quadratic Jordan algebra

quadratic Jordan algebra

Definition 1

.

Fix a

commutative ring

R

and an

R

-modules

J

and a

quadratic map

U

:

J

→

End

R

⁡

J

. Then the triple

⟨

J

,

U

⟩

is a

quadratic Jordan algebra

if (denoting the evaluation of

U

by

U

x

for

x

∈

J

)

1.

U

U

a

⁢

b

=

U

a

⁢

U

b

⁢

U

a

for all

a

,

b

∈

J

.

2.

The induced

bilinear map

U

a

,

b

:=

U

a

+

b

-

U

a

-

U

b

gives rise to
an

endomorphism

V

a

,

b

on

J

defined by

V

a

,

b

⁢

x

=

U

a

,

x

⁢

b

which satisfies

U

a

⁢

V

b

,

a

=

V

a

,

b

⁢

U

a

.

3.

If

R

⊆

K

is a commutative ring

extension

of

R

then the extension

J

K

:=

K

⊗

R

J

with the extension

U

K

:=

1

K

⊗

U

, satisfies the first two axioms.

For a unital quadratic Jordan algebra we include the added assumption
that there exist some

1

∈

J

such that

U

1

is the

identity

endomorphism of

J

.

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

A

and defining the
product as

a

.

b

=

1

2

⁢

(

a

⁢

b

+

b

⁢

a

)

.

The

1

/

2

is optional (and avoided in the analogous

Lie bracket

definitions

[

a

,

b

]

=

a

⁢

b

-

b

⁢

a

), 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

=

a

⁢

b

+

b

⁢

a

=

a

⁢

b

-

b

⁢

a

=

[

a

,

b

]

. 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

∈

K

then a Jordan algebra over

K

is a quadratic Jordan algebra
where the quadratic map is given by

U

a

=

{

a

⁢

x

⁢

a

}

where

{

x

⁢

y

⁢

z

}

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

I

of a quadratic Jordan algebra

J

is an

inner quadratic
ideal

, or simply an

inner ideal

if

U

I

⁢

(

J

)

≤

I

, that is

U

i

⁢

(

x

)

∈

I

for all

i

∈

I

,

x

∈

J

.

A submodule

I

of a quadratic Jordan algebra

J

is an

outer quadratic
ideal

, or a

outer ideal

if

U

J

⁢

(

I

)

≤

I

, that is,

U

x

⁢

(

i

)

∈

I

for all

i

∈

I

,

x

∈

J

.

If the quadratic Jordan algebra is derived from a Jordan algebra then

U

i

⁢

(

x

)

=

{

i

⁢

x

⁢

i

}

So we are asking for

{

i

⁢

J

⁢

i

}

≤

I

, and in a special
Jordan algebra we can further express this as

i

⁢

J

⁢

i

≤

I

.

References

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

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