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 te ...
, an alternative algebra is an
algebra
Algebra () is one of the areas of mathematics, 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 mathem ...
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. 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 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 ...
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 is
alternating
Alternating may refer to:
Mathematics
* Alternating algebra, an algebra in which odd-grade elements square to zero
* Alternating form, a function formula in algebra
* Alternating group, the group of even permutations of a finite set
* Alter ...
. The associator is a
trilinear map 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
:
, y, x
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
-
, x+y, x+y
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
=
, x+y, -y
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
=
, x, -y
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
-
, y, y
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
= 0
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,
:
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...
. The converse holds so long as the
characteristic of the base
field is not 2.
Examples
* Every associative algebra is alternative.
* 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 form a non-associative alternative algebra, a
normed division algebra of dimension 8 over the
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s.
* More generally, any
octonion algebra is alternative.
Non-examples
* The
sedenions and all higher
Cayley–Dickson algebras lose alternativity.
Properties
Artin's theorem states that in an alternative algebra the
subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
"Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear oper ...
generated by any two elements is
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 ...
.
[Schafer (1995) p. 29] 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.
A
corollary
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
of Artin's theorem is that alternative algebras are
power-associative, that is, the subalgebra generated by a single element is associative.
[Schafer (1995) p. 30] The converse need not hold: the sedenions are power-associative but not alternative.
The
Moufang identities
*
*
*
hold in any alternative algebra.
[
In a unital alternative algebra, multiplicative inverses are unique whenever they exist. Moreover, for any invertible element and all one has
:
This is equivalent to saying the associator ]Moufang loop Moufang is the family name of the following people:
* Christoph Moufang (1817–1890), a Roman Catholic cleric
* Ruth Moufang (1905–1977), a German mathematician, after whom several concepts in mathematics are named:
** Moufang–Lie algebra
** ...
. This ''loop of units'' in an alternative ring or algebra is analogous to the group of units
In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that
vu = uv = 1,
where is the multiplicative identity; the element is unique for th ...
in an associative ring or algebra.
Kleinfeld's theorem states that any simple non-associative alternative ring is a generalized octonion algebra over its center.[Zhevlakov, Slin'ko, Shestakov, Shirshov. (1982) p. 151]
The structure theory of alternative rings is presented in.[Zhevlakov, Slin'ko, Shestakov, Shirshov. (1982)]
Applications
The projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that ...
over any alternative division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
is a Moufang plane.
The close relationship of alternative algebras and composition algebra
In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies
:N(xy) = N(x)N(y)
for all and in .
A composition algebra includes an involuti ...
s was given by Guy Roos in 2008:[Guy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in ''Symmetries in Complex Analysis'' by Bruce Gilligan & Guy Roos, volume 468 of ''Contemporary Mathematics'', ]American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...
He shows (page 162) the relation for an algebra ''A'' with unit element ''e'' and an involutive anti-automorphism a \mapsto a^* such that ''a'' + ''a''* and ''aa''* are on the line spanned by ''e'' for all ''a'' in ''A''. Use the notation ''n''(''a'') = ''aa''*. Then if ''n'' is a non-singular mapping into the field of ''A'', and ''A'' is alternative, then (''A'',''n'') is a composition algebra.
See also
* Algebra over a field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and additio ...
* Maltsev algebra
In mathematics, a Malcev algebra (or Maltsev algebra or Moufang–Lie algebra) over a field is a nonassociative algebra that is antisymmetric, so that
:xy = -yx
and satisfies the Malcev identity
:(xy)(xz) = ((xy)z)x + ((yz)x)x + ((zx)x)y.
Th ...
* Zorn ring
References
*
*
External links
*
{{DEFAULTSORT:Alternative Algebra
Non-associative algebras