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 Involutions Directness Hermitian forms Splitting fields 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 ) . σ 2 = id A . 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 Involutions Directness Hermitian forms Splitting fields 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: A K = A ⊗ F K . σ K = σ ⊗ id K . ( A ,σ ) K = ( A K ,σ K ) . Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Involutions Directness Hermitian forms Splitting fields 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 )) = λ h ( 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 Involutions Directness Hermitian forms Splitting fields The adjoint involution Theorem For every λ –hermitian form h on V there exists a unique involution σ on End D ( V ) such that, for all f ∈ End D ( 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 ad h . 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 End D ( V ) . Proposition The involution ad h is isotropic ⇔ h is isotropic. Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Involutions Directness Hermitian forms Splitting fields 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 Involutions Directness Hermitian forms Splitting fields The adjoint involution Definition By Wedderburn’s Theorem for every F–algebra A, there exists a field extension K / F such A K ≃ End K ( 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 Involutions Directness Hermitian forms Splitting fields The adjoint involution We say ( A ,σ ) is metabolic if there exists an anisotropic ideal I ⊂ A such that dim F I = 1 2 dim F A . 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 A L 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 Involutions Directness Hermitian forms Splitting fields The adjoint involution char ( F ) ≠ 2 Algebras with Involution (of the first kind) �������� � � � � � � � � Orthogonal Symplectic Involutions Involutions �������� � � � � � � � � θ Orthogonal θ Symplectic θ Symplectic θ Orthogonal λ = 1 λ = − 1 λ = 1 λ = − 1 ��������������� ������� � � � � � � � � � � � � � � � � � � � � � λ -Hermitian forms � over ( D , θ ) (Up to a factor in F × ) Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Involutions Directness Hermitian forms Splitting fields The adjoint involution char ( F ) = 2 Algebras with Involution (of the first kind) �������� � � � � � � � � � Orthogonal Symplectic Involutions Involutions h non-alternating h alternating ��������� � � � � � � � � � Hermitian forms over ( D , θ ) (Up to a factor in F × )) Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Involutions Directness Hermitian forms Splitting fields The adjoint involution char ( F ) = 2 Algebras with Involution ���������� � � � � � � � � � � Direct Orthogonal Symplectic Involutions Involutions Involutions h non-alternating h alternating h direct ���������� � � � � � � � � � � Hermitian forms over ( D , θ ) (Up to a factor in F × ) Andrew Dolphin Decomposition of involutions in characteristic 2

Involutions and hermitian forms Direct hermitian forms Directness Direct involutions Splitting fields 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 ≃ ϕ � ψ � ρ where ϕ is direct, ψ is anisotropic and alternating and ρ is metabolic. Moreover ϕ and ψ are unique up to isometry and h an ≃ ϕ � ψ. 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 Direct hermitian forms Directness Direct involutions Splitting fields 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 Direct hermitian forms Directness Direct involutions Splitting fields 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 ad h is direct. An hermitian form h is alternating if and only if ad h 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 A L 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 A L 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

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