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

Petersson, Holger P.

doi:10.1081/agb-120027965

Cited by 20 publications

(17 citation statements)

References 37 publications

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“…Remark.A similar argument will give a proof of[4, (40.13)] in all characteristics, see[10].4.7. Proof of 4.2.…”

mentioning

confidence: 74%

The étale Tits process of Jordan algebras revisited

Petersson¹,

Thakur²

2004

Journal of Algebra

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“…Remark.A similar argument will give a proof of[4, (40.13)] in all characteristics, see[10].4.7. Proof of 4.2.…”

mentioning

confidence: 74%

The étale Tits process of Jordan algebras revisited

Petersson¹,

Thakur²

2004

Journal of Algebra

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“…For

G

a group of type

F_{4}

, one traditionally decomposes

b false(G false) \in H^{3} false(k, double-struckZ / 6 double-struckZ (2) false)

f_{3} false(G false) + g_{3} false(G false)

for

f_{3} false(G false) \in H^{3} false(k, double-struckZ / 2 double-struckZ (2) false)

and

g_{3} false(G false) \in H^{3} false(k, double-struckZ / 3 double-struckZ (2) false)

. There is furthermore another cohomological invariant

f_{5} false(G false) \in H^{5} false(k, double-struckZ / 2 double-struckZ (4) false)

, see [, 37.16] or [, p. 50] when

char k \neq 2

(in which case

f_{5} false(G false)

belongs to

H^{5} false(k, double-struckZ / 2 double-struckZ false)

) or [, § 4] for arbitrary

k

. (These statements rely on viewing each group of type

F_{4}

over

k

as the automorphism group of a uniquely determined Albert

k

‐algebra.…”

Section: Tits P‐indexes Of Exceptional Groupsmentioning

confidence: 99%

“…For

p = 2

, all three possible indexes occur over the 2‐special field

double-struckR

, so they are also 2‐indexes. Alternatively, one can handle the

p = 2

case by noting that groups of type

F_{4}

over a 2‐special field are of the form

Aut (J)

for an Albert algebra

J

containing a nonzero element with norm zero, that is, such that

J

is reduced as described in [, 1.7] or [, p. 47].…”

Section: Tits P‐indexes Of Exceptional Groupsmentioning

confidence: 99%

Tits p‐indexes of semisimple algebraic groups

Clercq

Garibaldi

2017

J. London Math. Soc.

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Abstract. The first author has recently shown that semisimple algebraic groups are classified up to motivic equivalence by the local versions of the classical Tits indexes over field extensions, known as Tits p-indexes. We provide in this article the complete description of the values of the Tits p-indexes over fields. From this exhaustive study, we also deduce criteria of motivic equivalence for semisimple groups of many types, hence giving a dictionary between classic algebraic structures, representation theory, cohomological invariants and Chow motives of the twisted flag varieties for those groups.The Tits index (sometimes called Satake diagram) of a semisimple linear algebraic group G over a field k includes as special cases the classical notions of Schur index of a central simple associative algebra and the Witt index of a quadratic form. It is a fundamental invariant of semisimple algebraic groups. However, for the purpose of stating and proving theorems about Chow motives with F p coefficients, one should consider not the Tits index of G, but rather the (Tits) p-index, meaning the Tits index of G L where L is an algebraic extension of k of degree not divisible by p, yet all the finite algebraic extensions of L have degree a power of p. Such an L is called a p-special closure of k in [8, §101.B] and all such fields are isomorphic as k-algebras, so the notion of Tits p-index over k is well defined.Let G be a semisimple algebraic group over k. As shown in [7], the Tits p-indexes of G on all fields extensions of k -the higher Tits p-indexes of G -determine the motivic equivalence class of G modulo p. The aim of this article is to determine the values of the Tits p-indexes of the absolutely simple algebraic groups, using as a starting point the known list of possible Tits indexes as in [45], [43], or [36]. Along the way, we give in some cases criteria for motivic equivalence for semisimple groups in terms of their algebraic and cohomological invariants.Date: 10 novembre 2015.

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“…Set K to be the compositum of E 2 and E 3 in some separable closure of k. Since E 2 is Galois over k, the degree [K : k] divides the product [E 2 : k][E 3 : k], hence K has degree dividing g. By construction, K kills g 3 (J ′ ), so J ′ is reduced over K. Since f i (J) agrees with f i (J ′ ) over K for i = 3, 5, the algebras J and J ′ are isomorphic over K by [SV,5.8.1], [Pe,4.1].…”

Section: Proof Of Theorem 03: Type Fmentioning

confidence: 99%

Totaro's Question on Zero-Cycles on $G_2$, $F_4$ and $E_6$ Torsors

Garibaldi

Hoffmann

2006

J. London Math. Soc.

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Abstract. In a 2004 paper, Totaro asked whether a G-torsor X that has a zero-cycle of degree d > 0 will necessarily have a closedétale point of degree dividing d, where G is a connected algebraic group. This question is closely related to several conjectures regarding exceptional algebraic groups. Totaro gave a positive answer to his question in the following cases: G simple, split, and of type G2, type F4, or simply connected of type E6. We extend the list of cases where the answer is "yes" to all groups of type G2 and some nonsplit groups of type F4 and E6. No assumption on the characteristic of the base field is made. The key tool is a lemma regarding linkage of Pfister forms.For certain linear algebraic groups G over a field k and certain homogeneous G-varieties X, Totaro asked in [To]: . It is possible that the answer to (0.2) is "yes" whenever G is semisimple and X is a G-torsor (and, in particular, is affine); no counterexamples are known. In contrast, for X projective, there are examples where the answer is "no" even when d = 1, see [Fl] and [Pa].Every group of type F 4 is the group of automorphisms of some uniquely determined Albert k-algebra J. We say that the group is reduced if the 2000 Mathematics Subject Classification. 11E72 (20G15).

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

Cited by 20 publications

References 37 publications

The étale Tits process of Jordan algebras revisited

The étale Tits process of Jordan algebras revisited

Tits p‐indexes of semisimple algebraic groups

Totaro's Question on Zero-Cycles on $G_2$, $F_4$ and $E_6$ Torsors

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