Quadratic forms of dimension 8 with trivial discriminant and Clifford algebra of index 4

Masquelein, Alexandre; Quéguiner-Mathieu, Anne; Tignol, Jean-Pierre

doi:10.1007/s00013-009-0019-2

Cited by 5 publications

(1 citation statement)

References 11 publications

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“…It is relevant to mention that in characteristic = 2, the behavior of J τ on simple components of C 0 (V ), even for the simplest case τ = id, has been of importance in many applications of Clifford algebras in the literature (see, e.g., [3], [16], [11], [13]). …”

Section: Introductionmentioning

confidence: 99%

Involutions of a Clifford Algebra Induced by Involutions of Orthogonal Group in Characteristic 2

Mahmoudi

Nokhodkar

2015

Communications in Algebra

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Section: Introductionmentioning

confidence: 99%

Involutions of a Clifford Algebra Induced by Involutions of Orthogonal Group in Characteristic 2

Mahmoudi

Nokhodkar

2015

Communications in Algebra

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Cohomological Invariants of Central Simple Algebras with Involution

Tignol

2010

Developments in Mathematics

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Cohomological invariants for orthogonal involutions on degree 8 algebras

Quéguiner-Mathieu¹,

Tignol²

2011

J K-Theor

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Abstract. Using triality, we define a relative Arason invariant for orthogonal involutions on a -possibly division-central simple algebra of degree 8. This invariant detects hyperbolicity, but it does not detect isomorphism. We produce explicit examples, in index 4 and 8, of non isomorphic involutions with trivial relative Arason invariant.The discriminant and the Clifford algebra are classical invariants of quadratic forms over a field F of characteristic different from 2. Up to similarity, the discriminant classifies quadratic forms of dimension 2, while the even part of the Clifford algebra classifies forms of dimension 4. In [2], Arason defined an invariant e 3 , for even-dimensional quadratic forms with trivial discriminant and split Clifford algebra, which has values in H 3 (F, µ 2 ). Again, this invariant is classifying in dimension 8, that is for forms similar to a 3-fold Pfister form. The main purpose of this paper is to study the same question for orthogonal involutions on central simple algebras of degree 8.To see the relation between quadratic forms and involutions, recall that every (nondegenerate) quadratic form q on an F -vector space V defines an adjoint involution ad q on the endomorphism algebra End F V , and every orthogonal involution on End F V has the form ad q for some (nondegenerate) quadratic form q on V , uniquely determined up to a scalar factor, see [18, p. 1]. Therefore, orthogonal involutions on central simple algebras can be viewed as twisted forms (in the sense of Galois cohomology) of quadratic forms up to scalars. Analogues for orthogonal involutions of the discriminant and the (even) Clifford algebra were defined by Jacobson and Tits. By [18, (7.4) and § 15.B], the discriminant classifies orthogonal involutions on a given quaternion algebra, and the Clifford algebra classifies orthogonal involutions on a given biquaternion algebra.On the other hand, it was shown in [4, § 3.4] that the Arason invariant does not extend to orthogonal involutions if the underlying algebra is a degree 8 division algebra. In this paper, we define a relative Arason invariant for orthogonal involutions on a degree 8 (possibly division) central simple algebra A over F . Our construction, based on triality [18, § 42], is specific to this degree. Namely, we assign to any pair of involutions (σ, σ ′ ) on A with trivial discriminant and Clifford invariant, a degree 3 cohomology class e 3

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Quadratic forms of dimension 8 with trivial discriminant and Clifford algebra of index 4

Cited by 5 publications

References 11 publications

Involutions of a Clifford Algebra Induced by Involutions of Orthogonal Group in Characteristic 2

Involutions of a Clifford Algebra Induced by Involutions of Orthogonal Group in Characteristic 2

Cohomological Invariants of Central Simple Algebras with Involution

Cohomological invariants for orthogonal involutions on degree 8 algebras

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