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Mutation (algebra)

Let A be an algebra over a field F with multiplication (not assumed to be associative) denoted by juxtaposition. For an element aofA, 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 algebratoA, it suffices to study left mutations.^[1]

IfA is a unital algebra and a is invertible, we refer to the isotopebya.

Properties[edit]

• IfA is associative then so is any homotope of A, and any mutation of AisLie-admissible.
• IfAisalternative then so is any homotope of A, and any mutation of AisMalcev-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[edit]

AJordan 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 yinA the mutation^[3]orhomotope^[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]Ifyisnuclear then the isotope by y is isomorphic to the original.^[6]

References[edit]

• Elduque, Alberto; Myung, Hyo Chyl (1994). Mutations of Alternative Algebras. Mathematics and Its Applications. Vol. 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. Vol. 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.