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.Springer (1973) p.10

• 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.Springer (1973) p.11

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.Springer (1973) p.16 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.Springer (1973) p.18

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.

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.Springer (1973) p.33

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.Springer (1973) p.66

A Jordan algebra is defined by the identitySchafer (1995) p.91Okubo (2005) p.13

:$x(x^2\; y)\; =\; x^2\; (x\; y)\; .$

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.

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 beSpringer (1973) p.72McCrimmon (2004) p.83

• 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.Springer (1973) p.74

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.Springer (1973) p.76

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.McCrimmon (1977)McCrimmon (1978)

A J-structure has a Peirce decomposition into subspaces determined by idempotent elements.Springer (1973) p.90 Let a be an idempotent of the J-structure (V,j,e), that is, a² = a. Let Q be the quadratic map. Define

:$\backslash phi\_a(t,u)\; =\; Q(ta\; +\; u(e-a))\; .$

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

:$V\_a\; =\; \backslash left\backslash lbrace\{\; x\; \backslash in\; V\; :\; \backslash phi\_a(t,u)\; x\; =\; t^2\; x\; \}\backslash right\backslash rbrace$

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

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

and these form a direct sum decomposition of V. This is the Peirce decomposition for the idempotent a.Springer (1973) p.92

If we drop the condition on the distinguished element e, we obtain "J-structures without identity".Springer (1973) p.21 These are related to isotopes of Jordan algebras.Springer (1973) p.22

{{reflist}}

• {{cite journal | last=McCrimmon | first=Kevin |authorlink=Kevin McCrimmon | title=Axioms for inversion in Jordan algebras | zbl=0421.17013 | journal=J. Algebra | volume=47 | pages=201–222 | year=1977 | doi=10.1016/0021-8693(77)90221-6| doi-access=free }}
• {{cite journal | last1=McCrimmon | first1=Kevin | title=Jordan algebras and their applications | journal=Bull. Am. Math. Soc. | volume=84 | pages=612–627 | year=1978 | zbl=0421.17010 | mr=0466235 | url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-84/issue-4/Jordan-algebras-and-their-applications/bams/1183540925.pdf | doi=10.1090/S0002-9904-1978-14503-0 | doi-access=free }}
• {{cite book | last1=McCrimmon | first1=Kevin | title=A taste of Jordan algebras | publisher=Springer-Verlag | location=Berlin, New York | series=Universitext | isbn=978-0-387-95447-9 | doi=10.1007/b97489 | url=http://www.math.virginia.edu/Faculty/McCrimmon/ | year=2004 | mr=2014924 | zbl=1044.17001 | access-date=2014-05-18 | archive-url=https://web.archive.org/web/20121116162444/http://www.math.virginia.edu/Faculty/McCrimmon/ | archive-date=2012-11-16 | url-status=dead }}
• {{cite book |last=Okubo |first=Susumu |title=Introduction to Octonion and Other Non-Associative Algebras in Physics | year=2005 | origyear=1995 | publisher=Cambridge University Press | isbn=0-521-01792-0 | doi=10.1017/CBO9780511524479 | zbl=0841.17001 | series=Montroll Memorial Lecture Series in Mathematical Physics | volume=2 }}
• {{cite book |first=Richard D. |last=Schafer |title=An Introduction to Nonassociative Algebras |year=1995 |origyear=1966 |publisher=Dover |isbn=0-486-68813-5 |url=https://archive.org/details/introductiontono0000scha |zbl=0145.25601 |url-access=registration }}
• {{cite book | last=Springer | first=T.A. | title=Jordan algebras and algebraic groups | zbl=0259.17003 | series=Ergebnisse der Mathematik und ihrer Grenzgebiete | volume=75 | location=Berlin-Heidelberg-New York | publisher=Springer-Verlag | year=1973 | authorlink=T.A. Springer | isbn=3-540-06104-5}}

Category:Algebraic structures