Moufang identities
Introduction
The Moufang identities, named after the German mathematician Ruth Moufang, are a set of identities that hold in a loop if and only if it is a Moufang loop. These identities are fundamental in the study of non-associative algebraic structures, particularly in the field of quasigroups and loops.
Definition
A loop is said to satisfy the Moufang identities if for all elements x, y, and z in the loop, the following identities hold:
1. (xy)(zx) = x(yz)x 2. (yx)(xz) = y(xzx) 3. x(yz)(xw) = (xy)(zxw) 4. (wyx)(zx) = w(yxz)x
These identities are often expressed in a more compact form using the Cayley-Dickson notation, a notation system used to describe certain algebraic structures.
Properties
Moufang loops have several important properties that distinguish them from other types of loops. These properties are direct consequences of the Moufang identities and play a crucial role in the structure theory of Moufang loops.
Power-associativity
Every Moufang loop is power-associative. In a power-associative loop, every element generates a subloop that is associative. This property is a direct consequence of the first Moufang identity.
Alternative laws
Moufang loops satisfy the alternative laws. This means that for every pair of elements x and y in the loop, the subloops generated by x and y are associative. This property follows from the second and third Moufang identities.
Flexible
Moufang loops are flexible. This means that for all elements x and y in the loop, the identity (xy)x = x(yx) holds. This property is a direct consequence of the fourth Moufang identity.
Examples
There are several examples of Moufang loops in various areas of mathematics. Some of the most notable examples include the following:
Octonions
The octonions form a Moufang loop under multiplication. This is a non-associative algebraic structure that extends the complex numbers and quaternions.
Projective planes
In the theory of projective planes, Moufang loops arise in the context of Moufang planes. These are projective planes that satisfy certain additional axioms related to the Moufang identities.
Division algebras
Every finite-dimensional division algebra over a field forms a Moufang loop under multiplication. This is a consequence of the Artin-Steinitz theorem.