J-structure

In mathematics, a J-structure is an algebraic structure over a field related to a Jordan algebra. The concept was introduced by Springer (1973) 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.^[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

$j(e+x) + j(e+j(x)) = e$

• 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

$P(x) = - (d j)_x^{-1} .$

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]

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

$x^* = Q(x,e)e - x$

and an inversion map j by

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

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 e¹,...,e^r. Then (K^r, Q, e^r) is a J-structure of degree 2. It is denoted O₂.^[5]

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.^[6]

A Jordan algebra is defined by the identity^[7][8]

$x(x^2 y) = x^2 (x y) .$

An alternative characterisation is that for all invertible x we have

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

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.

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^[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]

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.^[13][14]

Peirce decomposition

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

$\phi_a(t,u) = Q(ta + u(e-a)) .$

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

$V_a = \left\lbrace{ x \in V : \phi_a(t,u) x = t^2 x }\right\rbrace$

$V'_a = \left\lbrace{ x \in V : \phi_a(t,u) x = tu x }\right\rbrace$

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

and these form a direct sum decomposition of V. This is the Peirce decomposition for the idempotent a.^[16]

Generalisations

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]

References

1. Springer (1973) p.10
2. Springer (1973) p.11
3. Springer (1973) p.16
4. Springer (1973) p.18
5. Springer (1973) p.33
6. Springer (1973) p.66
7. Schafer (1995) p.91
8. Okubo (2005) p.13
9. Springer (1973) p.72
10. McCrimmon (2004) p.83
11. Springer (1973) p.74
12. Springer (1973) p.76
13. McCrimmon (1977)
14. McCrimmon (1978)
15. Springer (1973) p.90
16. Springer (1973) p.92
17. Springer (1973) p.21
18. Springer (1973) p.22

• 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.
• 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.