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Isotopy of an algebra
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an isotopy from a possibly <a href="/facts/Non-associative_algebra/Vk9Kjmrs">non-associative algebra</a> <i>A</i> to another is a triple of <a href="/facts/Bijection,_injection_and_surjection/ayC7YE3z">bijective</a> <a href="/facts/Linear_map/Q1PBKNVV">linear maps</a> (<i>a</i>, <i>b</i>, <i>c</i>) such that if <i>xy</i> = <i>z</i> then <i>a</i>(<i>x</i>)<i>b</i>(<i>y</i>) = <i>c</i>(<i>z</i>). This is similar to the definition of an <a href="/facts/Isotopy_of_loops/EqW922k8">isotopy of loops</a>, except that it must also preserve the linear structure of the algebra. For <i>a</i> = <i>b</i> = <i>c</i> 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. </p><p>Isotopy of algebras was introduced by <a href="/facts/A._A._Albert/whzaVMMg">Albert</a> (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 <i>a</i>, <i>b</i>, <i>c</i> such that if <i>xyz</i> = 1 then <i>a</i>(<i>x</i>)<i>b</i>(<i>y</i>)<i>c</i>(<i>z</i>) = 1. For <a href="/facts/Alternative_algebra/3boGwrqQ">alternative</a> <a href="/facts/Division_algebra/xp5Bepx3">division algebras</a> such as the <a href="/facts/Octonion/nC2sIz3p">octonions</a> the two definitions of isotopy are equivalent, but in general they are not. </p>