J-structure

In mathematics, a J-structure is an algebraic structure over a field related to a Jordan algebra. The concept was introduced by to develop a theory of Jordan algebras using linear algebraic groups and axioms taking the Jordan inversion as basic operation and Hua's identity as a basic relation. There is a classification of simple structures deriving from the classification of semisimple algebraic groups. Over fields of characteristic not equal to 2, the theory of J-structures is essentially the same as that of Jordan algebras.
Definition.
Let "V" be a finite-dimensional vector space over a field "K" and "j" a rational map from "V" to itself, expressible in the form "n"/"N" with "n" a polynomial map from "V" to itself and "N" a polynomial in "K"["V"]. Let "H" be the subset of GL("V") × GL("V") containing the pairs ("g","h") such that "g"∘"j" = "j"∘"h": it is a closed subgroup of the product and the projection onto the first factor, the set of "g" which occur, is the "structure group" of "j", denoted "G"'("j").
A "J-structure" is a triple ("V","j","e") where "V" is a vector space over "K", "j" is a birational map from "V" to itself and "e" is a non-zero element of "V" satisfying the following conditions.
The "norm" associated to a J-structure ("V","j","e") is the numerator "N" of "j", normalised so that "N"("e") = 1. The "degree" of the J-structure is the degree of "N" as a homogeneous polynomial map.
The "quadratic map" of the structure is a map "P" from "V" to End("V") defined in terms of the differential d"j" at an invertible "x". We put
The quadratic map turns out to be a quadratic polynomial map on "V".
The subgroup of the structure group "G" generated by the invertible quadratic maps is the "inner structure group" of the J-structure. It is a closed connected normal subgroup.
J-structures from quadratic forms.
Let "K" have characteristic not equal to 2. Let "Q" be a quadratic form on the vector space "V" over "K" with associated bilinear form "Q"("x","y") = "Q"("x"+"y") − "Q"("x") − "Q"("y") and distinguished element "e" such that "Q"("e".) is not trivial. We define a reflection map "x"* by
and an inversion map "j" by
Then ("V","j","e") is a J-structure.
Example.
Let "Q" be the usual sum of squares quadratic function on "K""r" for fixed integer "r", equipped with the standard basis "e"1...,"e""r". Then ("K""r", "Q", "e""r") is a J-structure of degree 2. It is denoted O2.
Link with Jordan algebras.
In characteristic not equal to 2, which we assume in this section, the theory of J-structures is essentially the same as that of Jordan algebras.
Let "A" be a finite-dimensional commutative non-associative algebra over "K" with identity "e". Let "L"("x") denote multiplication on the left by "x". There is a unique birational map "i" on "A" such that "i"("x")."x" = "e" if "i" is regular on "x": it is homogeneous of degree −1 and an involution with "i"("e") = "e". It may be defined by "i"("x") = "L"("x")−1."e". We call "i" the "inversion" on "A".
A Jordan algebra is defined by the identity
An alternative characterisation is that for all invertible "x" we have
If "A" is a Jordan algebra, then ("A","i","e") is a J-structure. If ("V","j","e") is a J-structure, then there exists a unique Jordan algebra structure on "V" with identity "e" with inversion "j".
Link with quadratic Jordan algebras.
In general characteristic, which we assume in this section, J-structures are related to quadratic Jordan algebras. We take a quadratic Jordan algebra to be a finite dimensional vector space "V" with a quadratic map "Q" from "V" to End("V") and a distinguished element "e". We let "Q" also denote the bilinear map "Q"("x","y") = "Q"("x"+"y") − "Q"("x") − "Q"("y"). The properties of a quadratic Jordan algebra will be
We call "Q"("x")"e" the "square" of "x". If the squaring is dominant (has Zariski dense image) then the algebra is termed "separable".
There is a unique birational involution "i" such that "Q"("x")"i" "x" = "x" if "Q" is regular at "x". As before, "i" is the "inversion", definable by "i"("x") = "Q"("x")−1 "x".
If ("V","j","e") is a J-structure, with quadratic map "Q" then ("V","Q","e") is a quadratic Jordan algebra. In the opposite direction, if ("V","Q","e") is a separable quadratic Jordan algebra with inversion "i", then ("V","i","e") is a J-structure.
H-structure.
McCrimmon proposed a notion of "H-structure" by dropping the density axiom and strengthening the third (a form of Hua's identity) to hold in all isotopes. The resulting structure is categorically equivalent to a quadratic Jordan algebra.
Peirce decomposition.
A J-structure has a Peirce decomposition into subspaces determined by idempotent elements. Let "a" be an idempotent of the J-structure ("V","j","e"), that is, "a"2 = "a". Let "Q" be the quadratic map. Define
This is invertible for non-zero "t","u" in "K" and so φ defines a morphism from the algebraic torus GL1 × GL1 to the inner structure group "G"1. There are subspaces
and these form a direct sum decomposition of "V". This is the Peirce decomposition for the idempotent "a".
Generalisations.
If we drop the condition on the distinguished element "e", we obtain "J-structures without identity". These are related to isotopes of Jordan algebras.