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'' '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 d''j'' at an invertible ''x''.Springer (1973) p.16 We put
:$P(x) = - (d j)_x^ .$

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.Springer (1973) p.18

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

:$j(x) = Q(x)^ x^* .$

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 ''e''¹,...,''e''^''r''. Then (''K''^''r'', ''Q'', ''e''^''r'') is a J-structure of degree 2. It is denoted O₂.Springer (1973) p.33

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''.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

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

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 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

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.McCrimmon (1977)McCrimmon (1978)

Peirce decomposition

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, ''a''² = ''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 ''G''₁. There are subspaces

:$V_a = \left\lbrace\right\rbrace$
:$V'_a = \left\lbrace\right\rbrace$
:$V_ = \left\lbrace\right\rbrace$

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

Generalisations

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

References

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* {{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

Algebraic structures