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

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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.[1]

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

Properties

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Jordan algebras

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A Jordan algebra is a commutative algebra satisfying the Jordan identity  . The Jordan triple product is defined by

 

For y in A the mutation[3] or homotope[4] 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.[5] If y is nuclear then the isotope by y is isomorphic to the original.[6]

References

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  1. ^ a b c Elduque & Myung (1994) p. 34
  2. ^ González, S. (1992). "Homotope algebra of a Bernstein algebra". In Myung, Hyo Chul (ed.). Proceedings of the fifth international conference on hadronic mechanics and nonpotential interactions, held at the University of Northern Iowa, Cedar Falls, Iowa, USA, August 13–17, 1990. Part 1: Mathematics. New York: Nova Science Publishers. pp. 149–159. Zbl 0787.17029.
  3. ^ Koecher (1999) p. 76
  4. ^ McCrimmon (2004) p. 86
  5. ^ McCrimmon (2004) p. 71
  6. ^ McCrimmon (2004) p. 72