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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 *x(xy) = (xx)y *(yx)x = y(xx) 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 : ,y,z= (xy)z - x(yz). 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 toSchafer (1995) p. 27 : ,x,y= 0 : ,x,x= 0. Both of these identities together imply that : ,y,x=
, x, 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 ...
+ , y, x- , x+y, x+y= , x+y, -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 ...
-
, 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 x and y. This is equivalent to the '' flexible identity''Schafer (1995) p. 28 :(xy)x = x(yx). 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: x(xy) = (xx)y *right alternative identity: (yx)x = y(xx) *flexible identity: (xy)x = x(yx). is alternative and therefore satisfies all three identities. An alternating associator is always totally skew-symmetric. That is, :
_, x_, 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 ...
= \sgn(\sigma) _1,x_2,x_3/math> for any
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 pro ...
\sigma. 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 measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. * More generally, any
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 ...
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 operat ...
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 x,y,z in an alternative algebra associate (i.e., ,y,z= 0), 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 *a(x(ay)) = (axa)y *((xa)y)a = x(aya) *(ax)(ya) = a(xy)a hold in any alternative algebra. In a unital alternative algebra, multiplicative inverses are unique whenever they exist. Moreover, for any invertible element x and all y one has :y = x^(xy). This is equivalent to saying the associator ^,x,y/math> vanishes for all such x and y. If x and y are invertible then xy is also invertible with inverse (xy)^ = y^x^. The set of all invertible elements is therefore closed under multiplication and forms a
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 thi ...
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 Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
.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 d ...
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 In geometry, a Moufang plane, named for Ruth Moufang, is a type of projective plane, more specifically a special type of translation plane. A translation plane is a projective plane that has a ''translation line'', that is, a line with the proper ...
. 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 involution ...
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, meeting ...
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 addition ...
* Maltsev algebra * Zorn ring


References

* *


External links

* {{DEFAULTSORT:Alternative Algebra Non-associative algebras