J-structure

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 is a homogeneous birational involution of degree −1
• j is regular at e and j(e) = e
• if j is regular at x, e + x and e + j(x) then

${\displaystyle j(e+x)+j(e+j(x))=e}$

• the orbit G e of e under the structure group G = G(j) is a Zariski open subset of V.

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

${\displaystyle P(x)=-(dj)_{x}^{-1}.}$

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

${\displaystyle x^{*}=Q(x,e)e-x}$

and an inversion map j by

${\displaystyle j(x)=Q(x)^{-1}x^{*}.}$

Then (V,j,e) is a J-structure.

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]

${\displaystyle x(x^{2}y)=x^{2}(xy).}$

An alternative characterisation is that for all invertible x we have

${\displaystyle x^{-1}(xy)=x(x^{-1}y).}$

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]

• Q(e) = id_V, Q(x,e)y = Q(x,y)e
• Q(Q(x)y) = Q(x)Q(y)Q(x)
• Q(x)Q(y,z)x = Q(Q(x)y,x)z

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]

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

${\displaystyle \phi _{a}(t,u)=Q(ta+u(e-a)).}$

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

${\displaystyle V_{a}=\left\lbrace {x\in V:\phi _{a}(t,u)x=t^{2}x}\right\rbrace }$

${\displaystyle V'_{a}=\left\lbrace {x\in V:\phi _{a}(t,u)x=tux}\right\rbrace }$

${\displaystyle V_{e-a}=\left\lbrace {x\in V:\phi _{a}(t,u)x=u^{2}x}\right\rbrace }$

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]

• McCrimmon, Kevin (1977). "Axioms for inversion in Jordan algebras". J. Algebra. 47: 201–222. doi:10.1016/0021-8693(77)90221-6. Zbl 0421.17013.
• McCrimmon, Kevin (1978). "Jordan algebras and their applications". Bull. Am. Math. Soc. 84: 612–627. doi:10.1090/S0002-9904-1978-14503-0. MR 0466235. Zbl 0421.17010.
• McCrimmon, Kevin (2004). A taste of Jordan algebras. Universitext. Berlin, New York: Springer-Verlag. doi:10.1007/b97489. ISBN 978-0-387-95447-9. MR 2014924. Zbl 1044.17001. Archived from the original on 2012-11-16. Retrieved 2014-05-18.
• Okubo, Susumu (2005) [1995]. Introduction to Octonion and Other Non-Associative Algebras in Physics. Montroll Memorial Lecture Series in Mathematical Physics. 2. Cambridge University Press. doi:10.1017/CBO9780511524479. ISBN 0-521-01792-0. Zbl 0841.17001.
• Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601.
• Springer, T.A. (1973). Jordan algebras and algebraic groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. 75. Berlin-Heidelberg-New York: Springer-Verlag. ISBN 3-540-06104-5. Zbl 0259.17003.