mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an isotopy from a possibly

non-associative algebra A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' if ...

''A'' to another is a triple of

bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...

linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...

s such that if then . This is similar to the definition of an

isotopy of loops In the mathematical field of abstract algebra, isotopy is an equivalence relation used to classify the algebraic notion of loop. Isotopy for loops and quasigroups was introduced by , based on his slightly earlier definition of isotopy for algebra ...

, except that it must also preserve the linear structure of the algebra. For 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 , 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 then . For

alternative Alternative or alternate may refer to: Arts, entertainment and media * Alternative (''Kamen Rider''), a character in the Japanese TV series ''Kamen Rider Ryuki'' * ''The Alternative'' (film), a 1978 Australian television film * ''The Alternative ...

division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fie ...

s such as the

octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...

s the two definitions of isotopy are equivalent, but in general they are not.

Examples

*If is an isomorphism then the triple is an isotopy. Conversely, if the algebras have identity elements 1 that are preserved by the maps ''a'' and ''b'' of an isotopy, then 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 is an isotopy. Similarly we could take ''b'' and ''c'' to be right multiplication by some invertible element and take ''a'' to be the identity. These form two commuting subgroups of the autotopy group, and the full autotopy group is generated by these two subgroups and the automorphism group. *If an algebra (not assumed to be associative) with an identity element is isotopic to an associative algebra with an identity element, then the two algebras are isomorphic. In particular two associative algebras with identity elements are isotopic if and only if they are isomorphic. However associative algebras with identity elements can be isotopic to algebras without identity elements. *The autotopy group of the octonions is the

spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a L ...

Spin₈, much larger than its automorphism group ''G''₂. *If ''B'' is a

mutation In biology, a mutation is an alteration in the nucleic acid sequence of the genome of an organism, virus, or extrachromosomal DNA. Viral genomes contain either DNA or RNA. Mutations result from errors during DNA or viral replication, mi ...

of the associative algebra ''A'' by an invertible element, then there is an isotopy from ''A'' to ''B''. *If ''a'', ''b'', and ''c'' are any invertible linear maps of an algebra, and one defines a new product , then the algebra defined by this new product is isotopic to the original algebra. For example, the complex numbers with the product ''x'' is isotopic to the complex numbers with the usual product, even though it is not commutative and has no identity element.

References

* * * * *{{citation, first=R. A. , last=Wilson, url=http://www.maths.qmul.ac.uk/~raw/talks_files/octonions.pdf, title=Octonions, year=2008, series=Pure Mathematics Seminar notes Non-associative algebras