Discriminant and Clifford algebras

Quéguiner-Mathieu, Anne; Tignol, Jean-Pierre

doi:10.1007/s002090100385

Cited by 10 publications

(7 citation statements)

References 14 publications

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“…On the other hand, taking for A an indecomposable algebra of degree 8 yields examples where (b) holds but (a) does not (see [9,Ex. 3.6]), whereas Proposition 1.1 shows that (a) and (b) are equivalent when deg A = 12.…”

Section: Decomposabilitymentioning

confidence: 99%

Decomposability of orthogonal involutions in degree 12

Quéguiner-Mathieu

Tignol

2020

Pacific J. Math.

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A theorem of Pfister asserts that every 12-dimensional quadratic form with trivial discriminant and trivial Clifford invariant over a field of characteristic different from 2 decomposes as a tensor product of a binary quadratic form and a 6-dimensional quadratic form with trivial discriminant. The main result of the paper extends Pfister's result to orthogonal involutions : every central simple algebra of degree 12 with orthogonal involution of trivial discriminant and trivial Clifford invariant decomposes into a tensor product of a quaternion algebra and a central simple algebra of degree 6 with orthogonal involutions. This decomposition is used to establish a criterion for the existence of orthogonal involutions with trivial invariants on algebras of degree 12, and to calculate the f 3 -invariant of the involution if the algebra has index 2.

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Section: Decomposabilitymentioning

confidence: 99%

Decomposability of orthogonal involutions in degree 12

Quéguiner-Mathieu

Tignol

2020

Pacific J. Math.

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“…In particular, any triple which includes a division algebra cannot be obtained from this proposition. Consider for instance the algebra with involution (A, σ ) described in [34,Example 3.6], and let (B, τ ) and (C, γ ) be the two components of its Clifford algebra. As explained there, A is a indecomposable division algebra, and one component of its Clifford algebra, say B, has index 2.…”

Section: Explicit Examplesmentioning

confidence: 99%

TheJ-invariant, Tits algebras and triality

Quéguiner-Mathieu

Semenov

Zainoulline

2012

Journal of Pure and Applied Algebra

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a b s t r a c tIn the present paper we set up a connection between the indices of the Tits algebras of a semisimple linear algebraic group G and the degree one indices of its motivic J-invariant.Our main technical tools are the second Chern class map and Grothendieck's γ -filtration.As an application we provide lower and upper bounds for the degree one indices of the J-invariant of an algebra A with orthogonal involution σ and describe all possible values of the J-invariant in the trialitarian case, i.e., when degree of A equals 8. Moreover, we establish several relations between the J-invariant of (A, σ ) and the J-invariant of the corresponding quadratic form over the function field of the Severi-Brauer variety of A.

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“…Using either the proposition in the Appendix of [24], or [3, 3.4(2)], one may easily check that (A, σ) decomposes as…”

Section: When This Isomorphism Holds We Havementioning

confidence: 99%

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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Discriminant and Clifford algebras

Cited by 10 publications

References 14 publications

Decomposability of orthogonal involutions in degree 12

Decomposability of orthogonal involutions in degree 12

TheJ-invariant, Tits algebras and triality

Cohomological invariants for orthogonal involutions on degree 8 algebras

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