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.
Contents
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
)
{\displaystyle A(a)}
to be the algebra with multiplication
x
∗
y
=
(
x
a
)
y
.
{\displaystyle x*y=(xa)y.\,}
Similarly define the left (a,b) mutation
A
(
a
,
b
)
{\displaystyle A(a,b)}
x
∗
y
=
(
x
a
)
y
−
(
y
b
)
x
.
{\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.
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.
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
A Jordan algebra is a commutative algebra satisfying the Jordan identity
(
x
y
)
(
x
x
)
=
x
(
y
(
x
x
)
)
{\displaystyle (xy)(xx)=x(y(xx))}
. The Jordan triple product is defined by
{
a
,
b
,
c
}
=
(
a
b
)
c
+
(