In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, an alternative algebra is an
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
in which multiplication need not be
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
, only
alternative
Alternative or alternate may refer to:
Arts, entertainment and media
* Alternative (''Kamen Rider''), a character in the Japanese TV series ''Kamen Rider Ryuki''
* ''The Alternative'' (film), a 1978 Australian television film
* ''The Alternative ...
. That is, one must have
*
*
for all ''x'' and ''y'' in the algebra.
Every
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
is obviously alternative, but so too are some strictly
non-associative algebra
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary operation, binary multiplication operation is not assumed to be associative operation, associative. That is, an algebraic structure ''A'' is a non-ass ...
s such as the
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 or blackboard bold \mathbb O. Octonions hav ...
s.
The associator
Alternative algebras are so named because they are the algebras for which the
associator In abstract algebra, the term associator is used in different ways as a measure of the associativity, non-associativity of an algebraic structure. Associators are commonly studied as triple systems.
Ring theory
For a non-associative ring or non-as ...
is
alternating. The associator is a
trilinear map
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function
:f\colon V_1 \times \cdots \times V_n \to W\text
where V_1,\ldots,V_n and W a ...
given by
:
.
By definition, a
multilinear map
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function
:f\colon V_1 \times \cdots \times V_n \to W\text
where V_1,\ldots,V_n and W ar ...
is alternating if it
vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent to
[Schafer (1995) p. 27]
:
:
Both of these identities together imply that
:
for all
and
. This is equivalent to the ''
flexible identity''
[Schafer (1995) p. 28]
:
The associator of an alternative algebra is therefore alternating.
Conversely
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposit ...
, 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,
: