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.

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 ${\displaystyle A(a)}$ to be the algebra with multiplication

${\displaystyle x*y=(xa)y.\,}$

Similarly define the left (a,b) mutation ${\displaystyle A(a,b)}$

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

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.^[1]

If A is a unital algebra and a is invertible, we refer to the isotope by a.

• 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.^[1]
• Any isotope of a Hurwitz algebra is isomorphic to the original.^[1]
• A homotope of a Bernstein algebra by an element of non-zero weight is again a Bernstein algebra.^[2]

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

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

For y in A the mutation^[3] or homotope^[4] A^y is defined as the vector space A with multiplication

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

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.^[5] If y is nuclear then the isotope by y is isomorphic to the original.^[6]

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• Koecher, Max (1999) [1962]. Krieg, Aloys; Walcher, Sebastian (eds.). The Minnesota Notes on Jordan Algebras and Their Applications. Lecture Notes in Mathematics. Vol. 1710 (reprint ed.). Springer-Verlag. ISBN 3-540-66360-6. Zbl 1072.17513.
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