Springer forms and the first Tits construction of exceptional Jordan division algebras

Petersson, Holger P.; Racine, Michel L.

doi:10.1007/bf01158039

Cited by 31 publications

(29 citation statements)

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“…In particular, ,7 must be a pure second Tits construction. Summing up, we have arrived at an example of a pure second Tits construction Albert division algebra which is arguably more accessible than the one given by Albert [2] as well as the Albert algebra of generic matrices [11]. …”

Section: Examples: the Unital Tits Processmentioning

confidence: 99%

“…We also obtain several characterizations of those absolutely simple Jordan algebras of degree 3 and dimension 9 whose octonion algebras are split (2.11). Furthermore, the explicit description we have derived for the reduced model leads to a (comparatively) straightforward construction of Albert division algebras with nonzero invariants mod 2 (4.11), forcing them to be pure second Tits constructions in the sense of [17].…”

Section: Introductionmentioning

confidence: 99%

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Reduced models of Albert algebras

Petersson

Racine

1996

Math Z

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Section: Examples: the Unital Tits Processmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Reduced models of Albert algebras

Petersson

Racine

1996

Math Z

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“…But then, by [PR4,Corollary 3] and [PR2,(4.8)], it follows that J 2 contains an isomorphic copy of L. Therefore, J 1 and J 2 contain the same cubic subfields. Now, in J 1 x = 0 1 0 is a Kummer element with x 3 = µ.…”

Section: Theorem 3 ([Pr3 Theorem 2 ])mentioning

confidence: 83%

Kummer Elements and the mod-3 Invariant of Albert Algebras

Thakur

2001

Journal of Algebra

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Let k be a field with characteristic not 2 and 3. Assume that k contains the cube roots of unity. Let J be a Tits'-first-construction Albert division algebra over k. In this paper we relate Kummer elements in J with the mod-3 invariant g 3 J . We prove that if x ∈ J is a Kummer element with x 3 = λ, then J J D λ for some D, a degree-3 central division algebra over k. We show that if J 1 = J A µ and J 2 = J B ν are Tits' first-construction Albert division algebras withAcademic Press

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“…Notice that our definition of q E differs from the one in [11,14] by a sign, bringing us back to the original normalization due to Springer, cf. Springer and Veldkamp [17, (6.5)].…”

Section: Springer Formsmentioning

confidence: 96%