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 to be the algebra with multiplication
Similarly define the left (a,b) mutation
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.Elduque & Myung (1994) p. 34
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 . The Jordan triple product is defined by
For y in A the mutationKoecher (1999) p. 76 or homotopeMcCrimmon (2004) p. 86 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.McCrimmon (2004) p. 71 If y is nuclear then the isotope by y is isomorphic to the original.McCrimmon (2004) p. 72
References