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 [math]\displaystyle{ A(a) }[/math] to be the algebra with multiplication
- [math]\displaystyle{ x * y = (xa)y. \, }[/math]
Similarly define the left (a,b) mutation [math]\displaystyle{ A(a,b) }[/math]
- [math]\displaystyle{ x * y = (xa)y - (yb)x. \, }[/math]
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 [math]\displaystyle{ (xy)(xx) = x(y(xx)) }[/math]. The Jordan triple product is defined by
- [math]\displaystyle{ \{a,b,c\}=(ab)c+(cb)a -(ac)b. \, }[/math]
For y in A the mutation[3] or homotope[4] Ay is defined as the vector space A with multiplication
- [math]\displaystyle{ a\circ b= \{a,y,b\}. \, }[/math]
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
- ↑ 1.0 1.1 1.2 Elduque & Myung (1994) p. 34
- ↑ 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.
- ↑ Koecher (1999) p. 76
- ↑ McCrimmon (2004) p. 86
- ↑ McCrimmon (2004) p. 71
- ↑ 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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