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 ${\displaystyle A(a)}$ 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.

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

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]

References

1. 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

• 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. Zbl 0874.16002.
• 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. Zbl 1072.17513.
• McCrimmon, Kevin (2004). A taste of Jordan algebras. Universitext. Berlin, New York: Springer-Verlag. doi:10.1007/b97489. ISBN 0-387-95447-3. MR 2014924.
• 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. MR 1356224. Archived from the original on 2012-11-16. Retrieved 2014-02-04.