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Quadratic quaternion forms, involutions and triality

Max-Albert Knus, Oliver Villa

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Abstract

Quadratic quaternion forms, introduced by Seip-Hornix (1965), are special cases of generalized quadratic forms over algebras with involutions. We apply the formalism of these generalized qua- dratic forms to give a characteristic free version of difierent results related to hermitian forms over quaternions: 1) An exact sequence of Lewis 2) Involutions of central simple algebras of exponent 2. 3) Triality for 4-dimensional quadratic quaternion forms. 1991 Mathematics Subject Classiflcation: 11E39, 11E88 Keywords and Phrases: Quadratic quaternion forms, Involutions, Tri- ality

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What this paper is about

Quadratic quaternion forms, introduced by Seip-Hornix (1965), are special cases of generalized quadratic forms over algebras with involutions. We apply the formalism of these generalized qua- dratic forms to give a characteristic free version of difierent results related to hermitian forms over quaternions: 1) An exact sequence of Lewis 2) Involutions of central simple algebras of exponent 2. 3) Triality for 4-dimensional quadratic quaternion forms. 1991 Mathematics Subject Classiflcation: 11E39, 11E88 Keywords and Phrases: Quadratic quaternion forms, Involutions, Tri- ality

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Available abstract

Quadratic quaternion forms, introduced by Seip-Hornix (1965), are special cases of generalized quadratic forms over algebras with involutions. We apply the formalism of these generalized qua- dratic forms to give a characteristic free version of difierent results related to hermitian forms over quaternions: 1) An exact sequence of Lewis 2) Involutions of central simple algebras of exponent 2. 3) Triality for 4-dimensional quadratic quaternion forms. 1991 Mathematics Subject Classiflcation: 11E39, 11E88 Keywords and Phrases: Quadratic quaternion forms, Involutions, Tri- ality

Key concepts: Quaternion, Triality, Mathematics, Quadratic equation, Pure mathematics, Hermitian matrix, Quadratic form (statistics), Simple (philosophy)

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