Michel L. Racine's research while affiliated with University of Ottawa and other places
What is this page?
This page lists the scientific contributions of an author, who either does not have a ResearchGate profile, or has not yet added these contributions to their profile.
It was automatically created by ResearchGate to create a record of this author's body of work. We create such pages to advance our goal of creating and maintaining the most comprehensive scientific repository possible. In doing so, we process publicly available (personal) data relating to the author as a member of the scientific community.
If you're a ResearchGate member, you can follow this page to keep up with this author's work.
If you are this author, and you don't want us to display this page anymore, please let us know.
It was automatically created by ResearchGate to create a record of this author's body of work. We create such pages to advance our goal of creating and maintaining the most comprehensive scientific repository possible. In doing so, we process publicly available (personal) data relating to the author as a member of the scientific community.
If you're a ResearchGate member, you can follow this page to keep up with this author's work.
If you are this author, and you don't want us to display this page anymore, please let us know.
Publications (23)
This document presents the solutions to the exercises in the book "Albert algebras over commutative rings" published by Cambridge University Press, 2024.
Albert algebras, a specific kind of Jordan algebra, are naturally distinguished objects among commutative nonassociative algebras and also arise naturally in the context of simple affine group schemes of type $\mathsf {F}_4$ , $\mathsf {E}_6$ , or $\mathsf {E}_7$ . We study these objects over an arbitrary base ring R , with particular attention to...
Albert algebras, a specific kind of Jordan algebra, are naturally distinguished objects among commutative non-associative algebras and also arise naturally in the context of simple affine group schemes of type $F_4$, $E_6$, or $E_7$. We study these objects over an arbitrary base ring $R$, with particular attention to the case of the integers. We pr...
We define Lie multiplication derivations of an arbitrary non-associative algebra A over any commutative ring and, following an approach due to K. McCrimmon, de-scribe them completely if A is alternative. Using this description, we propose a new definition of inner derivations for alternative algebras, among which Schafer's stan-dard derivations and...
A natural octonion algebra structure on the symmetric elements of trace 0 of central simple associative algebras of degree 3 with involution of the second kind is obtained.
A natural octonion algebra structure on the symmetric el- ements of trace 0 of central simple associative algebras of degree 3 with involution of the second kind is obtained.
We 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.
We 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.
In this report, we are concerned with the problem of enumerating and classifying Albert algebras over an arbitrary base field k, for simplicity assumed to be of characteristic not 2 and 3. This assumption, though actually unnecessary, allows us to keep prerequisites from the theory of Jordan algebras at a minimum; in particular, plain old linear Jo...
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 S3 the standard polynomial of degree 3.
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 Jo...
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...
Citations
... Moreover, their automorphism group is an exceptional algebraic group of type F 4 , and their cubic norms have isometry groups of type E 6 . For some recent developments, see [2][3][4][5][6]. ...
... For details on Albert algebras and their cohomological invariants, we refer the reader to the survey article ( [5]) or to ([4]). Given an Albert algebra A over a field k, the mod-2 invariant f 3 (A) ∈ H 3 (k, Z/2Z) is defined as the Arason invariant of the norm form n C of the octonion algebra C of A, i.e. the coordinate algebra of the reduced model of A (see [9]). Let the reduced model of A be written as H 3 (C, Γ) (see [9]), for ...
Reference: The cyclicity problem for Albert algebras
... Proposition 3.5 (see also [18]), ensures that for a rank 3 Jordan algebra ...
... The significance of the preceding result is underscored by the fact that pure first Tits constructions exist in abundance. For example, all Albert division algebras over the iterated Laurent series field in several variables with complex coefficients are pure first Tits constructions ( [PR86c]), see also 13.7 below for more details. ...
Reference: A SURVEY ON ALBERT ALGEBRAS
... We remind the reader that H. Petersson and M. Racine extended the construction of f 3 , f 5 , g 3 to fields of characteristic 2 and 3 (see [16], [17], [18]). ...
... It is known that every Albert algebra over a field is obtained by the first Tits construction or second Tits construction (which we have not described here) (see [McC70,Theorem 10] or [PeR86b,Theorem 3.1(i)]). Both constructions have been extended from the case of algebras over a field to an arbitrary base ring [PeR86a]. ...
Reference: Albert algebras over and other rings
... Remark 12.5 (dichotomy of fields and the Tits construction). The classification of Albert algebras over a field F of characteristic ≠ 3 has a fundamentally different flavor depending on whether or not 3 ( , Z/3) is zero, as indicated by [Rost], [PeR96], or [Gar09,Section 8]. If 3 ( , Z/3) = 0 -as is the case for global fields, p-adic fields, and the real numbers -every Albert F-algebra is reduced, that is, of the form Her 3 ( , Γ) for some C and Γ, and is not a division algebra. ...
Reference: Albert algebras over and other rings
... Proposition A. 16. The groups J and † 0 have the same image in Aut( 6 ). ...
... We remind the reader that H. Petersson and M. Racine extended the construction of f 3 , f 5 , g 3 to fields of characteristic 2 and 3 (see [16], [17], [18]). ...
... Note that these are relations that also hold for a cubic form with adjoint and base point (N, , 1) [15,17]. ...
