homotopes and isotopes of algebras

homotopes and isotopes of algebras

1

Geometric motivations

In the study projective geometries a first step in developing
properties of the geometry is to uncover

coordinates

of the geometry. For example, Desargues’

theorem

(and Pappaus’ theorem) are methods to uncover

division rings

(and fields) which can be used to coordinatize every line in the geometry – that is, every line will be in a

1-1 correspondence

with a fixed
division ring (plus a formal

∞

point) called the

coordinate ring

of the geometry. Form this
one can often coordinatize the entire geometry, that is, represent the
geometry as

subspaces

of a

vector space

over the division ring.

There is a common problem with these constructions: the coordinates depend
on fixing reference points in the geometry. If we change the reference
points we change the

product

in the coordinate ring as well. Fortunately,
nice geometries generally do not allow for too much variance to occur with
a change of reference points. However, to track these geometric changes we
create an

algebraic

process to restructure the multiplication of a fixed
ring. The process is called a

homotope

of the

algebra

or an

isotope

when the process is reversible. These geometrically inspired
terms

reflect

the original applications of the process, but in some theorems
they now find

multiple

other applications unrelated to geometry.

2

Associative homotopes and isotopes

Given a unital

associative algebra

A

and an

element

a

∈

A

, we define the

homotope

of

A

as the algebra

A

(

a

)

which retains the same module

structure

of

A

but where we replace the multiplication by

x

⋅

a

y

:=

x

⁢

a

⁢

y

,

x

,

y

∈

A

.

By the

assumption

of

associativity

x

⁢

a

⁢

y

makes sense. A homotope is an

isotope

if

a

is

invertible

. We now prove all isotopes are

isomorphic

algebras.

Proposition 1

.

If

a

is invertible in

A

then

A

(

a

)

is isomorphic as algebras to

A

via the map

x

↦

x

⁢

a

-

1

. Furthermore, the

identity

of

A

(

a

)

is

a

-

1

.

Proof.

As

a

is invertible, the map we describe is an

isomorphism

of modules.
To verify the multiplication let

x

,

y

∈

A

. Then

x

⁢

a

-

1

⋅

a

y

⁢

a

-

1

=

x

⁢

a

-

1

⁢

a

⁢

y

⁢

a

-

1

=

(

x

⁢

y

)

⁢

a

-

1

.

Furthermore,

x

⋅

a

a

-

1

=

x

⁢

a

⁢

a

-

1

=

x

=

a

-

1

⁢

a

⁢

x

=

a

-

1

⋅

a

x

so

a

-

1

is the identity of

A

(

a

)

.
∎

3

Non-associative homotopes and isotopes

Returning to projective geometries, it is known that

projective planes

can
include rather exceptional examples such as planes which do not

satisfy

Desargues’ theorem (and consequently they do not satisfy Pappaus’ theorem).
This means it is not possible to use the standard approach to apply coordinates rings to the lines of the geometry. An example of these include certain

Moufang planes

. This however is simply a lack of flexibility in the
definition of coordinates. For it is known that using

Octonion

’s as a

division
algebra

(non-associative) the Moufang planes can be coordinatized. Therefore,
the geometric process of isotopes and homotopes will force a construction
of isotopes and homotopes of

non-associative algebras

.

For example, in a unital

Jordan algebra

J

, a homotope/isotope can be built
by means of the

Jordan triple product

:

x

∘

a

y

:=

{

x

,

a

,

y

}

=

(

x

.

a

)

.

y

+

(

a

.

x

)

.

y

-

(

x

.

y

)

.

a

which in a

special Jordan algebra

takes the more familiar form:

x

∘

a

y

:=

1

2

⁢

(

x

⁢

a

⁢

y

+

y

⁢

a

⁢

x

)

.

We call this product the

a

-homotope of

J

and denote it by

J

(

a

)

.
When

a

is invertible in the

Jordan product

then we call this an isotope of

J

. Unlike associative algebras, an isotope of a Jordan algebra need not
be an isomorphic algebra.

4

Non-geometric applications of homotopes

Given an element

x

∈

A

, we say

x

is left/right

quasi-invertible

if for all

a

∈

A

,

1

+

a

⁢

x

and

x

⁢

a

+

1

are left and right invertible respectively. It is a classic theorem that the

Jacobson radical

is
comprised of all left quasi-invertible elements. McCrimmon showed
that expressed in homotopes we can instead say:

x

is quasi-invertible
if

1

+

x

is quasi-invertible in every homotope of

A

. This gives the

characterization

:

The

radical

of an associative algebra is the set of elements

x

∈

A

which
are (left) quasi-invertible in every homotope of

A

.

It is this version of an element description of Jacobson radicals which can be generalized to non-associative algebras. First one must define homotopes and isotopes for the given non-associative algebra.

Theorem 2

(McCrimmon, Jacobson).

The radical of a Jordan algebra

J

is equivalently defined as:

1.

The largest

solvable

ideal in

J

.

2.

The set of all elements

x

∈

J

for which

1

+

x

is invertible in
every homotope of

J

. We call such

x

properly quasi-invertible

.

3.

The

intersection

of all

maximal

inner ideals

of

J

.

5

Principal homotopes

So far we have used only a subset of all possible homotopes because we have
insisted thus far that we use elements from the orignal algebra to induce the
homotope. These are what Albert calls a

principal homotope

. However,
there are often reasons to allow for external parameters. For example, if
a non-associative algebra

A

is embedded in an

associative envelope

U

⁢

(

A

)

,
one may use an element

u

∈

U

⁢

(

A

)

to induce a homotope or isotope of

A

.
Another possible approach is to modify the terms of the product with

involutions

*

, for example,

x

∘

a

y

:=

1

2

⁢

(

x

⁢

a

⁢

y

*

+

y

⁢

a

⁢

x

*

)

.

Such constructions require a more general treatment of homotopes and isotopes
which leads to the general definition:

Definition 3

.

Given a non-associative

K

-algebra

A

, a homotope of

A

is triple of

K

-

endomorphisms

f

,

g

,

h

of

A

such that:

x

⁢

f

∘

y

⁢

g

=

(

x

∘

y

)

⁢

h

,

x

,

y

∈

A

.

A homotope is an isotope if the maps

f

,

g

and

h

are invertible.

References

1

Jacobson, Nathan

Structure Theory of Jordan Algebras

, The University of
Arkansas Lecture Notes in Mathematics, vol. 5, Fayetteville, 1981.

2

McCrimmon, Kevin

A Taste of Jordan Algebras

, Springer, New York, 2004.

Title

homotopes and isotopes of algebras

Canonical name

HomotopesAndIsotopesOfAlgebras

Date of creation

2013-03-22 16:27:24

Last modified on

2013-03-22 16:27:24

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

6

Author

Algeboy (12884)

Entry type

Derivation

Classification

msc 17A01

Related topic

QuasiRegularity

Defines

homotope

Defines

isotope

Defines

quasi-invertible

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