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 . 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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Octonion algebra

In mathematics, an octonion algebra or Cayley algebra over a field F is a composition algebra over F that has dimension 8 over F. In other words, it is a unital non-associative algebra A over F with a non-degenerate quadratic form N (called the norm form) such that for all x and y in A. The most well-known example of an octonion algebra is the classical octonions, which are an octonion algebra over R, the field of real numbers. The split-octonions also form an octonion algebra over R.

Flexible algebra

In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity: for any two elements a and b of the set. A magma (that is, a set equipped with a binary operation) is flexible if the binary operation with which it is equipped is flexible. Similarly, a nonassociative algebra is flexible if its multiplication operator is flexible. Every commutative or associative operation is flexible, so flexibility becomes important for binary operations that are neither commutative nor associative, e.

Bol loop

In mathematics and abstract algebra, a Bol loop is an algebraic structure generalizing the notion of group. Bol loops are named for the Dutch mathematician Gerrit Bol who introduced them in . A loop, L, is said to be a left Bol loop if it satisfies the identity for every a,b,c in L, while L is said to be a right Bol loop if it satisfies for every a,b,c in L. These identities can be seen as weakened forms of associativity, or a strengthened form of (left or right) alternativity.