Mutation (algebra) explained

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

A(a) to be the algebra with multiplicationx * 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

See 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] A^y is defined as the vector space A with multiplication

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

• Book: Mutations of Alternative Algebras . 278 . Mathematics and Its Applications . Alberto . Elduque . Hyo Chyl . Myung . . 1994 . 0792327357 .
• Book: Jacobson, Nathan . Finite-dimensional division algebras over fields . 0874.16002 . Berlin . . 3-540-57029-2 . 1996 .
• Book: Koecher, Max . The Minnesota Notes on Jordan Algebras and Their Applications . 1710 . Lecture Notes in Mathematics . Aloys . Krieg . Sebastian . Walcher . reprint . . 1999 . 3-540-66360-6 . 1072.17513 . 1962 .
• Book: McCrimmon, Kevin . A taste of Jordan algebras . . Berlin, New York . Universitext . 0-387-95447-3 . 10.1007/b97489 . 2004 . 2014924.
• Book: Okubo, Susumo . Introduction to Octonion and Other Non-Associative Algebras in Physics . Cambridge University Press . Berlin, New York . Montroll Memorial Lecture Series in Mathematical Physics . 0-521-47215-6 . 1995 . 1356224 . 2014-02-04 . https://web.archive.org/web/20121116162444/http://www.math.virginia.edu/Faculty/McCrimmon/ . 2012-11-16 . dead .

Notes and References

1. Elduque & Myung (1994) p. 34
2. González . S. . Homotope algebra of a Bernstein algebra . 0787.17029 . Myung . Hyo Chul . 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 . 149–159 . 1992 .
3. Koecher (1999) p. 76
4. McCrimmon (2004) p. 86
5. McCrimmon (2004) p. 71
6. McCrimmon (2004) p. 72