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 (not assumed to be associative) denoted by juxtaposition. For an element a of A, define the left a-homotope $$A(a)$$ to be the algebra with multiplication


 * $$x * y = (xa)y. \, $$

Similarly define the left (a,b) mutation $$A(a,b)$$


 * $$x * y = (xa)y - (yb)x. \, $$

Right homotope and mutation are defined analogously. Since the right (p,q) mutation of A is the left (−q, −p) 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

 * If A is associative then so is any homotope of A, and any mutation of A is Lie-admissible.
 * If A is alternative then so is any homotope of A, and any mutation of A is Malcev-admissible.
 * Any isotope of a Hurwitz algebra is isomorphic to the original.
 * A homotope of a Bernstein algebra by an element of non-zero weight is again a Bernstein algebra.

Jordan algebras
A Jordan algebra is a commutative algebra satisfying the Jordan identity $$(xy)(xx) = x(y(xx))$$. The Jordan triple product is defined by


 * $$ \{a,b,c\}=(ab)c+(cb)a -(ac)b. \, $$

For y in A the mutation or homotope Ay is defined as the vector space A with multiplication


 * $$ a\circ b= \{a,y,b\}. \,$$

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.