Michel L. Racine

Springer Forms and the First Tits Construction of Exceptional Jordan Division Algebras.

In this paper, a certain quadratic form, originally due to Springer [15], which may be associated with any separable cubic subfield living inside an exceptional simple Jordan algebra is related to the coordinate algebra of an appropriate scalar extension. We use this relation to show that, in the presence of the third roots of unity, exceptional Jordan division algebras arising from the first Tits construction are precisely those where reducing fields and splitting fields agree, and that all isotopes of a first construction exceptional division algebra are isomorphic.

An elementary approach to the Serre-Rost invariant of Albert algebras

In the present paper, we give a proof for the existence of this invariant, called the Serre-Rost invariant in the sequel, that is more elementary than Rost’s. Our approach takes up another suggestion of Serre [26] and is inspired by the concept of chain equivalence [23, p. 143] in the algebraic theory of quadratic forms (see 4.2, 4.13 for details). The proof we obtain in this way works uniformly in all characteristics except 3. (In characteristic 3, Serre has shown how to define the invariant in a different way; see 4.24 for comments). In order to make our presentation comparatively selfcontained, we include without proof some preliminary material from elementary Galois cohomology (Sec. 1) and the theory of algebras of degree 3 (Sec. 2) that will be needed in the subsequent development. Rather than striving for maximum generality, we confine ourselves to what is indispensable for

Reduced models of Albert algebras

SummaryWe prove existence and uniqueness of reduced models for arbitrary Albert algebras and relate them to the Tits process. This relationship yields explicit noncohomological realizations of the invariants mod 2 due to Serre and Rost. We also construct nontrivial examples of Albert division algebras with nonvanishing invariants mod 2.

Classification of algebras arising from the Tits process

In the preceding paper [lo] we generalized the second Tits construction of simple exceptional Jordan algebras [6] to a construction over an arbitrary ring of scalars, which we called the Tits process, and claimed that it plays for Jordan rings a role akin to that of the Cayley-Dickson process for alternative rings. In this paper we assume that the scalars form a field F and determine the semisimple Jordan algebras arising from the Tits process. We see that, except for predictable problems in low characteristics, all simple Jordan algebras of degree 3 are obtainable by starting from Fl and repeating the Tits process. Review Let @ be a unital commutative associative ring,

Minimal identities of symmetric matrices

a @-module. Recall that JV = (N, #, 1) is a cubic norm structure on f if (1) N: f -+ @ is cubic form, # : 2 + f is a quadratic map, 1 is an element of 2, (2) # is an udjoint for N:

Minimal identities for jordan algebras of degree 2

Let Hn (F) denote the subspace of symmetric matrices of Mn (F), the full matrix algebra with coefficients in a field F . The subspace Hn (F) c Mn (F) does not have any polynomial identity of degree less than 2n . Let T (Xl Xk) (1)Xa(l)Xa(2) ... Xa(k) aE5k I 1, T2in is an identity of Hn(F) . If the characteristic of F does not divide e(n)! and if n

CUBIC SUBFIELDS OF EXCEPTIONAL SIMPLE JORDAN ALGEBRAS

3, then any homogeneous polynomial identity of Hn(F) of degree 2n is a consequence of T2n . The case n = 3 is also dealt with. The proofs are algebraic, but an equivalent formulation of the first result in graph-theoretical terms is given.

The Serre-Rost invariant of Albert algebras in characteristic three

In this note we give a basis for the space of multilinear Jordan polynomials of degree 5 which are identities for all Jordan algebras of degree 2 that Is for all algebras J(Q,1) obtained from a quadratic form Q These basic identities are all derived by linearizing the identity for S 3 the standard polynomial of degree 3.

Point spaces in exceptional quadratic Jordan algebras

Let E/k be a cubic field extension and J a simple exceptional Jordan algebra of degree 3 over k. Then E is a reducing field of J if and only if E is isomorphic to a (maximal) subfield of some isotope of J. If k has characteristic not 2 or 3 and contains the third roots of unity then every simple exceptional Jordan division algebra of degree 3 over k contains a cyclic cubic subfield.

*-Polynomial Identities of Matrices with the Symplectic Involution: The Low Degrees

0. Introduction. The authors [5] have recently developed an elementary approach to the Serre-Rost invariant of Albert algebras that is valid in all characteristics except 3. In this special case, Serre [9] has defined the invariant in a different way and established its existence by using Rost’s original results [6] in characteristic zero and reducing them mod 3. It is the purpose of the present note to show that the elementary approach of [5] survives in characteristic 3 as well once the necessary modifications of the cohomological set-up as indicated in [8] have been carried out.