In the theory of algebras over a field, mutation is a construction of a new

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...

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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''F'' with multiplication (not assumed to be

associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

) 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 In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring is the ring ...

to ''A'', it suffices to study left mutations.Elduque & Myung (1994) p. 34 If ''A'' is a

unital algebra In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and additio ...

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. * Any isotope of a Hurwitz algebra is isomorphic to the original. * A homotope of a Bernstein algebra by an element of non-zero weight is again a Bernstein algebra.

Jordan algebras

Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan al ...

is a commutative algebra satisfying the ''Jordan identity''

(xy)(xx) = x(y(xx))

. The

Jordan triple product In algebra, a triple system (or ternar) is a vector space ''V'' over a field F together with a F-trilinear map : (\cdot,\cdot,\cdot) \colon V\times V \times V\to V. The most important examples are Lie triple systems and Jordan triple systems. The ...

is defined by :

\=(ab)c+(cb)a -(ac)b. \,

For ''y'' in ''A'' the ''mutation''Koecher (1999) p. 76 or ''homotope''McCrimmon (2004) p. 86 ''A''^''y'' is defined as the vector space ''A'' with multiplication :

a\circ 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.McCrimmon (2004) p. 71 If ''y'' is

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then the isotope by ''y'' is isomorphic to the original.McCrimmon (2004) p. 72

References

* * * * * {{Authority control Non-associative algebras