Concept

Alternative algebra

In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have for all x and y in the algebra. Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions. Alternative algebras are so named because they are the algebras for which the associator is alternating. The associator is a trilinear map given by By definition, a multilinear map is alternating if it vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent to Both of these identities together imply that for all and . This is equivalent to the flexible identity The associator of an alternative algebra is therefore alternating. Conversely, any algebra whose associator is alternating is clearly alternative. By symmetry, any algebra which satisfies any two of: left alternative identity: right alternative identity: flexible identity: is alternative and therefore satisfies all three identities. An alternating associator is always totally skew-symmetric. That is, for any permutation . The converse holds so long as the characteristic of the base field is not 2. Every associative algebra is alternative. The octonions form a non-associative alternative algebra, a normed division algebra of dimension 8 over the real numbers. More generally, any octonion algebra is alternative. The sedenions and all higher Cayley–Dickson algebras lose alternativity. Artin's theorem states that in an alternative algebra the subalgebra generated by any two elements is associative. Conversely, any algebra for which this is true is clearly alternative. It follows that expressions involving only two variables can be written unambiguously without parentheses in an alternative algebra. A generalization of Artin's theorem states that whenever three elements in an alternative algebra associate (i.e., ), the subalgebra generated by those elements is associative.

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Related concepts (17)
Octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface O or blackboard bold . Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.
Non-associative algebra
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × A → A which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation.
Power associativity
In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity. An algebra (or more generally a magma) is said to be power-associative if the subalgebra generated by any element is associative. Concretely, this means that if an element is performed an operation by itself several times, it doesn't matter in which order the operations are carried out, so for instance .
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