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. Let H be the subset of GL × GL containing the pairs 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'.
A J-structure is a triple 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 is the numerator N of j, normalised so that N = 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 defined in terms of the differential dj 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 = Q − Q − Q and distinguished element e such that Q is not trivial. We define a reflection map x* by
and an inversion map j by
Then is a J-structure.

Example

Let Q be the usual sum of squares quadratic function on Kr for fixed integer r, equipped with the standard basis e1,...,er. Then 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 denote multiplication on the left by x. There is a unique birational map i on A such that i.x = e if i is regular on x: it is homogeneous of degree −1 and an involution with i = e. It may be defined by i = L−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 is a J-structure. If 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 and a distinguished element e. We let Q also denote the bilinear map Q = Q − Q − Q. The properties of a quadratic Jordan algebra will be
We call Q'e the square of x. If the squaring is dominant then the algebra is termed separable.
There is a unique birational involution i such that Qi x = x if Q is regular at x. As before, i is the inversion, definable by i = Q−1 x.
If is a J-structure, with quadratic map Q then is a quadratic Jordan algebra. In the opposite direction, if is a separable quadratic Jordan algebra with inversion i, then is a J-structure.

H-structure

McCrimmon proposed a notion of H-structure by dropping the density axiom and strengthening the third 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, that is, a2 = 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 G1. 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.