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.

## Contents

* 1 Definitions
* 2 Properties
* 3 Jordan algebras
* 4 References

## 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.[1]

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.[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]

## Jordan algebras

Main article: Mutation (Jordan algebra)

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

## References

1. ↑ 1.0 1.1 1.2 Elduque & Myung (1994) p. 34
2. ↑ González, S. (1992). "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". in Myung, Hyo Chul. New York: Nova Science Publishers. pp. 149–159.
3. ↑ Koecher (1999) p. 76
4. ↑ McCrimmon (2004) p. 86
5. ↑ McCrimmon (2004) p. 71
6. ↑ McCrimmon (2004) p. 72

* Elduque, Alberto; Myung, Hyo Chyl (1994). Mutations of Alternative Algebras. Mathematics and Its Applications. 278. Springer-Verlag. ISBN 0792327357.
* Jacobson, Nathan (1996). Finite-dimensional division algebras over fields. Berlin: Springer-Verlag. ISBN 3-540-57029-2.
* Koecher, Max (1999) [1962]. Krieg, Aloys; Walcher, Sebastian. eds. The Minnesota Notes on Jordan Algebras and Their Applications. Lecture Notes in Mathematics. 1710 (reprint ed.). Springer-Verlag. ISBN 3-540-66360-6.
* McCrimmon, Kevin (2004). A taste of Jordan algebras. Universitext. Berlin, New York: Springer-Verlag. doi:10.1007/b97489. ISBN 0-387-95447-3.
* Okubo, Susumo (1995). Introduction to Octonion and Other Non-Associative Algebras in Physics. Montroll Memorial Lecture Series in Mathematical Physics. Berlin, New York: Cambridge University Press. ISBN 0-521-47215-6. http://www.math.virginia.edu/Faculty/McCrimmon/. Retrieved 2014-02-04.

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