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Mutation (algebra)

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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.

Contents

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 = ( x a ) y .

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

x ∗ y = ( x a ) y − ( y b ) 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 ( x y ) ( x x ) = x ( y ( x x ) ) . The Jordan triple product is defined by

    { a , b , c } = ( a b ) c + ( c b ) a − ( a c ) b .

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

    a ∘ 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.

    References

    Mutation (algebra) Wikipedia
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