Mutation (algebra)


In the theory of algebras over a field, mutation is a construction of a new binary operation related to the multiplication of the algebra. In specific cases the resulting algebra may be referred to as a homotope or an isotope of the original.

Definitions

Let A be an algebra over a field F with multiplication denoted by juxtaposition. For an element a of A, define the left a-homotope to be the algebra with multiplication
Similarly define the left mutation
Right homotope and mutation are defined analogously. Since the right mutation of A is the left mutation of the opposite algebra to A, it suffices to study left mutations.
If A is a unital algebra and a is invertible, we refer to the isotope by a.

Properties

A Jordan algebra is a commutative algebra satisfying the Jordan identity. The Jordan triple product is defined by
For y in A the mutation or homotope Ay is defined as the vector space A with multiplication
and if y is invertible this is referred to as an isotope. A homotope of a Jordan algebra is again a Jordan algebra: isotopy defines an equivalence relation. If y is nuclear then the isotope by y is isomorphic to the original.