quadratic Jordan algebra

Definition 1.

Fix a commutative ring R and an R-modules J and a quadratic map U:JEndRJ. Then the triple J,U is a quadratic Jordan algebra if (denoting the evaluation of U by Ux for xJ)

  1. 1.

    UUab=UaUbUa for all a,bJ.

  2. 2.

    The induced bilinear map Ua,b:=Ua+b-Ua-Ub gives rise to an endomorphismMathworldPlanetmathPlanetmathPlanetmath Va,b on J defined by Va,bx=Ua,xb which satisfies

  3. 3.

    If RK is a commutative ring extension of R then the extension JK:=KRJ with the extension UK:=1KU, satisfies the first two axioms.

For a unital quadratic Jordan algebra we include the added assumption that there exist some 1J such that U1 is the identityPlanetmathPlanetmathPlanetmath endomorphism of J.

The concept of a quadratic Jordan algebra was developed by McCrimmon to introduce uniform methods in the study of Jordan algebrasMathworldPlanetmathPlanetmath over characteristic 2. In a strict sense they are not algebrasMathworldPlanetmathPlanetmathPlanetmath as they do not have a bilinear productMathworldPlanetmath; however, their connection to Jordan algebras motivates this terminology.

A common construction for Jordan algebras, so called special Jordan algebra, is by means of using a submodule of an associative algebra A and defining the product as


The 1/2 is optional (and avoided in the analogous Lie bracket definitions [a,b]=ab-ba), in characteristic 2 we can opt to remove it. The result is the usual special Jordan product is also the usual Lie bracket, a.b=ab+ba=ab-ba=[a,b]. So we can treat these algebras as Jordan or Lie algebrasMathworldPlanetmath.

However, the axioms of an abstract Jordan algebra are insufficient to conclude that every Jordan algebra is special (indeed exceptional Jordan algebras called Albert algebras of dimensionMathworldPlanetmathPlanetmathPlanetmath 27 exist and are not special Jordan algebras.) So general Jordan algebra over characteristic 2 may have different structure than a Lie algebra of characteristic 2. To make these algebras manageable, McCrimmon appealed to the quadratic definition given above.

Proposition 2.

If 1/2K then a Jordan algebra over K is a quadratic Jordan algebra where the quadratic map is given by Ua={axa} where {xyz} is the Jordan triple product.

A bonus to this definition is that it highlights the fundamental tools in the study of Jordan algebras. For example, instead of using ideals of the Jordan product it is common to use quadratic ideals, for instance, in the definition of the solvablePlanetmathPlanetmath radicalPlanetmathPlanetmathPlanetmathPlanetmath of a Jordan algebra.

Definition 3.

A submodule I of a quadratic Jordan algebra J is an inner quadratic ideal, or simply an inner ideal if UI(J)I, that is Ui(x)I for all iI, xJ.

A submodule I of a quadratic Jordan algebra J is an outer quadratic ideal, or a outer ideal if UJ(I)I, that is, Ux(i)I for all iI, xJ.

If the quadratic Jordan algebra is derived from a Jordan algebra then Ui(x)={ixi} So we are asking for {iJi}I, and in a special Jordan algebra we can further express this as iJiI.


  • 1 Jacobson, Nathan Structure Theory of Jordan Algebras, The University of Arkansas Lecture Notes in Mathematics, vol. 5, Fayetteville, 1981.
Title quadratic Jordan algebra
Canonical name QuadraticJordanAlgebra
Date of creation 2013-03-22 16:27:58
Last modified on 2013-03-22 16:27:58
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 6
Author Algeboy (12884)
Entry type Derivation
Classification msc 17C05
Related topic QuadraticMap2
Defines quadratic Jordan algebra
Defines inner ideal
Defines outer ideal
Defines quadratic ideal