PlanetMath: quadratic Jordan algebra

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

quadratic Jordan algebra

(Derivation)

Definition 1 Fix a commutative ring and an -modules and a quadratic map $U:J\rightarrow {\mathrm{End}}_R J$ . Then the triple $\langle J,U\rangle$ is a quadratic Jordan algebra if (denoting the evaluation of by for $x\in J$ )

$U_{U_a b}=U_a U_b U_a$ for all $a,b\in J$ .
The induced bilinear map $U_{a,b}:=U_{a+b}-U_a-U_b$ gives rise to an endomorphism $V_{a,b}$ on defined by $V_{a,b} x=U_{a,x}b$ which satisfies
$\displaystyle U_a V_{b,a}=V_{a,b} U_a.$
If $R\subseteq K$ is a commutative ring extension of then the extension $J_K:=K\otimes_R J$ with the extension $U_K:=1_K\otimes U$ , satisfies the first two axioms.

For a unital quadratic Jordan algebra we include the added assumption that there exist some $1\in J$ such that is the identity endomorphism of .

The concept of a quadratic Jordan algebra was developed by McCrimmon to introduce uniform methods in the study of Jordan algebras over characteristic 2. In a strict sense they are not algebras as they do not have a bilinear product; 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 and defining the product as

$\displaystyle a.b=\frac{1}{2}(ab+ba).$

The

is optional (and avoided in the analogous Lie bracket definitions

), in characteristic 2 we can opt to remove it. The result is the usual special Jordan product is also the usual Lie bracket,

. So we can treat these algebras as Jordan or Lie algebras.

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 dimension 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 $1/2\in K$ then a Jordan algebra over is a quadratic Jordan algebra where the quadratic map is given by $U_a=\{axa\}$ 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 solvable radical of a Jordan algebra.

Definition 3 A submodule of a quadratic Jordan algebra is an inner quadratic ideal, or simply an inner ideal if $U_I(J)\leq I$ , that is $U_i(x)\in I$ for all $i\in I$ , $x\in J$ .

A submodule of a quadratic Jordan algebra is an outer quadratic ideal, or a outer ideal if $U_J(I)\leq I$ , that is, $U_x(i)\in I$ for all $i\in I$ , $x\in J$ .

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