Decomposition of involutions in characteristic 2 Andrew Dolphin - - PowerPoint PPT Presentation

Involutions and hermitian forms Directness Splitting fields

Decomposition of involutions in characteristic 2

Andrew Dolphin

Universität Konstanz

May, 2011

Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Directness Splitting fields Involutions Hermitian forms The adjoint involution

Let F be a field (of arbitrary characteristic) and A a central simple algebra over F. Definition An involution (of the first kind) on A is an F–linear map σ ∶ A → A such that for all x,y ∈ A and a ∈ F: σ(xy) = σ(y)σ(x). σ2 = idA. We say an involution is isotropic if there exists an a ∈ A/{0} such that σ(a)a = 0, and anisotropic otherwise.

Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Directness Splitting fields Involutions Hermitian forms The adjoint involution

We will refer to a pair (A,σ) as an F-algebra with involution. We wish to study the effect of passing to a field extension on the anisotropy of algebra with involution. For an F–algebra with involution (A,σ) and a field extension K/F we denote: AK = A ⊗F K. σK = σ ⊗ idK. (A,σ)K = (AK,σK).

Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Directness Splitting fields Involutions Hermitian forms The adjoint involution

Let (D,θ) be an F–division algebra with involution and λ = ±1. Definition Let V be a right D–vector space. A λ–hermitian form on V, with respect to (D,θ), is a non-singular bi-additive map h ∶ V × V → D subject to the following conditions for all x,y ∈ V and d ∈ D: θ(h(x,y)) = λh(y,x). h(x,yd) = h(x,y)d. We say h is hermitian if λ = 1 and skew–hermitian if λ = −1. We say h is isotropic if there exists x ∈ V/{0} such that h(x,x) = 0, and anisotropic otherwise.

Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Directness Splitting fields Involutions Hermitian forms The adjoint involution

Theorem For every λ–hermitian form h on V there exists a unique involution σ on EndD(V) such that, for all f ∈ EndD(V) and x,y ∈ V, h(x,f(y)) = h(σ(f)(x),y). The involution σ is called the adjoint involution with respect to h, and we denote it adh. This gives a one-to-one correspondence between hermitian and skew-hermitian forms on V (with respect to θ) up to a factor in F × and involutions on EndD(V). Proposition The involution adh is isotropic ⇔ h is isotropic.

Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Directness Splitting fields Involutions Hermitian forms The adjoint involution

Definition We call the set of elements Alt(A,σ) = {σ(a) − a ∣ a ∈ A} the alternating elements of (A,σ). We call a λ–hermitian form on (D,θ) alternating if h(x,x) ∈ Alt(D,θ) for all x ∈ V.

Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Directness Splitting fields Involutions Hermitian forms The adjoint involution

Definition By Wedderburn’s Theorem for every F–algebra A, there exists a field extension K/F such AK ≃ EndK(V). We call such a K a splitting field. We call an algebra with involution (A,σ) symplectic if (A,σ)K is isomorphic to the adjoint algebra of an alternating bilinear form, for some splitting field K . Otherwise we call (A,σ) orthogonal. This definition is independent of the choice of K.

Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Directness Splitting fields Involutions Hermitian forms The adjoint involution

We say (A,σ) is metabolic if there exists an anisotropic ideal I ⊂ A such that dimFI = 1

2dimFA.

Theorem (Karpenko) Assume that the characteristic of F is different from 2. Let (A,σ) be a non-metabolic F–algebra with involution. Then there exists a field extension L/F such that AL is split and (A,σ)L not metabolic if and only if (A,σ) is orthogonal. This is not true if the characteristic of F is 2.

Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Directness Splitting fields Involutions Hermitian forms The adjoint involution

char(F) ≠ 2

Algebras with Involution (of the first kind) Orthogonal Involutions

• Symplectic

Involutions

• θ Orthogonal

λ = 1

• θ Symplectic

λ = −1 θ Symplectic λ = 1 θ Orthogonal λ = −1

• λ-Hermitian forms
• ver (D, θ)

(Up to a factor in F ×)

• Andrew Dolphin

Decomposition of involutions in characteristic 2

Involutions and hermitian forms Directness Splitting fields Involutions Hermitian forms The adjoint involution

char(F) = 2

Algebras with Involution (of the first kind) Orthogonal Involutions

• Symplectic

Involutions

• h non-alternating

h alternating Hermitian forms

• ver (D, θ)

(Up to a factor in F ×))

• Andrew Dolphin

Decomposition of involutions in characteristic 2

Involutions and hermitian forms Directness Splitting fields Involutions Hermitian forms The adjoint involution

char(F) = 2

Algebras with Involution Direct Involutions

• Orthogonal

Involutions Symplectic Involutions

• h direct

h non-alternating h alternating Hermitian forms

• ver (D, θ)

(Up to a factor in F ×)

• Andrew Dolphin

Decomposition of involutions in characteristic 2

Involutions and hermitian forms Directness Splitting fields Direct hermitian forms Direct involutions

Definition We call an hermitian form h over (D,θ) direct if h(x,x) ∉ Alt(D,θ) for all x ∈ V/{0}. Theorem (D.) Let h be a hermitian form. Then there exists a decomposition h ≃ ϕψρ where ϕ is direct, ψ is anisotropic and alternating and ρ is metabolic. Moreover ϕ and ψ are unique up to isometry and han ≃ ϕψ. We call ϕ the direct part of h, and ψ the alternating part of h.

Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Directness Splitting fields Direct hermitian forms Direct involutions

This concept is trivial in char(F) ≠ 2. Proposition Assume char(F) ≠ 2. A λ–hermitian form h is direct ⇔ h is anisotropic and λ = 1. A λ–hermitian form h is alternating ⇔ λ = −1. So for the rest of the talk, we assume char(F) = 2.

Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Directness Splitting fields Direct hermitian forms Direct involutions

Definition We call an involution (A,σ) direct if σ(a)a ∉ Alt(A,σ) for all a ∈ A/{0}. Proposition An involution (A,σ) is symplectic if and only if σ(a)a ∈ Alt(A,σ) for all a ∈ A. Theorem An hermitian form h is direct if and only if adh is direct. An hermitian form h is alternating if and only if adh is sympletic.

Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Directness Splitting fields

Theorem (D.) Let (A,σ) be an anisotropic F–algebra with involution. Then there exists a field extension L/F such that AL is split and (A,σ)L anisotropic if and only if (A,σ) is direct. “Only if” follows from the decomposition theorem and the following fact: Proposition Let (A,σ) be an F–algebra with symplectic involution and L/F a field extension such that AL is split. Then (A,σ)L is hyperbolic.

Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Directness Splitting fields

Lemma Let (A,σ) be an anisotropic F–algebra with involution and K/F a quadratic separable extension. Then (A,σ)K is direct if (A,σ) is direct. In particular (A,σ)K is anisotropic. We prove the “if” part of our main theorem. Sketch Proof Every central simple F–algebra of even degree is Brauer equivalent to a product of quaternion algebras (Albert, 1930s) Every quaternion algebra splits over a quadratic separable extension. Hence every central simple algebra splits over a series of quadratic separable extensions. Apply lemma.

Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Directness Splitting fields

Proof. We may write L = F(δ) with δ ∈ L/F such that δ2 + δ + α = 0 for some α ∈ F ×. Let a ∈ AL be such that σL(a)a = 0. Then a = b ⊗ 1 + c ⊗ δ for some b,c ∈ AL and we get = σ(b)b ⊗ 1 + (σ(b)c + σ(c)b) ⊗ δ + σ(c)c ⊗ δ2 = (σ(b)b + σ(c)c) ⊗ 1 + (σ(b)c + σ(c)b + ασ(c)c) ⊗ δ. Comparing coefficients of the basis vectors 1⊗1 and 1⊗δ gives σ(b)b + σ(c)c = σ(b)c + σ(c)b + ασ(c)c = 0. 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 anisotropic.

Andrew Dolphin Decomposition of involutions in characteristic 2