title: Mutation (algebra)
text: # 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 $A^y$ 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.
  
  