Isotopy of an algebra
triple of maps with certain properties
In mathematics, an isotopy from a possibly non-associative algebra A to another is a triple of bijective linear maps (a, b, c) such that if xy = z then a(x)b(y) = c(z). This is similar to the definition of an isotopy of loops, except that it must also preserve the linear structure of the algebra. For a = b = c this is the same as an isomorphism. The autotopy group of an algebra is the group of all isotopies to itself (sometimes called autotopies), which contains the group of automorphisms as a subgroup.
Isotopy of algebras was introduced by Albert (1942), who was inspired by work of Steenrod.
Some authors use a slightly different definition that an isotopy is a triple of bijective linear maps a, b, c such that if xyz = 1 then a(x)b(y)c(z) = 1. For alternative division algebras such as the octonions the two definitions of isotopy are equivalent, but in general they are not.
Examples
If a = b = c is an isomorphism then the triple (a, b, c) is an isotopy. Conversely, if the algebras have identity elements 1 that are preserved by the maps a and b of an isotopy, then a = b = c is an isomorphism.
If A is an associative algebra with identity and a and c are left multiplication by some fixed invertible element, and b is the identity then (a, b, c) is an isotopy.
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