homotopes and isotopes of algebras

1 Geometric motivations

In the study projective geometries a first step in developingproperties 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 fixeddivision ring (plus a formal $\mathrm{\infty }$ point) called the coordinate ring of the geometry. Form thisone can often coordinatize the entire geometry, that is, represent thegeometry as subspaces of a vector space over the division ring.

There is a common problem with these constructions: the coordinates dependon fixing reference points in the geometry. If we change the referencepoints we change the product in the coordinate ring as well. Fortunately,nice geometries generally do not allow for too much variance to occur witha change of reference points. However, to track these geometric changes wecreate an algebraic process to restructure the multiplication of a fixedring. The process is called a homotope of the algebra or an isotope when the process is reversible. These geometrically inspiredterms reflect the original applications of the process, but in some theoremsthey now find multiple other applications unrelated to geometry.

2 Associative homotopes and isotopes

Given a unital associative algebra $A$ and an element $a\in A$, we define thehomotope of $A$ as the algebra $A^{(a)}$ which retains the same modulestructure of $A$ but where we replace the multiplication by

$$x{\cdot }_{a}y:=xay,x,y\in A.$$

By the assumption of associativity $xay$ makes sense. A homotope is an isotope if $a$ is invertible. We now prove all isotopes are isomorphicalgebras.

Proposition 1.

If $a$ is invertible in $A$ then $A^{\mathrm{(}a\mathrm{)}}$ is isomorphic as algebras to $A$via the map $x\mathrm{\mapsto }x\mathit{}{a}^{\mathrm{-}\mathrm{1}}$. Furthermore, the identity of $A^{\mathrm{(}a\mathrm{)}}$ is$a^{\mathrm{-}\mathrm{1}}$.

Proof.

As $a$ is invertible, the map we describe is an isomorphism of modules.To verify the multiplication let $x,y\in A$. Then

$$x{a}^{-1}{\cdot }_{a}y{a}^{-1}=x{a}^{-1}ay{a}^{-1}=(xy)a^{-1}.$$

Furthermore, $x{\cdot }_a{a}^{-1}=xa{a}^{-1}=x=a^{-1}ax=a^{-1}{\cdot }_{a}x$ 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 caninclude rather exceptional examples such as planes which do not satisfyDesargues’ 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 certainMoufang planes. This however is simply a lack of flexibility in thedefinition of coordinates. For it is known that using Octonion’s as a divisionalgebra (non-associative) the Moufang planes can be coordinatized. Therefore,the geometric process of isotopes and homotopes will force a constructionof isotopes and homotopes of non-associative algebras.

For example, in a unital Jordan algebra $J$, a homotope/isotope can be builtby means of the Jordan triple product:

$$x{\circ }_{a}y:=\{x,a,y\}=(x.a).y+(a.x).y-(x.y).a$$

which in a special Jordan algebra takes the more familiar form:

$$x{\circ }_{a}y:=\frac{1}{2}(xay+yax).$$

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 notbe an isomorphic algebra.

4 Non-geometric applications of homotopes

Given an element $x\in A$, we say $x$ is left/right quasi-invertibleif for all $a\in A$, $1+ax$ and $xa+1$ are left and right invertible respectively. It is a classic theorem that the Jacobson radical iscomprised of all left quasi-invertible elements. McCrimmon showedthat expressed in homotopes we can instead say: $x$ is quasi-invertibleif $1+x$ is quasi-invertible in every homotope of $A$. This gives thecharacterization:

The radical of an associative algebra is the set of elements $x\in A$ whichare (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 $x\in J$ for which $1+x$ is invertible inevery 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 haveinsisted thus far that we use elements from the orignal algebra to induce thehomotope. These are what Albert calls a principal homotope. However,there are often reasons to allow for external parameters. For example, ifa non-associative algebra $A$ is embedded in an associative envelope $U(A)$,one may use an element $u\in U(A)$ to induce a homotope or isotope of $A$.Another possible approach is to modify the terms of the product withinvolutions $*$, for example,

$$x{\circ }_{a}y:=\frac{1}{2}(xa{y}^{*}+ya{x}^{*}).$$

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

Definition 3.

Given a non-associative $K$-algebra $A$, a homotope of $A$ is triple of$K$-endomorphisms $f\mathrm{,}g\mathrm{,}h$ of $A$ such that:

$$xf\circ yg=(x\circ y)h,x,y\in A.$$

A homotope is an isotope if the maps $f$, $g$ and $h$ are invertible.