DECOMPOSITION OF ALGEBRAS WITH INVOLUTION IN CHARACTERISTIC 2

ANDREW DOLPHIN

Abstract. In this paper we present a decomposition theorem for hermitian forms over fields of characteristic 2 refining the usual Witt decomposition in this case. We apply this decomposition to algebras with involution over fields of characteristic 2 to give a complete description of the effect of passing to a generic splitting field of the algebra on the isotropy of the involution.

1. Introduction

The decomposition theorem of Witt (see [15]) states that any regular symmetric quadratic form over a field of characteristic different from 2 uniquely decomposes into an orthogonal sum of an anisotropic quadratic form and a hyperbolic quadratic form. This theorem has since been generalised and extended in many ways and to many different contexts. Our interest will be in its generalisation to the theory of hermitian forms, where it says that every regular hermitian form over a division algebra with involution can be decomposed into an orthogonal sum of an anisotropic hermitian form and a metabolic hermitian form, but where the latter need not be uniquely determined if the characteristic of the underlying field is 2.

In this paper we will show that in the case of a division algebra over a base field F of characteristic 2 with an F –linear involution, an anisotropic hermitian form has a natural decomposition into what we will call its direct and alternating parts. As we shall see, the notion of a direct hermitian form allows the generalisation to hermitian form theory of several results on anisotropic bilinear forms, which do not generally hold for anisotropic hermitian forms.

In particular, in (6.7) we shall show that in characteristic 2 direct hermitian forms remain direct, and in particular anisotropic, over the separable closure of the base field. This is a generalisation of [9, Satz 10.2.1], a theorem on bilinear forms over fields of characteristic 2. We shall also see, in (6.3), that in characteristic 2, the direct hermitian forms are precisely those forms that remain anisotropic when the algebra is generically split.

Most of our final results will be stated in terms of algebras with involution, rather than hermitian forms. Any hermitian form over a division algebra is associated with an algebra with involution. In Section 2 we will briefly cover this association, as well as cover the basic definitions and results we need from the theory of algebras with involution and hermitian forms. In Section 3 we introduce the concept of direct hermitian forms and show that any anisotropic hermitian form decomposes into direct and alternating parts in (3.9).

2000 Mathematics Subject Classification. Primary 11E81; Secondary 11E39, 12F10. Key words and phrases. Central simple algebras; involutions; hermitian forms; characteristic two, generic splitting.

In Section 4 we translate the concepts of direct and alternating from the theory of hermitian forms to the theory of algebras with involution. We shall show that every involution over a base field of characteristic 2 has well-defined direct and alternating parts in (4.5).

Section 5 will be a brief digression into the study of the notion of the orthogonal sum of algebras with involution. This concept was introduced by Dejaiffe in [3] for fields of characteristic different from 2. We follow the more general presentation of the concept found in [2].

Finally, in Section 6 we will apply (3.9) in order to study the effect of generically splitting the algebra on the isotropy of an involution over that algebra. This is a topic of much recent study of fields of characteristic different from 2, see e.g. [8]. In particular, in characteristic different from 2, it is an open problem whether any anisotropic involution on a central simple algebra remains anisotropic after extension to some splitting field of A. As we shall see in (6.4), over fields of characteristic 2, there exists such a field precisely when the involution is direct. Moreover, we shall show that the anisotropic part of an algebra with involution after generically splitting the algebra is simply its direct part extended to the splitting field.

2. Algebras with involution

In this section we recall the basic definitions and results we will use on central simple algebras with involution and hermitian forms. We refer to [13] for a general reference on central simple algebras.

Throughout, let F be a field and A be a finite-dimensional simple F –algebra. The centre of A is a field E, and A can be viewed as a E–algebra. In this case dim_{E}(A) is a square, and the positive root of this integer is called the degree of A and is denoted deg_{E}(A). By Wedderburn’s theorem, A ≃ End_{D}(V) for F –division algebra D with centre E and a right D–vector space V. The degree of D is called the index of A and is denoted ind_{E}(A). We call any A with ind_{E}(A) = 1 split. We call a field extension L/E a splitting field of A if A ⊗_{F} L is split. If E = F, then we call the F –algebra A central simple.

An F –involution on A is an F –linear map σ : A → A such that σ(xy) = σ(y)σ(x) for all x, y ∈ A and σ^{2} = id_{A}. An F –algebra with involution is a pair (A, σ) of a finite-dimensional F –algebra A and an F –involution σ of A such that, with E being the centre of A, one has F = {x ∈ E | σ(x) = x}, and such that either A is simple or a product of two simple F –algebras that are mapped to each other by σ. In this situation, there are two possibilities: either E = F, so that A a central simple F –algebra, or E/F is a quadratic étale extension with σ restricting to the nontrivial F –automorphism of E. To distinguish these two situations, we speak of involutions of the first or second kind; more precisely, we say that the F –algebra with involution (A, σ) is of the first kind if E = F and of the second kind otherwise. Involutions of the second kind are also known as unitary involutions. For any field extension K/F we will use the notations A_{K} = A ⊗_{F} K, σ_{K} = σ ⊗ id_{K} and (A, σ)_{K} = (A_{K}, σ_{K}).

Let (A, σ) be an F –algebra with involution E be the centre of A. For λ ∈ E, let Sym_{λ}(A, σ) = {a ∈ A | λσ(a) = a} and Alt(A, σ) = {σ(a) − a | a ∈ A}. These are F –linear subspaces of A.

We call an idempotent e ∈ A hyperbolic with respect to σ if σ(e) = 1 − e. Moreover, we call an F –algebra with involution (A, σ) hyperbolic if A contains a hyperbolic idempotent with respect to σ.

An idempotent e ∈ A is called metabolic with respect to σ if σ(e)e = 0 and dim_{F} eA = \frac{1}{2}dim_{F} A. Note that, by [4, Corollary 4.3], we may substitute the condition that dim_{F} eA = \frac{1}{2}dim_{F} A for the condition that (1 − e)(1 − σ(e)) = 0 in this definition. An algebra with involution (A, σ) is called metabolic if A contains a metabolic idempotent with respect to σ.

An F –quaternion algebra is a central simple F -algebra of degree 2. The description of quaternion algebras over fields of any characteristic is give in [14, Section 8.11]. We recall it for fields of characteristic 2. If char(F) = 2, then given any α ∈ F and β ∈ F^{×}, there exists an F –quaternion algebra with an F –basis (1, i, j, k) subject to the relations that i^{2} + i = α, j^{2} = β and k = ij = ji + j; we denote this F –quaternion algebra by [α, β)_{F}. If char(F) = 2, by [14, Section 8.11], every quaternion algebra is isomorphic to [α, β)_{F} for some α ∈ F and β ∈ F^{×}. Note that [α, β)_{F} is split if α = u^{2} + u for some u ∈ F. In particular, any quaternion division algebra splits over a quadratic separable extension.

Every quaternion algebra Q has a so-called canonical involution (see [11, Proposition 2.21]). When the characteristic of F is 2 and (1, i, j, k) is an F –basis of Q with relations as above, this involution is given by

x_{0} + x_{1}i + x_{2}j + x_{3}k ⟼ x_{0} + x_{1}(i + 1) + x_{2}j + x_{3}k

for x_{0}, x_{1}, x_{2}, x_{3} ∈ F. This involution is symplectic, and is in fact the unique symplectic involution on Q.

Throughout the rest of this section, let (D, θ) be an F –division algebra with involution and E be the centre of D. Further, fix λ ∈ E such that λθ(λ) = 1.

A λ–hermitian form over (D, θ) is a pair (V, h) where V is a finite-dimensional right D-vector space and h is a non-degenerate bi-additive map h : V × V → D such that

h(x, yd) = h(x, y)d and h(y, x) = λθ(h(x, y))

holds for all x, y ∈ V and d ∈ D. Let ≃ denote isometry between hermitian forms and we denote the orthogonal sum of two hermitian forms (V, h) and (W, h′) as (V, h)⊥(W, h′).

Throughout the rest of this section we fix (V, h) to be a λ–hermitian form over an F –division algebra with involution (D, θ). We say that (V, h) is hermitian when λ = 1 and skew when λ = −1. Note that if (D, θ) is of the first kind, then (V, h) is always either hermitian or skew. If D is split and θ = Id_{D}, then (V, h) is a symmetric bilinear form. We say (V, h) represents a ∈ D if there exists some x ∈ V such that h(x, x) = a, and we say (V, h) represents a ∈ D non-trivially if there exists some x ∈ V \{0} such that h(x, x) = a. We say (V, h) is isotropic if it represents 0 non-trivially, and anisotropic otherwise. We call (V, h) alternating if h(x, x) ∈ Alt(D, θ) for all x ∈ V.

Let V be a finite-dimensional right D–vector space V and V^{∗} = End_{D}(V, D), the dual of V. Then there exists a regular λ–hermitian form ℍ_{λ}(V) = (V^{∗} ⊕ V, h_{λ}) over (D, θ), where

h_{λ} : (V^{∗} ⊕ V) × (V^{∗} ⊕ V) → D

is given by

h_{λ}(φ + x, ψ + y) = φ(y) + λθ(ψ(x)) for φ, ψ ∈ V^{∗} and x, y ∈ V.

We call a λ-hermitian form over (D, θ) hyperbolic if it is isometric to ℍ_{λ}(V) for some right D–vector space V.

Let S ⊂ V. We define the orthogonal complement S^{⊥} of S with respect to a fixed λ–hermitian form (V, h) as

S^{⊥} = {x ∈ V | h(x, s) = 0 for all s ∈ S}.

The λ–hermitian form (V, h) is called metabolic if there exists a subspace S ⊂ V such that S = S^{⊥}.

For a_{1}, …, a_{n} ∈ D^{×} such that a_{i} ∈ Sym_{λ}(A, σ), for i = 1 …, n, we denote by ⟨a_{1}, …, a_{n}⟩_{(D,θ,λ)} the λ-hermitian form (D^{n}, h) where

h : D^{n} × D^{n} → D, (x, y) ↦ ∑_{i=1}^{n} θ(x_{i})a_{i}y_{i}.

We call such a form a diagonal form. We call a hermitian form diagonalisable if it is isometric to a diagonal form.

Proposition 2.1. The λ–hermitian form (V, h) is diagonalisable, except if D = F and either char(F) ≠ 2 and (V, h) is a skew-symmetric bilinear space, or char(F) = 2 and (V, h) is a hyperbolic symmetric bilinear space.

Proof. See [10, Proposition 6.2.4]. □

Proposition 2.2. Let (V, h) be a λ–hermitian form over (D, θ). There is a unique F –involution σ on End_{D}(V) such that

h(f(x), y) = h(x, σ(f)(y)) for all x, y ∈ V and f ∈ A.

This involution σ is uniquely determined by h. Further, (End_{D}(V), ad_{h}) is an F – algebra with involution of the same kind as (D, θ) and A is Brauer equivalent to D.

Proof. See, for example, [11, Theorem 4.1]. □

In the situation of (2.2), we call σ the adjoint involution to h and denote it by ad_{h}, and we further write Ad(V, h) = (End_{D}(V), ad_{h}).

Let L be a splitting field of the F –algebra A. An involution (A, σ) of the first kind is said to be symplectic if (A, σ)_{L} ≃ Ad(V, b), where (V, b) is an alternating bilinear form, and orthogonal otherwise. This definition is independent of the choice of the splitting field L (see [11, Section 2.A]).

Proposition 2.3. Assume char(F) = 2 and (A, σ) is of the first kind. Then (V, h) is alternating if and only if Ad(V, h) is symplectic.

Proof. See [11, Theorem 4.2] □

Proposition 2.4. (V, h) is hyperbolic (resp. metabolic) if and only if Ad(V, h) is hyperbolic (resp. metabolic).

Proof. See [11, Proposition 6.7] for the statement on hyperbolicity and [4, Theorem 4.8] for the statement on metabolicity in the case of involutions of the first kind. The argument there remains valid for unitary involutions. □

We will often use a single letter to denote a hermitian form.

Proposition 2.5. Any λ–hermitian form has a decomposition η ≃ φ⊥ρ, where φ is an anisotropic λ–hermitian form and ρ is a metabolic λ–hermitian form, both over (D, θ). Moreover φ is unique up to isometry.

Proof. See [10, Proposition 6.1.1] and [10, Proposition 6.1.4]. □

In the situation of (2.5) we call φ the anisotropic part of η, denoted η_{an}.

Let (A, σ) be an F –algebra with involution. We call an algebra (B, τ) with involution an anisotropic part of (A, σ), if (B, τ) ≃ Ad(η_{an}) for some hermitian space η such that Ad(η) ≃ (A, σ).

Proposition 2.6. Let (A, σ) be an F –algebra with involution. Then the anisotropic part of (A, σ) is uniquely determined up to isomorphism by (A, σ).

Proof. See [4, Proposition 3.6] for the case of involutions of the first kind. The argument there remains valid for unitary involutions. □

We may therefore speak of the anisotropic part of (A, σ), which we denote by (A, σ)_{an}.

3. Alternating and direct Hermitian forms

In this section we shall show a decomposition theorem for hermitian forms. This result will be only of interest of fields of characteristic 2, and for involutions of the first kind. In all other cases, the decomposition theorem will be trivial (see (3.12)). The decomposition will be in terms of alternating forms, and forms that we will call direct hermitian forms. Firstly we will show a couple of results on alternating forms.

We now fix some notation for the rest of this section. Let (D, θ) be an F –division algebra with involution and E be the centre of D. Further, fix λ ∈ E such that λθ(λ) = 1.

Proposition 3.1. Assume char(F) ≠ 2. Then a λ–hermitian form over (D, θ) is alternating if and only if it is skew.

Proof. Since char(F) ≠ 2, by [11, Section 2.A.] and [11, Proposition 2.17] we have Sym_{−1}(D, θ) = Alt(A, θ) and the result is clear. □

Proposition 3.2. Let η be an alternating hermitian form over (D, θ). If (D, θ) = (F, id_{F}) then η is a hyperbolic form. Otherwise, η ≃ ⟨a_{1}, …, a_{n}⟩_{(D,θ,λ)} for some a_{1}, …, a_{n} ∈ Alt(D, θ).

Proof. See [4, Proposition 3.6] for the case of involutions of the first kind. The argument there is easily adapted to the case of unitary involutions. □

We will call an hermitian form (V, h) direct if h(x, x) ∉ Alt(D, θ) for all x ∈ V \ {0}. Direct hermitian forms are necessarily anisotropic and non-alternating.

Proposition 3.3. Let φ be a symmetric bilinear form over a field F. Then φ is direct if and only if φ is anisotropic.

Proof. This follows easily from the fact that Alt(F, id_{F}) = {0}. □

Proposition 3.4. Assume that char(F) ≠ 2. Then an anisotropic λ–hermitian form over (D, θ) is direct if and only if λ ≠ −1.

Proof. Let η = (V, h) be an anisotropic λ–hermitian form over (D, θ). If η is skew, then η is not direct by (3.1). Conversely, suppose η is not direct. Then

λh(x, x) = θ(h(x, x)) = −h(x, x) ≠ 0,

as η is anisotropic, so λ = −1. □

Proposition 3.5. Assume η ≃ ⟨a⟩_{(D,θ,λ)} for some a ∈ Sym_{λ}(D, θ). If a ∈ Alt(D, θ) then η is alternating. Otherwise η is direct.

Proof. If a ∈ Alt(D, θ) then η is alternating by (3.2).

Otherwise, η represents elements of the form θ(b)ab for some b ∈ D. We have that θ(b)ab ∈ Alt(D, θ) if and only if a ∈ Alt(D, θ), hence η is direct. □

Proposition 3.6. Let η be an anisotropic λ–hermitian form over (D, θ) and suppose that there exist λ–hermitian forms φ and ψ over (D, θ) such that φ is direct, ψ is alternating and η ≃ φ⊥ψ. Let a ∈ Alt(D, θ) be an element represented nontrivially by η. Then a is represented non-trivially by ψ.

Proof. Let φ = (U, h′) and ψ = (W, b) for right D–vector spaces U, W and let x ∈ V \{0} be such that h(x, x) = a. Then x = y + w for y ∈ U and w ∈ W, and

h(x, x) = h′(y, y) + b(w, w) = a.

However, a − b(w, w) ∈ Alt(D, θ), hence y = 0 as φ is direct. Since x ≠ 0, we must have that a is represented non-trivally by ψ. □

Corollary 3.7. The sum of a direct hermitian form and an anisotropic alternating form over the same F –division algebra with involution is anisotropic.

Lemma 3.8. Let φ, ψ and ρ be λ-hermitian forms over (D, θ). If φ and ψ are anisotropic, then φ⊥ρ ≃ ψ⊥ρ implies that φ ≃ ψ.

Proof. We have that ρ⊥ρ is metabolic by [10, Proposition 3.7.8], and φ⊥ρ⊥ρ ≃ ψ⊥ρ⊥ρ. Hence the result follows from [10, Proposition 6.1.4]. □

Theorem 3.9. Let η be an anisotropic λ–hermitian form over (D, θ). Then η ≃ φ⊥ψ for some direct λ-hermitian form φ and an alternating λ-hermitian ψ form, over (D, θ). Moreover φ and ψ are unique up to isometry.

Proof. First we show existence of such a decomposition. If η does not represent any non-zero element in Alt(D, θ), then η is direct, so we are done.

Assume now that η represents 0 ≠ a ∈ Alt(D, θ). Then

η ≃ η′⊥⟨a⟩_{(D,θ)}

for some λ-hermitian form η′ by [10, Lemma 3.6.2]. The existence the required a decomposition then follows by induction on the dimension of V and the fact that the sum of two alternating forms is an alternating form.

We now show the uniqueness of the decomposition. Let φ_{1} and φ_{2} be direct λ-hermitian forms and ψ_{1} and ψ_{2} be alternating anisotropic λ-hermitian forms, all over (D, θ) such that φ_{1}⊥ψ_{1} ≃ φ_{2}⊥ψ_{2}.

If φ_{1}⊥ψ_{1} represents no alternating elements, then both ψ_{1} and ψ_{2} must be trivial and we are done.

Otherwise, let a ∈ Alt(D, θ) be an element represented by φ_{1}⊥ψ_{1}, and hence also by φ_{2}⊥ψ_{2}. By (3.6), a is represented by ψ_{1} and ψ_{2}. Hence

ψ_{1} ≃ ψ′_{1}⊥⟨a⟩_{(D,θ,λ)} and ψ_{2} ≃ ψ′_{2}⊥⟨a⟩_{(D,θ,λ)}

where ψ′_{1} and ψ′_{2} are alternating λ–hermitian forms of smaller dimension. Therefore

φ_{1}⊥ψ′_{1}⊥⟨a⟩_{(D,θ,λ)} ≃ φ_{2}⊥ψ′_{2}⊥⟨a⟩_{(D,θ,λ)},

and hence by (3.8) we have

φ_{1}⊥ψ′_{1} ≃ φ_{2}⊥ψ′_{2}.

Repeating this process, we see that we eventually arrive at

φ_{1}⊥ψ′′_{1} ≃ φ_{2}⊥ψ′′_{2}

where ψ′′_{1} and ψ′′_{2} are alternating λ–hermitian forms and such that φ_{1}⊥ψ′′_{1} does not represent any alternating elements. In which case both ψ′′_{1} and ψ′′_{2} must be trivial and φ_{1} ≃ φ_{2}. That ψ_{1} ≃ ψ_{2} now follows from (3.8). □

Corollary 3.10. Let η be a λ-hermitian form over an F –division algebra (D, θ). Then there exists a decomposition

η ≃ φ⊥ψ⊥ρ,

for some direct λ-hermitian form φ, an anisotropic alternating λ-hermitian form ψ and ρ a metabolic hermitian form, over (D, θ). Moreover, φ and ψ are unique up to isometry.

Proof. By (2.5) we have a decomposition η ≃ η_{an}⊥ρ where ρ is a metabolic hermitian form and η_{an} is a λ–hermitian form that is uniquely defined. We can then apply (3.9) to η_{an} in order to give the full decomposition. □

In the situation of (3.10), we call the direct form φ the direct part of η, denoted η_{dir}, and we call the alternating form ψ the alternating part of η, denoted η_{alt}.

Note that a subform of a direct form is always direct, and that a subform of an alternating form is always alternating. However, while a sum of alternating hermitian forms is always alternating, a sum of two direct forms need not be direct, even in the case where the sum is anisotropic, as the following example shows.

Example 3.11. Let Q be an F –quaternion division algebra and γ is the canonical involution on Q. There exists j ∈ Q \ F such that j^{2} ∈ F^{×} and γ(j) = j, and ⟨j, j + 1⟩_{(Q,γ)} is an anisotropic sum of two direct hermitian forms that is not direct.

Proof. Since j, j + 1 ∉ Alt(Q, γ) = F, the forms ⟨j⟩_{(Q,γ)} and ⟨j + 1⟩_{(Q,γ)} are direct by (3.5).

It is clear that h is not direct as it represents 1 ∈ Alt(Q, γ). Finally, note that the

change of basis matrix ( \begin{matrix}1 & 1 + β^{−1}j \\ 1 & 1 \end{matrix} ) gives the isometry h ≃ ⟨1, 1 + β^{−1}j⟩_{(Q,γ)}.

Since ⟨1⟩_{(Q,γ)} is alternating and ⟨1 + β^{−1}j⟩_{(Q,γ)} is direct, it follows from (3.7) that h is anisotropic. □

While the results in this section have been given free from any assumption on the characteristic of the underlying field and on the kind of involution, the following propositions show that these results are trivial when the characteristic is different from 2 or the involution is of unitary type.

Proposition 3.12. Assume char(F) ≠ 2 or that (D, θ) is unitary. Let η be a λ-hermitian form over (D, θ). Then η_{alt} ≃ η_{an} and η_{dir} is trivial if and only if λ = −1. Otherwise, η_{dir} ≃ η_{an} and η_{alt} is trivial.

Proof. For char(F) ≠ 2, this follows directly from (3.1) and (3.4).

Assume char(F) = 2 and that (A, σ) is unitary. Then we have that Sym_{1}(A, σ) = Alt(A, σ) by [11, Proposition 2.17]. Let a be any element represented by η, then θ(a) = λa. So a ∈ Sym_{1}(A, σ) = Alt(A, σ) for all elements a ∈ D represented by η if and only if λ = 1. □

4. Decomposition of algebras with involution

In this section we will use the results of Section 3 to decompose algebras with involution into direct and alternating parts. We call an F –algebra with involution direct if it is isomorphic to the adjoint involution of some direct λ–hermitian form.

Proposition 4.1. Let (A, σ) be an F –algebra with involution and assume that char(F) = 2 or (A, σ) is unitary. Then (A, σ) is direct if and only if (A, σ) is anisotropic.

Proof. First note that if (A, σ) is isotropic, then any involution that it is the adjoint to is isotropic, and hence not direct.

Assume that char(F) ≠ 2 and (A, σ) of the first kind. Then (A, σ) isomorphic to the adjoint involution of an anisotropic 1–hermitian form by [11, Theorem 4.2], and hence is direct by (3.4).

Now assume that (A, σ) is unitary and over a field of arbitary characteristic. Let (V, h) be an anisotropic λ–hermitian form over an F –division algebra with involution (D, θ), such that Ad(V, h) ≃ (A, σ). Then by [14, Lemma 7.6.6] and the remarks that follow, there exists a u ∈ E^{×}, where E is the centre of D, such that (V, uh) is a 1-hermitian form over (D, θ). Note that for all f ∈ End_{D}(V) we have

uh(ad_{h}(x), y) = uh(x, ad_{h}(f)(y)),

as u is invertible, and hence Ad(V, h) = Ad(V, uh) by (2.2). We have that (V, uh) is direct by (3.12), and therefore (A, σ) is direct. □

Hence, the concept of a direct involution is trivial if the characteristic of the underlying field is not 2, or the involution is of the second kind. Therefore, we shall assume throughout the rest of this section that char(F) = 2, and that (A, σ) is an F –algebra with involution of the first kind. In this setting we must have that λ = 1 for any λ–hermitian form over an F –division algebra of the first kind, so we will drop λ from all notation.

We shall characterise direct involutions of the first kind without reference to hermitian forms. First we show the following characterisation of symplectic involutions for comparison, as the alternating part of an involution is always symplectic.

Proposition 4.2. (A, σ) is symplectic if and only if σ(a)a ∈ Alt(A, σ) for all a ∈ A.

Proof. Since char(F) = 2, (A, σ) is symplectic if and only if 1 ∈ Alt(A, σ) by [11, Proposition 2.6]. If σ(a)a ∈ Alt(A, σ) for all a ∈ A then in particular 1 = σ(1) · 1 ∈ Alt(A, σ), and hence (A, σ) is symplectic.

If 1 ∈ Alt(A, σ) then 1 = σ(b) − b for some b ∈ A. Then, for all a ∈ A,

σ(a)a = σ(a)(σ(b) − b)a = σ(σ(a)ba) − σ(a)ba ∈ Alt(A, σ).
□

Proposition 4.3. (A, σ) is direct if and only if σ(a)a ∉ Alt(A, σ) for all a ∈ A\{0}.

Proof. Let (V, h) be a hermitian space over an F –division algebra with involution (D, θ) such that Ad(V, h) ≃ (A, σ).

Assume that there exist a ∈ A\{0} and b ∈ A such that σ(a)a = σ(b) − b. Then for all x ∈ V

h(a(x), a(x)) = h((σ(a)a)(x), x)
= h(σ(b)(x), x) − h(b(x), x) = h(x, b(x)) − h(b(x), x)
= θ(h(b(x), x)) − h(b(x), x) ∈ Alt(D, θ).

In particular, since a ≠ 0, we must have that a(x) ≠ 0 for some x ∈ V, therefore (V, h) is not direct, and (A, σ) is not direct.

Now assume that (V, h) is not direct. Then there exists some x ∈ V \{0} such that h(x, x) = d ∈ Alt(D, θ). Let e ∈ End_{D}(V) be the restriction map e : V → dD, and let σ′ be the restriction of σ to the F –algebra A′ = eAe. Then (A′, σ′) is a symplectic involution, as it is adjoint to ⟨d⟩_{(D,θ)}. Hence for all a′ ∈ A′\{0} we have σ′(a′)a′ ∈ Alt(A′, σ′) by (4.2). In particular σ′(e)e = σ′(b′) − b′ for some b′ ∈ A′.

Then for all e ∈ A\{0} we have that

σ(e) = σ′(b′) − b′ = σ(ebe) − ebe

for some b ∈ A. That is σ(e)e ∈ Alt(A, σ). □

We call an F –algebra with involution (B, τ) a direct part of (A, σ) if (B, τ) ≃ Ad(η_{dir}) for some hermitian form η such that Ad(η) ≃ (A, σ). Similarly we call an F –algebra with involution (C, ρ) an alternating part of (A, σ) if (C, ρ) ≃ Ad(η_{alt}).

For u ∈ A, let Int(u) ∈ End_{F} (A) denote the inner automorphism, that is the map A → A given by a ↦ uau^{−1} for all a ∈ A.

Lemma 4.4. For every involution σ′ of the first kind on A, there exists some u ∈ A^{×}, uniquely determined up to a factor in F^{×}, such that

σ′ = Int(u) ∘ σ and σ(u) = u.

Further, Alt(A, σ′) = u · Alt(A, σ).

Proof. See [11, Proposition 2.7]. □

Proposition 4.5. Let (V, h) be a hermitian form over an F –division algebra (D, θ) and let (W, b) be a hermitian form with over an F –division algebra with involution (D, θ′). If Ad(V, h) ≃ Ad(W, b), then Ad((V, h)_{dir}) ≃ Ad((W, b)_{dir}) and Ad((V, h)_{alt}) ≃ Ad((W, b)_{alt}).

Proof. By (4.4), there exists some u ∈ D^{×} such that θ′ = Int(u) ∘ θ and θ(u) = u. We shall first show that (V, uh) is a hermitian form over (D, θ′). We only need to check that uh(y, x) = θ′(uh(x, y)), for all x, y ∈ V. We have

uh(y, x) = uθ(h(x, y)) = u · u^{−1}θ′(h(x, y))u = θ′(h(x, y))u
= θ′(h(x, y))θ′(u) = θ′(uh(x, y)),

for all x, y ∈ V, as required.

Now note that for all f ∈ End_{D}(V) we have

uh(ad_{h}(x), y) = uh(x, ad_{h}(f)(y)),

as u is invertible, and hence Ad(V, h) = Ad(V, uh) by (2.2). By assumption we have Ad(V, h) ≃ Ad(W, b), and hence Ad(V, uh) ≃ Ad(W, b).

Since (V, uh) and (W, b) are both hermitian forms over (D, θ′) with the same adjoint algebra with involution, it follows from [11, Theorem 4.2] that V = W and μuh = b for some μ ∈ F^{×}. Since a ∈ Alt(D, θ′) if and only if μa ∈ Alt(D, θ′) for all μ ∈ F, it follows that Ad((V, uh)_{dir}) ≃ Ad((W, b)_{dir}) and Ad((V, uh)_{alt}) ≃ Ad((W, b)_{alt}).

Finally, since Alt(D, θ′) = u · Alt(D, θ) by (4.4), it follows that Ad((V, h)_{dir}) ≃ Ad((V, uh)_{dir}) and Ad((V, h)_{alt}) ≃ Ad((V, uh)_{alt}). □

Corollary 4.6. The direct and alternating parts of (A, σ) are uniquely determined up to isomorphism by (A, σ).

We may therefore speak of the direct part and the alternating part of (A, σ), which we denote by (A, σ)_{dir} and (A, σ)_{alt} respectively. Note that for any (A, σ), (A, σ)_{alt} is necessarily symplectic, (A, σ)_{dir} is necessarily orthogonal, and both are necessarily anisotropic.

Example 4.7. Assume char(F) = 2. Let Q be an F –quaternion algebra and γ it’s canonical involution. There exists j ∈ Q \ F such that j^{2} ∈ F^{×} and γ(j) = j. Let A = M_{2}(Q) and σ be the F –involution on A given by

σ ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}γ(a) & γ(c)j^{−1} \\ jγ(b) & jγ(d)j^{−1} \end{matrix} ) .

Then (A, σ)_{dir} = (Q, Int(j) ∘ γ) and (A, σ)_{alt} = (Q, γ).

Proof. We have that (A, σ) = Ad(⟨1, j⟩_{(Q,γ)}). We have (⟨1, j⟩_{(Q,γ)})_{alt} = ⟨1⟩_{(Q,γ)} and (⟨1, j⟩_{(Q,γ)})_{dir} = ⟨j⟩_{(Q,γ)}. The result now follows by taking the adjoint involutions of these hermitian forms. □

For further discussion of this example, see [4, Section 6].

5. Orthogonal Sums

In this section we will briefly discuss the related topic of orthogonal sums of algebras with involution.

A central simple F –algebra with involution (A, σ) is an orthogonal sum of central simple F –algebras with involution (A_{1}, σ_{1}) and (A_{2}, σ_{2}) if there are idempotents e_{1}, e_{2} ∈ A such that e_{1} + e_{2} = 1 and σ(e_{1}) = e_{1}, σ(e_{2}) = e_{2} and for i = 1, 2, there is an F –algebra isomorphism

φ_{i} : A_{i} →^{∼} e_{i}Ae_{i}

such that φ_{i} ∘ σ_{i} = σ ∘ φ_{i}. We may use φ_{1} and φ_{2} to identify A_{1} and A_{2} to subsets of A.

Note that an orthogonal sum is not uniquely determined up to isomorphism by its summands. However, if two algebras with involution (A, σ) and (A′, σ′) are the direct sum of the same summands, then A ≃ A′ and

σ(a) = (u_{1}e_{1} + u_{2}e_{2})σ′(a)(u_{1}^{−1}e_{1} + u_{2}^{−1}e_{2})

for u_{1}, u_{2} ∈ F^{×}, u_{1} ≠ u_{2}.

We say two F –algebras with involution (A_{1}, σ_{1}) and (A_{2}, σ_{2}) are compatible if the following conditions all hold:

(1) A_{1} and A_{2} are Brauer equivalent,

(2) If E is the centre of A_{1} and A_{2}, then σ_{1}|_{E} = σ_{2}|_{E}, (3) If char(F) ≠ 2 and the involutions are of the first kind, then (A_{1}, σ_{1}) and

(A_{2}, σ_{2}) are either both orthogonal or both symplectic. The following propositions are corrections of [2, Proposition 1.2 and 1.3]. The

results there hold in the case of char(F) ≠ 2, but they are incorrectly stated for char(F) = 2.

Proposition 5.1. If an F –algebra with involution (A, σ) is an orthogonal sum of (A_{1}, σ_{1}) and (A_{2}, σ_{2}), then (A_{1}, σ_{1}) and (A_{2}, σ_{2}) are compatible. Moreover deg A = deg A_{1} + deg A_{2}.

Assume (A, σ) is of the first kind. If char(F) ≠ 2, then (A, σ) has the same type as (A_{1}, σ_{1}) and (A_{2}, σ_{2}). If char(F) = 2, then (A, σ) is symplectic if and only if (A_{1}, σ_{1}) and (A_{2}, σ_{2}) are both symplectic.

Proof. The proof in [2, Proposition 1.2] holds for every statement, except for the statement on the types of (A, σ), (A_{1}, σ_{1}) and (A_{2}, σ_{2}) when char(F) = 2. Assume we are in this case.

As in [2, Proposition 1.2] we may identify (A, σ) = Ad(η), (A_{1}, σ_{1}) = Ad(η_{1}) and (A_{2}, σ_{2}) = Ad(η_{2}) where η, η_{1} and η_{2} are λ–hermitian forms over some F –division algebra (D, θ) such that η = η_{1} + η_{2}. The result now follows as η is alternating if and only if both η_{1} and η_{2} are alternating. □

Proposition 5.2. Let (A_{1}σ_{1}) and (A_{2}, σ_{2}) be compatible F –algebras with involution. Then there exists a central simple F –algebra with involution (A, σ) such that (A, σ) is isomorphic to an orthogonal sum (A_{1}, σ_{1}) and (A_{2}, σ_{2}).

Proof. When char(F) ≠ 2 the proof in [2, Proposition 1.3] holds. When char(F) = 2 we may adapt the proof in [2, Proposition 1.3] as in (5.1) to allow for the sum of two algebras with involution of different type. □

Lemma 5.3. An orthogonal sum of metabolic algebras with involution is metabolic.

Proof. With notation as in the proof of (5.1) we may identify any F –algebra with involution (A, σ) that is a direct sum of two F –algebras with involution with Ad(η_{1}+ η_{2}) for λ–hermitian forms η_{1} and η_{2}. Then by (2.4) η_{1} and η_{2} are metabolic, hence η_{1}⊥η_{2} is metabolic by [10, Corollary 3.7.7], and therefore, again by (2.4), (A, σ) is metabolic. □

Lemma 5.4. Every F –algebra with involution (A, σ) decomposes as an orthogonal sum of (A, σ)_{an} and (A′, σ′) where (A′, σ′) is some metabolic F –algebra with involution.

Proof. We may identify (A, σ) = Ad(η) for a λ–hermitian form η over an F –division algebra with involution (D, θ). The result now follows from (2.5). □

Proposition 5.5. Every F –algebra with involution (A, σ) decomposes as an orthogonal sum of (A, σ)_{dir}, (A, σ)_{alt} and (A′, σ′), where (A′, σ′) is some metabolic F –algebra with involution. Further (A, σ)_{an} is isomorphic to an orthogonal sum of (A, σ)_{dir} and (A, σ)_{alt}.

Proof. This follows directly from (5.4) and (3.10). □

Note that while (A, σ)_{dir} and (A, σ)_{alt} are determined up to isomorophism by (4.6), the metabolic involution (A′, σ′) is not generally.

6. Generic splitting over fields of characteristic 2

In this section we will apply the decomposition theorems from the previous sections in order to describe the effect of passing to the generic splitting field of an algebra on the isotropy of involutions on that algebra.

Let A be a central simple F –algebra. A field extension L/F is called a generic splitting field if for any splitting field L′ there exists an F –place of L to L′. It was shown in [1] that every central simple algebra has a generic splitting field.

In [7] the following conjecture is given.

Conjecture 6.1. Let char(F) ≠ 2 and let (A, σ) be a central simple F –algebra with anisotropic orthogonal involution. Let L be a generic splitting field of A. Then (A, σ)_{L} is anisotropic.

We shall show, in (6.4), that the corresponding conjecture for fields of characteristic 2 is false in general, but holds precisely when the involution is direct.

Since our decomposition theorems only have weight over fields of characteristic 2 and involutions of the first kind, throughout the rest of this section we assume F is a field with char(F) = 2 and all involutions are of the first kind. We will show that the anisotropic part of an algebra with involution over a generic splitting field is precisely its direct part extended to the same field.

Proposition 6.2. Let L/F be a separable quadratic extension, and let (A, σ) be a direct F –algebra with involution. Then (A, σ)_{L} is direct. In particular (A, σ)_{L} is anisotropic.

Proof. We may write L = F (δ) with δ ∈ L\F such that δ^{2} + δ + α = 0 for some α ∈ F^{×}. Let a ∈ A_{L} be such that σ_{L}(a)a ∈ Alt(A, σ)_{L}. Then a = b ⊗ 1 + c ⊗ δ for some b, c ∈ A_{L} and we get

σ_{L}(a)a = σ(b)b ⊗ 1 + (σ(b)c + σ(c)b) ⊗ δ + σ(c)c ⊗ δ^{2}
= (σ(b)b + σ(c)c) ⊗ 1 + (σ(b)c + σ(c)b + ασ(c)c) ⊗ δ.

We also have that σ_{L}(a)a = σ_{L}(d) − d for some d ∈ A_{L}. Then d = e ⊗ 1 + f ⊗ δ for some e, f ∈ A_{L} and we get

σ_{L}(a)a = (σ(e) − e) ⊗ 1 + (σ(f) − f) ⊗ δ.

Comparing coefficients of the basis vectors 1 ⊗ 1 and 1 ⊗ δ gives

σ(b)b + σ(c)c = σ(e) − e ∈ Alt(A, σ)
σ(b)c + σ(c)b + ασ(c)c = σ(f) − f ∈ Alt(A, σ).

However,

σ(b)c + σ(c)b = σ(σ(c)b) + σ(c)b ∈ Alt(A, σ)

hence σ(c)c ∈ Alt(A, σ), and hence c = 0, as (A, σ) is direct. It follows that we must have b = 0 as (A, σ) is direct. Hence a = 0, and (A, σ)_{L} is direct. □

Theorem 6.3. Let (A, σ) be an F –algebra with involution and let L be a generic splitting field of A. Then

((A, σ)_{L})_{an} ≃ ((A, σ)_{dir})_{L}.

Proof. By [6, Theorem 9.1.8], every central simple F -algebra with involution is Brauer equivalent to a product of quaternion algebras. Since every quaternion algebra splits over a separable quadratic field extension, it follows that A splits over a field constructed from series of successive quadratic separable field extensions, which we call L′. We have that ((A, σ)_{dir})_{L′} is anisotropic by (6.2).

By (5.5), we have that (A, σ)_{L} is some orthogonal sum of ((A, σ)_{dir})_{L}, ((A, σ)_{alt})_{L} and (A′, σ′)_{L}, where (A′, σ′) is a metabolic F –algebra with involution. Clearly (A′, σ′)_{L} is metabolic and ((A, σ)_{alt})_{L} is metabolic as any symplectic involution is metabolic over any splitting field by [4, Corollary 4.7]. Hence any sum of ((A, σ)_{alt})_{L} and (A′, σ′)_{L} is metabolic by (5.4).

Finally, we must have ((A, σ)_{dir})_{L} is anisotropic as ((A, σ)_{dir})_{L′} is anisotropic and there exists an F –place from L to L′. Hence ((A, σ)_{L})_{an} ≃ ((A, σ)_{dir})_{L} follows from (5.5). □

Theorem 6.4. Let (A, σ) be an anisotropic F –algebra with involution and L a generic splitting field of A.

(1) (A, σ)_{L} is metabolic if and only if (A, σ) is symplectic.

(2) (A, σ)_{L} is anisotropic if and only if (A, σ) is direct.

Proof. If (A, σ) is symplectic then (A, σ)_{L} is metabolic by [4, Corollary 4.7]. If (A, σ) is not symplectic then (A, σ)_{dir} is non-trivial, and hence (A, σ)_{L} is not metabolic by (6.3).

Similarly, if (A, σ) is direct then (A, σ)_{L} is anisotropic by (6.3). If (A, σ) is not direct then (A, σ)_{alt} is non-trivial, and hence (A, σ)_{L} is not anisotropic by (6.3). □

Corollary 6.5. Let (A, σ) be an F –algebra with involution and L a generic splitting field of A. Then (A, σ)_{L} is metabolic if and only if (A, σ)_{an} = (A, σ)_{alt}.

Proof. If (A, σ)_{an} = (A, σ)_{alt} then ((A, σ)_{an})_{L} is metabolic by (6.4), and hence (A, σ)_{L} is metabolic by (5.3). Otherwise (A, σ)_{dir} is non-trivial, and (A, σ)_{L} is not metabolic by (6.3). □

Remark 6.6. Note that (6.5) shows that non-metabolic orthogonal algebra with involution can become metabolic over a generic splitting field. Take (A, σ) to be any anisotropic symplectic involution, and (B, τ) to be a compatible metabolic orthogonal involution. Then any orthogonal sum of (A, σ) and (B, τ) will be orthogonal by (5.4), and become metabolic over any splitting field by (6.5).

Theorem 6.7. Let E be the separable closure of F. Then (A, σ)_{E} is direct if and only if (A, σ) is direct.

Proof. Note that any algebra splits over a separable closure. Therefore if (A, σ) is not direct, then (A, σ)_{E} is isotropic by (6.4).

Now assume (A, σ) is direct. We consider the effect of passing to a separable extension K/F, and show that (A, σ)_{K} is anisotropic. If K/F is a quadratic separable extension, then this follows from (6.2).

Assume now that K/F is an odd degree extension. Let L be a generic splitting field of A (if A is already split, then F = L). Then there exists a field L′/F such that L′/L is of odd degree and K embeds into L′ over F.

By (6.4), we have that (A, σ)_{L} is anisotropic, and by Springer’s theorem, see [5, Corollary 18.5], it follows that (A, σ)_{L′} is anisotropic. L′ is also a splitting field for A_{K}, hence if (A, σ)_{K} were not direct, it would become isotropic over the extension L′/K by (6.4). But (A, σ)_{L′} is anisotropic, hence (A, σ)_{K} is direct

The result now follows, as any separable extension is a series of quadratic separable extensions and odd degree extensions by basic Galois theory. □

Note that the question of whether any general anisotropic algebra with involution stays anisotropic over any odd degree extension, an analogue of Springer’s theorem, is still open. Partial results have been found, such as [12], over fields of characteristic different from 2. (6.7) shows that an analogue of Springer’s theorem holds for direct algebras with involution over fields of characteristic 2.

Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft (project Quadratic Forms and Invariants, BE 2614/3-1) and by the Zukunftskolleg, Universität Konstanz.

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E-mail address: andrew.dolphin@uni-konstanz.de

Universität Konstanz, Fachbreich Mathematik und Statistik, D-78457 Konstanz, Germany