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.
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.[1]
j(e+x)+j(e+j(x))=e
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.[2]
The quadratic map of the structure is a map P from V to End(V) defined in terms of the differential dj at an invertible x.[3] We put
P(x)=-(d
| -1 | |
| j) | |
| x |
.
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.[4]
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
x*=Q(x,e)e-x
and an inversion map j by
j(x)=Q(x)-1x*.
Then (V,j,e) is a J-structure.
Let Q be the usual sum of squares quadratic function on Kr for fixed integer r, equipped with the standard basis e1,...,er. Then (Kr, Q, er) is a J-structure of degree 2. It is denoted O2.[5]
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.[6]
A Jordan algebra is defined by the identity[7] [8]
x(x2y)=x2(xy).
An alternative characterisation is that for all invertible x we have
x-1(xy)=x(x-1y).
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.
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[9] [10]
We call Q(x)e the square of x. If the squaring is dominant (has Zariski dense image) then the algebra is termed separable.[11]
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.[12]
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.[13] [14]
A J-structure has a Peirce decomposition into subspaces determined by idempotent elements.[15] Let a be an idempotent of the J-structure (V,j,e), that is, a2 = a. Let Q be the quadratic map. Define
\phia(t,u)=Q(ta+u(e-a)).
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
Va=\left\lbrace{x\inV:\phia(t,u)x=t2x}\right\rbrace
V'a=\left\lbrace{x\inV:\phia(t,u)x=tux}\right\rbrace
Ve-a=\left\lbrace{x\inV:\phia(t,u)x=u2x}\right\rbrace
and these form a direct sum decomposition of V. This is the Peirce decomposition for the idempotent a.[16]
If we drop the condition on the distinguished element e, we obtain "J-structures without identity".[17] These are related to isotopes of Jordan algebras.[18]
. Jordan algebras and algebraic groups . 0259.17003 . Ergebnisse der Mathematik und ihrer Grenzgebiete . 75 . Berlin-Heidelberg-New York . . 1973 . T.A. Springer . 3-540-06104-5.