The canonical quadratic pair on a Clifford algebra and triality

Dolphin, Andrew E.; Quéguiner-Mathieu, Anne

doi:10.1007/s11856-021-2128-y

“…Likewise, Dolphin-Quéguiner-Mathieu show in [7,Prop. 3.6] that a canonical quadratic pair (τ 0 , g 0 ) can be defined on C 0 (V, q) when dim V ≡ 0 mod 8 by associating to τ 0 the following semitrace:…”

Section: Clifford Groups and Lie Algebrasmentioning

confidence: 99%

“…4. A noteworthy feature of our discussion is that the restriction on the characteristic of the base field is made obsolete thanks to the ground-breaking paper [7] of Dolphin-Quéguiner-Mathieu, in which a canonical quadratic pair is defined on Clifford algebras (and where the cohomological approach to the definition of ∂ in arbitrary characteristic is given [7,Th. 4.11]).…”

Section: Introductionmentioning

confidence: 99%

Trialitarian Triples

Barry¹,

Tignol²

2023

Preprint

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Trialitarian triples are triples of central simple algebras of degree 8 with orthogonal involution that provide a convenient structure for the representation of trialitarian algebraic groups as automorphism groups. This paper explicitly describes the canonical "trialitarian" isomorphisms between the spin groups of the algebras with involution involved in a trialitarian triple, using a rationally defined shift operator that cyclically permutes the algebras. The construction relies on compositions of quadratic spaces of dimension 8, which yield all the trialitarian triples of split algebras. No restriction on the characteristic of the base field is needed.

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“…as expected. Likewise, Dolphin-Quéguiner-Mathieu show in [7,Prop. 3.6] that a canonical quadratic pair .…”

Section: Clifford Algebrasmentioning

confidence: 99%

“…The purpose of the first subsections of this section is to recall succinctly the Clifford groups of even-dimensional quadratic spaces and their twisted analogues (in the sense of Galois cohomology), which are defined in arbitrary characteristic through central simple algebras with quadratic pair. Most of the material is taken from [13], but we incorporate a few complements that are made possible by the definition of canonical quadratic pairs on Clifford algebras by Dolphin-Quéguiner-Mathieu [7].…”

Section: Clifford Groups and Lie Algebrasmentioning

confidence: 99%

“…By contrast, the definition of @ in Section 4.3 below is entirely explicit and cohomology-free, and the definition of the trialitarian isomorphisms follows easily in Section 4.4. A noteworthy feature of our discussion is that the restriction on the characteristic of the base field is made obsolete thanks to the ground-breaking paper [7] of Dolphin-Quéguiner-Mathieu, in which a canonical quadratic pair is defined on Clifford algebras (and where the cohomological approach to the definition of @ in arbitrary characteristic is given [7,Th. 4.11]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Trialitarian triples

Barry,

Tignol

2023

Doc. Math.

0

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Trialitarian triples are triples of central simple algebras of degree 8 with orthogonal involution that provide a convenient structure for the representation of trialitarian algebraic groups as automorphism groups. This paper explicitly describes the canonical "trialitarian" isomorphisms between the spin groups of the algebras with involution involved in a trialitarian triple, using a rationally defined shift operator that cyclically permutes the algebras. The construction relies on compositions of quadratic spaces of dimension 8, which yield all the trialitarian triples of split algebras. No restriction on the characteristic of the base field is needed.

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Minimal quadratic forms for the function field of a conic in characteristic 2

Chapman,

Quéguiner-Mathieu

2024

Amitsur Centennial Symposium

0

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In this note, we construct explicit examples of F Q F_Q -minimal quadratic forms of dimension 5 5 and 7 7 , where F Q F_Q is the function field of a conic over a field F F of characteristic 2 2 . The construction uses the fact that any set of n n cyclic p p algebras over a field of characteristic p p can be described using only n + 1 n+1 elements of the base field. It also uses a general result that provides an upper bound on the Witt index of an orthogonal sum of two regular anisotropic quadratic forms over a henselian valued field.

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The canonical quadratic pair on a Clifford algebra and triality

Cited by 3 publications

References 18 publications

Trialitarian Triples

Trialitarian Triples

Trialitarian triples

Minimal quadratic forms for the function field of a conic in characteristic 2

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