The étale Tits process of Jordan algebras revisited

Petersson, Holger P.; Thakur, Maneesh

doi:10.1016/j.jalgebra.2002.10.002

Cited by 13 publications

(18 citation statements)

References 21 publications

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“…If L ∼ = k×k splits, theétale Tits process becomes theétale first Tits construction J(E, α) for some α ∈ k × as in 1.12. Combining [PR2, Theorem 1] with [PT,1.6] and [PR7,(1.10.2)], we conclude:…”

Section: 11mentioning

confidence: 74%

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Structure Theorems for Jordan Algebras of Degree Three Over Fields of Arbitrary Characteristic

Petersson

2004

Communications in Algebra

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Classical results, like the construction of a 3-fold Pfister form attached to any central simple associative algebra of degree 3 with involution of the second kind [HKRT], or the SkolemNoether theorem for Albert algebras and their 9-dimensional separable subalgebras [PaST], which originally were derived only over fields of characteristic not 2 (or 3), are extended here to base fields of arbitrary characteristic. The methods we use are quite different from the ones originally employed and, in many cases, lead to expanded versions of the aforementioned results that continue to be valid in any characteristic. IntroductionThanks to their close connection with the Galois cohomology of classical and exceptional groups, Jordan algebras of degree 3 have attracted considerable attention over the last couple of years. The results on the (cohomological) invariants mod 2, based to a large extent on the construction of Haile-Knus-Rost-Tignol [HKRT] attaching a 3-fold Pfister form to any central simple associative algebra of degree 3 with involution of the second kind, are particularly noteworthy in this context, as is the Skolem-Noether theorem of Parimala-Sridharan-Thakur [PaST] for Albert algebras and their 9-dimensional separable subalgebras. Invariably, however, these results, a systematic account of which may be found in [KMRT, § §19, 30,[37][38][39][40], are confined to base fields of characteristic not 2; sometimes even characteristic 3 has to be excluded.In the present paper, an approach to the subject will be developed that yields expanded versions of the aforementioned results over fields of arbitrary characteristic. The methodological framework of our approach is mainly Jordan-theoretic in nature and relies heavily on the Tits process [PR3] for Jordan algebras of degree 3. Another key ingredient is the explicit, characteristic-free description due to 1.8, 2.4] and [PR7, 3.8, 3.9]) of the 3-fold Pfister form attached to a central simple associative algebra of degree 3 with involution of the second kind. Finally, the insight, which goes back to Racine [Ra1], that in this generality the role usually played by 1 Supported by Deutsche Forschungsgemeinschaft. The hospitality of the University of Ottawa, where most results of the paper were obtained, is gratefully acknowledged.

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Section: 11mentioning

confidence: 74%

“…We adapt the proof of [PT,4.5] to the present set-up and first assume that J 1 = H(B, τ ) is reduced, having the form H 3 (K, g) for some diagonal matrix g ∈ GL 3 (k). This implies B = M 3 (K), and w = diag(1, 1, λ) ∈ B does the job.…”

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confidence: 99%

Structure Theorems for Jordan Algebras of Degree Three Over Fields of Arbitrary Characteristic

Petersson

2004

Communications in Algebra

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“…Skolem-Noether type results might come in handy (as they did in the pure case) in tackling general first constructions. However, there is always an obstruction for Skolem-Noether type extension theorem for isomorphisms of cubic subfields (see [20]). One could study the k-subgroups of Str(A) for an Albert algebra and study branching rules for the representation on A for these subgroups.…”

Section: Discussionmentioning

confidence: 99%

Automorphisms of Albert algebras and a conjecture of Tits and Weiss

Thakur

2012

Trans. Amer. Math. Soc.

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Let k be an arbitrary field. The main aim of this paper is to prove the Tits-Weiss conjecture for Albert division algebras over k which are pure first Tits constructions. This conjecture asserts that for an Albert division algebra A over a field k, every norm similarity of A is inner modulo scalar multiplications. It is known that k-forms of E 8 with index E 78 8,2 and anisotropic kernel a strict inner k-form of E 6 correspond bijectively (via Moufang hexagons) to Albert division algebras over k. The Kneser-Tits problem for a form of E 8 as above is equivalent to the Tits-Weiss conjecture (see [31]). Hence we provide a solution to the Kneser-Tits problem for forms of E 8 arising from pure first Tits construction Albert division algebras. As an application, we prove that for G = Aut(A), G(k)/R = 1, where A is a pure first construction Albert division algebra over k and R stands for R-equivalence in the sense of Manin ([10]).

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“…In the remainder of this section, we will be interested in a specialization of the Tits process that is originally due to Petersson and Racine [14] and was taken up later by Petersson and Thakur [16]. Let L (respectively E) be a quadratic (respectively cubic) étale k-algebra and write ι for the non-trivial k-automorphism of L. Then we may specialize the Tits process to K = L, B = E ⊗ L, τ = 1 E ⊗ ι.…”

Section: The éTale Tits Processmentioning

confidence: 99%

“…Cyclic trisotopies and compositions of rank at most 2 are then enumerated in Section 6. We also relate their associated cubic norm structures to simple associative algebras of degree 3 with involution by a slight generalization of the étale Tits process, going back to Petersson and Racine [14], the terminology being due to Petersson and Thakur [16]. The significance of these results derives from the fact that every cyclic trisotopy of rank > 2 up to weak isomorphism contains a cyclic sub-trisotopy of rank equal to 2.…”

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confidence: 99%

Cyclic compositions and trisotopies

Petersson

2007

Journal of Algebra

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Cyclic compositions in the sense of Springer [T.A. Springer, Oktaven, Jordan-Algebren und Ausnahmegruppen, Universität Göttingen, 1963] and Knus, Merkurjev, Rost and Tignol [M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The Book of Involutions, Amer. Math. Soc. Colloq. Publ., vol. 44, Amer. Math. Soc., Providence, RI, 1998. With a preface in French by J. Tits] are investigated by means of cyclic trisotopies, a concept originally due to Albert [A.A. Albert, A construction of exceptional Jordan division algebras, Ann. of Math.(2) 67 (1958) 1-28]. Using the quadrupling of composition algebras, we enumerate cyclic trisotopies and compositions in a rational manner, i.e., without extending the base field. We relate cyclic trisotopies explicitly to simple associative algebras of degree 3 with involution and to the Tits process of cubic Jordan algebras.

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The étale Tits process of Jordan algebras revisited

Cited by 13 publications

References 21 publications

Structure Theorems for Jordan Algebras of Degree Three Over Fields of Arbitrary Characteristic

Structure Theorems for Jordan Algebras of Degree Three Over Fields of Arbitrary Characteristic

Automorphisms of Albert algebras and a conjecture of Tits and Weiss

Cyclic compositions and trisotopies

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