Moufang loop

In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang (1935). Smooth Moufang loops have an associated algebra, the Malcev algebra, similar in some ways to how a Lie group has an associated Lie algebra.

A Moufang loop is a loop Q that satisfies the four following (non equivalent) identities for all x, y, z in Q (the binary operation in Q is denoted by juxtaposition):

These identities are known as Moufang identities.

Moufang loops differ from groups in that they need not be associative. A Moufang loop that is associative is a group. The Moufang identities may be viewed as weaker forms of associativity.

By setting various elements to the identity, the Moufang identities imply

Moufang's theorem states that when three elements x, y, and z in a Moufang loop obey the associative law: (xy)z = x(yz) then they generate an associative subloop; that is, a group. A corollary of this is that all Moufang loops are di-associative (i.e. the subloop generated by any two elements of a Moufang loop is associative and therefore a group). In particular, Moufang loops are power associative, so that exponents xⁿ are well-defined. When working with Moufang loops, it is common to drop the parenthesis in expressions with only two distinct elements. For example, the Moufang identities may be written unambiguously as

The Moufang identities can be written in terms of the left and right multiplication operators on Q. The first two identities state that

while the third identity says

for all ${\displaystyle x,y,z}$ in ${\displaystyle Q}$. Here ${\displaystyle B_{z}=L_{z}R_{z}=R_{z}L_{z}}$ is bimultiplication by ${\displaystyle z}$. The third Moufang identity is therefore equivalent to the statement that the triple ${\displaystyle (L_{z},R_{z},B_{z})}$ is an autotopy of ${\displaystyle Q}$ for all ${\displaystyle z}$ in ${\displaystyle Q}$.

...
Wikipedia