Decomposition of Mal'cev-Neumann division algebras with involution

Dherte, H.

doi:10.1007/bf02572343

Cited by 3 publications

(6 citation statements)

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“…4.3] the analogue of condition (2) does not appear. However, the argument given for (2) ⇒ (1) in that paper does in fact show that the analogue of this condition is equivalent to the other three conditions of [4,Thm. 4.3].…”

Section: S Decomposes Into a Tensor Product Of Quaternion Algebras Stmentioning

confidence: 93%

“…, √ a n ) be an elementary Abelian extension of F of degree 2 n and let A be an F -central simple algebra of degree 2 n containing K. If A decomposes into a tensor product of quaternion algebras as A = (a 1 , b 1 ) ⊗ F · · · ⊗ F (a n , b n ) for some b i ∈ F * , we say that A has a decomposition adapted to K into quaternion algebras. We now extend [4,Thm. 4.1] to the case of inertially split division algebras over a Henselian valued field of residue characteristic not 2.…”

Section: Decompositions Of Involutionsmentioning

confidence: 99%

“…In [4] Dherte studied decomposability of involutions on certain MalcevNeumann division algebras. More precisely, if K/F is an elementary Abelian 2-extension with Galois group G, and if f ∈ Z 2 (G, K * ), then one can construct the Malcev-Neumann series division ring S(A/K/F ), where A is the crossed product algebra (K/F, G, f ).…”

Section: Introductionmentioning

confidence: 99%

“…If S(A/K/F ) = ⊕ g∈G Kz g with z g z h = f (g, h)z gh , then there is an involution σ * on S(A/K/F ) with σ * | K = id and σ * (z g ) = z g . Dherte proved that S(A/K/F ) has a σ * -stable F -central quaternion algebra if and only if A has a σ-stable F -central quaternion algebra generated by "monomials," and also that (S(A/K/F ), σ * ) decomposes into a tensor product of quaternion algebras if and only if (A, σ) decomposes into a tensor product of quaternion algebras "adapted to K" (see [4,Thm. 4.1, pp.…”

Section: Introductionmentioning

confidence: 99%

“…636] for exact statements). A fundamental technique in [4] was valuation theory; the field F has a Henselian valuation with residue field F , and so this valuation extends to S(A/K/F ).…”

Section: Introductionmentioning

confidence: 99%

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Decomposition of involutions on inertially split division algebras

Morandi¹,

Sethuraman²

2000

Mathematische Zeitschrift

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Let F be a Henselian valued field with char(F ) = 2, and let S be an inertially split F -central division algebra with involution σ * that is trivial on an inertial lift in S of the field Z(S). We prove necessary and sufficient conditions for S to contain a σ * -stable quaternion F -subalgebra, and for (S, σ * ) to decompose into a tensor product of quaternion algebras. These conditions are in terms of decomposability of an associated residue central simple algebra I that arises from a Brauer group decomposition of S.

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Section: S Decomposes Into a Tensor Product Of Quaternion Algebras Stmentioning

confidence: 93%

Section: Decompositions Of Involutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…636] for exact statements). A fundamental technique in [4] was valuation theory; the field F has a Henselian valuation with residue field F , and so this valuation extends to S(A/K/F ).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Decomposition of involutions on inertially split division algebras

Morandi¹,

Sethuraman²

2000

Mathematische Zeitschrift

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Discriminant and Clifford algebras

Quéguiner-Mathieu

Tignol

2002

Mathematische Zeitschrift

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The centralizer of a square-central skew-symmetric unit in a central simple algebra with orthogonal involution carries a unitary involution. The discriminant algebra of this unitary involution is shown to be an orthogonal summand in one of the components of the Clifford algebra of the orthogonal involution. As an application, structure theorems for orthogonal involutions on central simple algebras of degree 8 are obtained.Throughout this paper, F denotes a field of characteristic different from 2. Let A be a central simple F -algebra of degree n = 4m, for some integer m, endowed with an involution σ of orthogonal type. In the first two sections, we assume that the algebra A contains an element θ such that σ(θ) = −θ and θ 2 = a ∈ F . We denote byÃ the centralizer of θ in A. Since θ is skew-symmetric, σ induces an involutionσ onÃ, and (Ã,σ) is a central simple algebra with unitary involution of degree 2m over theétale quadratic extension F (θ) of F .With this data, we may associate two different central simple algebras with involution, namely the Clifford algebra of (A, σ), and the discriminant algebra of (Ã,σ), both endowed with their canonical involution (see [14, §8, §10]). Our aim in this paper is to relate these two algebras with involution.Reversing the viewpoint, one can show that every central simple algebra over a quadratic extension of F of even degree 2m endowed with a unitary involution and of exponent 2, can be embedded in a central simple F -algebra

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