be an involution on A such that σ(B) = B and σ(z) = εz for some ε centralizing A such that σ(ε)ε = 1. If σ|_{B} is anisotropic, then σ is hyperbolic if and only if there exists r ∈ B such that σ(r)z = −εzr and σ(r)r = −εb.

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

(1)  \{ \begin{matrix}x^{2} + yzyz = x \\ xyz + yzx = yz, \end{matrix} \{ \begin{matrix}σ(x) = 1 − x \\ σ(y) = −ε^{−1}z^{−1}yz. \end{matrix}

We first show that y is invertible. We observe that the right ideal I = {w ∈ B : yzw = 0} is generated by a symmetric idempotent f (cf. [2, Cor. 1.8]). The relation yzf = 0 implies yzyzf = 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|_{B} 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 yz and consequently y is invertible. We note that from the second equation of the first system of (1) one can obtain:

(2)  (1 − x) = yzxz^{−1}y^{−1}.

Now take r = x(z^{−1}yz)^{−1}. Using (1), we obtain:

σ(r)r = σ(xz^{−1}y^{−1}z)xz^{−1}y^{−1}z = σ(z)σ(y^{−1})σ(z^{−1})σ(x)xz^{−1}y^{−1}z
= εz(−εz^{−1}y^{−1}z)(ε^{−1}z^{−1})(1 − x)xz^{−1}y^{−1}z
= −εy^{−1}(x − x^{2})z^{−1}y^{−1}z = −εy^{−1}(yzyz)z^{−1}y^{−1}z
= −εz^{2} = −εb,
σ(r)z = σ(xz^{−1}y^{−1}z)z = σ(z)σ(y^{−1})σ(z^{−1})σ(x)z
= εz(−εz^{−1}y^{−1}z)(ε^{−1}z^{−1})(1 − x)z
= −εy^{−1}(1 − x)z
= −εy^{−1}(yzxz^{−1}y^{−1})z (using (2))
= −εzr.

Conversely, suppose that there exists r ∈ B such that σ(r)z = −εzr and σ(r)r = −εb. Take t = rz^{−1}. We have t^{2} = rz^{−1}rz^{−1} = (−ε^{−1}z^{−1}σ(r)z)z^{−1}rz^{−1} = −ε^{−1}z^{−1}(−εb)z^{−1} = z^{−1}bz^{−1} = 1 and σ(t) = σ(z^{−1})σ(r) = (ε^{−1}z^{−1})(−εzrz^{−1}) = −t. Now, if e = \frac{1}{2}(1 + t), then e^{2} = e and σ(e) = 1 − e. Therefore σ is hyperbolic. □

Corollary 3.2. 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 k-automorphism τ. Then σ ⊗ τ is hyperbolic if and only if there exists an element r ∈ A such that r^{2} = d and σ(r) = r.

Proof. Use Theorem 3.1 by taking B := A, A := B ⊗ k(\sqrt{d}) and z := 1 ⊗ \sqrt{d}. Here ε = −1 and b = d. □

Theorem 3.3. 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. If an element r with the indicated properties exists then using Corollary 3.2 we deduce that (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 corollary 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

(3)  (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 iso-

morphism given in (3). We have σ_{1}(r_{0}) = r_{0} and r_{0}^{2} = d. Let r ∈ A = End(V) be the element defined by

(4)  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

(5)  (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 (5).

We have σ_{0}(r_{0}) = r_{0} and r_{0}^{2} = d. The element r defined in (4) satisfies σ(r) = r and r^{2} = d. □

4. Quadratic extensions of algebras and hyperbolic involutions

Corollary 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) = 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. Use Theorem 3.1 by taking B := C_{A}(λ), A := A and z := μ. Here we have ε = ε_{μ}. □

Corollary 4.2. 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 Corollary 4.1, 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 4.3. 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 + ε_{λ}λ^{2}σ(b)b = 0 and σ(a)b + ε_{λ}σ(b)a = 0, 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 [5, Prop. 18]).

Remark 4.5. 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 4.2 implies that the involution ρ is hyperbolic if and only if −ab ∈ K^{×}^{2}. This is also well known (see [4, Thm. 2.1]).

As an application of above results, we present an alternative proof for the following result stated in [4, Prop. 3.1]. As it is mentioned in [4], this result is an immediate consequence of [2, Cor. 2.5].

Corollary 4.6. 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 it is hyperbolic since the degree of the algebra is 2. Corollary 3.2 leads to a contradiction if Q_{1} is not split. If σ_{1} ⊗σ_{2}|_{K(\sqrt{c})} is anisotropic, Corollary 4.2 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 4.7. Let Q_{1} = (a, b)_{K} and Q_{2} = (c, d)_{K} be two Kquaternion 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 the form ⟨a, b, −c, −d, cd⟩ (or equivalently ⟨a, b, −abc, −abd, abcd⟩) is isotropic over K.

Proof. First suppose that σ is hyperbolic. Let denote L = K(λ). Consider the involution σ′ := σ_{1}|_{L} ⊗ σ_{2} of L ⊗ Q_{2}.

– Case 1: σ′ is isotropic. In this case, L⊗Q_{2} should be split. If Q_{2} is split as well, then the form ⟨−c, −d, cd⟩ is already isotropic and we are done. Otherwise σ_{2} is anisotropic and in this case according to [2, Thm. 3.3], Q_{2} contains an element r such that σ_{2}(r) = −r and r^{2} = a. This implies that the form ⟨c, d, −cd⟩ represents a, thus ⟨a, b, −c, −d, cd⟩ is isotropic.

– Case 2: σ′ is anisotropic. Using Theorem 3.1, there exists an element r ∈ L ⊗ Q_{2} such that σ′(r)(μ ⊗ 1) = −(μ ⊗ 1)r and σ′(r)r = −b ⊗ 1. We can write r = 1 ⊗ q + λ ⊗ q′ where q, q′ ∈ Q_{2}. The relation σ′(r)(μ ⊗ 1) = −(μ ⊗ 1)r implies that q is a pure quaternion and q′ is a scalar. Now the relation σ′(r)r = −b ⊗ 1 readily implies that the form ⟨a, −c, −d, cd⟩ represents −b, therefore ⟨a, b, −c, −d, cd⟩ is isotropic.

Conversely, suppose that ⟨a, b, −c, −d, cd⟩ is isotropic. If σ′ is isotropic too, it is hyperbolic, hence σ is hyperbolic as well. So suppose that σ′ is anisotropic. There exist x, y, z, w ∈ K such that ax^{2} − cy^{2} − dz^{2} + cdw^{2} = −b. Take:

r = (1 ⊗ (yi + zj + wij)) + (λ ⊗ x) ∈ L ⊗ Q_{2},

where i and j are generators of Q_{2} with i^{2} = c, j^{2} = d and ij = −ji. We have σ′(r)(μ ⊗ 1) = −(μ ⊗ 1)r and σ′(r)r = −b ⊗ 1. Theorem 3.1 now implies that σ is hyperbolic. □

Remark 4.8. The previous result is implicit in the literature. As pointed out by the referee, it is already given in [6, 15.21]. In an earlier version of this article, this result had been stated differently and contained some incorrect statements. I thank the referee for suggesting the current formulation of this proposition which fixed the previous version.

Acknowledgments. I am indebted to the referee for several useful suggestions which led to a significant improvement of the organization of this article. In particular, the statement of Theorem 3.1 which unified several results of an earlier version of this article, is due to the referee. 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