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Concept
Moufang loop
Summary
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 . 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 that satisfies the four following equivalent identities for all , , in (the binary operation in is denoted by juxtaposition):
These identities are known as Moufang identities.
Any group is an associative loop and therefore a Moufang loop.
The nonzero octonions form a nonassociative Moufang loop under octonion multiplication.
The subset of unit norm octonions (forming a 7-sphere in O) is closed under multiplication and therefore forms a Moufang loop.
The subset of unit norm integral octonions is a finite Moufang loop of order 240.
The basis octonions and their additive inverses form a finite Moufang loop of order 16.
The set of invertible split-octonions forms a nonassociative Moufang loop, as does the set of unit norm split-octonions. More generally, the set of invertible elements in any octonion algebra over a field F forms a Moufang loop, as does the subset of unit norm elements.
The set of all invertible elements in an alternative ring R forms a Moufang loop called the loop of units in R.
For any field F let M(F) denote the Moufang loop of unit norm elements in the (unique) split-octonion algebra over F. Let Z denote the center of M(F). If the characteristic of F is 2 then Z = {e}, otherwise Z = {±e}. The Paige loop over F is the loop M*(F) = M(F)/Z. Paige loops are nonassociative simple Moufang loops. All finite nonassociative simple Moufang loops are Paige loops over finite fields. The smallest Paige loop M*(2) has order 120.
A large class of nonassociative Moufang loops can be constructed as follows. Let G be an arbitrary group. Define a new element u not in G and let M(G,2) = G ∪ (G u). The product in M(G,2) is given by the usual product of elements in G together with and
It follows that and .
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