Hermitian forms II

Allen, Harry P

doi:10.1016/0021-8693(68)90076-8

Cited by 8 publications

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“…(1) The kernel of a. is a connected stably rational algebraic group defined over F. (2) For any field extension E/F the image of a(E) in E x equals G+(A^, 0^).…”

Section: Assume Thatmentioning

confidence: 99%

“…-Let E be a finite extension of the field F such that the involution a^ is hyperbolic. Since the discriminant of a^ is trivial, we may assume that L C E, therefore, L®pE == E x E and one of the two components of the Clifford algebra C(A;g, cr^), say G+ = C-^AE, as), splits ([2,Th. 3]).…”

Section: Type Cmentioning

confidence: 99%

“…Proof. -(1) => (2). Since the dimension of the hyperbolic e-hermitian form associated to a is even and the index of A is a 2-power, it follows that the algebra A must be split.…”

Section: Let X E N^/p Nrd(g) We Will Prove That X E Hyp(a A)mentioning

confidence: 99%

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R-equivalence and rationality problem for semisimple adjoint classical algebraic groups

Merkurjev

1996

Publications Mathématiques de l’Institut des Hautes Scientifiqu

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L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ R-EQUIVALENCE AND RATIONALITY PROBLEM FOR SEMISIMPLE ADJOINT CLASSICAL ALGEBRAIC GROUPS by A. S. MERKURJEV ABSTRACT.-The group of R-equivalence classes for all adjoint semisimple classical algebraic groups is computed. Examples of stably non-rational adjoint simple groups of type D^, n ^ 3, are presented. The complete stable birational classification of adjoint simple groups of rank 3 is given.

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“…(1) The kernel of a. is a connected stably rational algebraic group defined over F. (2) For any field extension E/F the image of a(E) in E x equals G+(A^, 0^).…”

Section: Assume Thatmentioning

confidence: 99%

Section: Type Cmentioning

confidence: 99%

See 1 more Smart Citation

R-equivalence and rationality problem for semisimple adjoint classical algebraic groups

Merkurjev

1996

Publications Mathématiques de l’Institut des Hautes Scientifiqu

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“…Then deg A ≡ 0 mod 4 and it was proven by Tits [Tit68,p. 40] and, independently, Allen [All68] that (C 0 (A, σ), σ) ∼ = (S, τ ) × (N, π) where S is split, N is Brauer-equivalent to A, and τ is isotropic. (Tits and Allen did not state that τ is isotropic, but it is clear from their constructions.)…”

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confidence: 99%

Clifford algebras of hyperbolic involutions

Garibaldi

2001

Math Z

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For (A, σ) a central simple algebra of even degree with hyperbolic orthogonal involution, we describe the canonically induced involution σ on the even Clifford algebra (C 0 (A, σ), σ) of (A, σ). When deg A ≡ 0 mod 8, A ∼ = M 2 (B) and the interesting part of σ is isomorphic to the canonical involution on an exterior power algebra of B. As a corollary, we get some properties of the involution on the exterior power algebra.Associated to any central simple algebra (A, σ) of even degree with orthogonal involution over a field F of characteristic / = 2 is its even Clifford algebra (C 0 (A, σ), σ). The even Clifford algebra is also an algebra (although not F -central) with involution (not necessarily orthogonal), see [KMRT98, 8.10], [Jac64, §III], or [Tit68]. Even when given a very specific description of (A, σ), a description of σ is generally unknown. P. Morandi pointed out to me that this is so even in the simplest case, which is when σ is hyperbolic. (Note that A has a hyperbolic involution if and only if it is isomorphic to M 2 (B) for some B. When this occurs, there is up to isomorphism only one hyperbolic involution on A.) In this paper, we describe σ in that case.We begin by looking at the case where A is split and σ is isotropic. In 1.1 we show that σ restricts to be hyperbolic on each component of C 0 (A, σ). This is quite easy to prove, but is very surprising and seems not to be in the literature (except in the case where σ is hyperbolic). As a consequence, we get that whenever σ is orthogonal, it has trivial discriminant.The result when A is split allows us to focus on the situation when A is nonsplit and σ is hyperbolic. Then deg A ≡ 0 mod 4 and it was proven by Tits [Tit68, p. 40] and, independently, Allen [All68] that (C 0 (A, σ), σ) ∼ =

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Isotropic Trialitarian Algebraic Groups

Garibaldi

1998

Journal of Algebra

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Hermitian forms II

Cited by 8 publications

References 10 publications

R-equivalence and rationality problem for semisimple adjoint classical algebraic groups

R-equivalence and rationality problem for semisimple adjoint classical algebraic groups

Clifford algebras of hyperbolic involutions

Isotropic Trialitarian Algebraic Groups

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