Non-associative Algebra
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A non-associative algebra (or distributive algebra) is an
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 ...
where the binary multiplication operation is not assumed to 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 ...
. That is, an
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
''A'' is a non-associative algebra over a field ''K'' if it is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
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 algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
s, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and i ...
operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (''ab'')(''cd''), (''a''(''bc''))''d'' and ''a''(''b''(''cd'')) may all yield different answers. While this use of ''non-associative'' means that associativity is not assumed, it does not mean that associativity is disallowed. In other words, "non-associative" means "not necessarily associative", just as "noncommutative" means "not necessarily commutative" for
noncommutative ring In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...
s. An algebra is '' unital'' or ''unitary'' if it has an
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
''e'' with ''ex'' = ''x'' = ''xe'' for all ''x'' in the algebra. For example, the octonions are unital, but
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
s never are. The nonassociative algebra structure of ''A'' may be studied by associating it with other associative algebras which are subalgebras of the full algebra of ''K''- endomorphisms of ''A'' as a ''K''-vector space. Two such are the derivation algebra and the (associative) enveloping algebra, the latter being in a sense "the smallest associative algebra containing ''A''". More generally, some authors consider the concept of a non-associative algebra over a commutative ring ''R'': An ''R''-module equipped with an ''R''-bilinear binary multiplication operation. If a structure obeys all of the ring axioms apart from associativity (for example, any ''R''-algebra), then it is naturally a \mathbb-algebra, so some authors refer to non-associative \mathbb-algebras as non-associative rings.


Algebras satisfying identities

Ring-like structures with two binary operations and no other restrictions are a broad class, one which is too general to study. For this reason, the best-known kinds of non-associative algebras satisfy identities, or properties, which simplify multiplication somewhat. These include the following ones.


Usual properties

Let , and denote arbitrary elements of the algebra over the field . Let powers to positive (non-zero) integer be recursively defined by and either (right powers) or (left powers) depending on authors. * Unital: there exist an element so that ; in that case we can define . *
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 ...
: . *
Commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
: . * Anticommutative: . * Jacobi identity: or depending on authors. * Jordan identity: or depending on authors. * Alternative: (left alternative) and (right alternative). * Flexible: . * th power associative with : for all integers so that . ** Third power associative: . ** Fourth power associative: (compare with ''fourth power commutative'' below). * Power associative: the subalgebra generated by any element is associative, i.e., ''th power associative'' for all . * th power commutative with : for all integers so that . ** Third power commutative: . ** Fourth power commutative: (compare with ''fourth power associative'' above). * Power commutative: the subalgebra generated by any element is commutative, i.e., ''th power commutative'' for all . *
Nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...
of index : the product of any elements, in any association, vanishes, but not for some elements: and there exist elements so that for a specific association. * Nil of index : ''power associative'' and and there exist an element so that .


Relations between properties

For of any characteristic: * ''Associative'' implies ''alternative''. * Any two out of the three properties ''left alternative'', ''right alternative'', and ''flexible'', imply the third one. ** Thus, ''alternative'' implies ''flexible''. * ''Alternative'' implies ''Jordan identity''. * ''Commutative'' implies ''flexible''. * ''Anticommutative'' implies ''flexible''. * ''Alternative'' implies ''power associative''. * ''Flexible'' implies ''third power associative''. * ''Second power associative'' and ''second power commutative'' are always true. * ''Third power associative'' and ''third power commutative'' are equivalent. * ''th power associative'' implies ''th power commutative''. * ''Nil of index 2'' implies ''anticommutative''. * ''Nil of index 2'' implies ''Jordan identity''. * ''Nilpotent of index 3'' implies ''Jacobi identity''. * ''Nilpotent of index '' implies ''nil of index '' with . * ''Unital'' and ''nil of index '' are incompatible. If or : * ''Jordan identity'' and ''commutative'' together imply ''power associative''. If : * ''Right alternative'' implies ''power associative''. ** Similarly, ''left alternative'' implies ''power associative''. * ''Unital'' and ''Jordan identity'' together imply ''flexible''. * ''Jordan identity'' and ''flexible'' together imply ''power associative''. * ''Commutative'' and ''anticommutative'' together imply ''nilpotent of index 2''. * ''Anticommutative'' implies ''nil of index 2''. * ''Unital'' and ''anticommutative'' are incompatible. If : * ''Unital'' and ''Jacobi identity'' are incompatible. If : * ''Commutative'' and (one of the two identities defining ''fourth power associative'') together imply ''power associative''. If : * ''Third power associative'' and (one of the two identities defining ''fourth power associative'') together imply ''power associative''. If : * ''Commutative'' and ''anticommutative'' are equivalent.


Associator

The associator on ''A'' is the ''K''-
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 ...
cdot,\cdot,\cdot: A \times A \times A \to A given by : . It measures the degree of nonassociativity of A, and can be used to conveniently express some possible identities satisfied by ''A''. Let , and denote arbitrary elements of the algebra. * Associative: . * Alternative: (left alternative) and (right alternative). ** It implies that permuting any two terms changes the sign: ; the converse holds only if . * Flexible: . ** It implies that permuting the extremal terms changes the sign: ; the converse holds only if . * Jordan identity: or depending on authors. * Third power associative: . The nucleus is the set of elements that associate with all others: that is, the in ''A'' such that : . The nucleus is an associative subring of ''A''.


Center

The center of ''A'' is the set of elements that commute and associate with everything in ''A'', that is the intersection of : C(A) = \ with the nucleus. It turns out that for elements of ''C(A)'' it is enough that two of the sets ( ,A,A ,n,A, ,A,n are \ for the third to also be the zero set.


Examples

*
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
R3 with multiplication given by the vector cross product is an example of an algebra which is anticommutative and not associative. The cross product also satisfies the Jacobi identity. *
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
s are algebras satisfying anticommutativity and the Jacobi identity. * Algebras of vector fields on a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
(if ''K'' is R or the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s C) or an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...
(for general ''K''); * Jordan algebras are algebras which satisfy the commutative law and the Jordan identity. * Every associative algebra gives rise to a Lie algebra by using the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
as Lie bracket. In fact every Lie algebra can either be constructed this way, or is a subalgebra of a Lie algebra so constructed. * Every associative algebra over a field of characteristic other than 2 gives rise to a Jordan algebra by defining a new multiplication ''x*y'' = (''xy''+''yx'')/2. In contrast to the Lie algebra case, not every Jordan algebra can be constructed this way. Those that can are called ''special''. * Alternative algebras are algebras satisfying the alternative property. The most important examples of alternative algebras are the octonions (an algebra over the reals), and generalizations of the octonions over other fields. All associative algebras are alternative. Up to isomorphism, the only finite-dimensional real alternative, division algebras (see below) are the reals, complexes, quaternions and octonions. * Power-associative algebras, are those algebras satisfying the power-associative identity. Examples include all associative algebras, all alternative algebras, Jordan algebras over a field other than
GF(2) (also denoted \mathbb F_2, or \mathbb Z/2\mathbb Z) is the finite field of two elements (GF is the initialism of ''Galois field'', another name for finite fields). Notations and \mathbb Z_2 may be encountered although they can be confused with ...
(see previous section), and the sedenions. * The
hyperbolic quaternion In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form :q = a + bi + cj + dk, \quad a,b,c,d \in \mathbb \! where the squares of i, j, and k are +1 and distinct elemen ...
algebra over R, which was an experimental algebra before the adoption of
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...
for
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
. More classes of algebras: *
Graded algebra In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
s. These include most of the algebras of interest to multilinear algebra, such as the
tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of bein ...
, symmetric algebra, and
exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
over a given
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
. Graded algebras can be generalized to filtered algebras. *
Division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a f ...
s, in which multiplicative inverses exist. The finite-dimensional alternative division algebras over the field of real numbers have been classified. They are 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 (dimension 1), the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s (dimension 2), the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
s (dimension 4), and the octonions (dimension 8). The quaternions and octonions are not commutative. Of these algebras, all are associative except for the octonions. *
Quadratic algebra In mathematics, a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important role in the theory of quantum groups. The most ...
s, which require that ''xx'' = ''re'' + ''sx'', for some elements ''r'' and ''s'' in the ground field, and ''e'' a unit for the algebra. Examples include all finite-dimensional alternative algebras, and the algebra of real 2-by-2 matrices. Up to isomorphism the only alternative, quadratic real algebras without divisors of zero are the reals, complexes, quaternions, and octonions. * The Cayley–Dickson algebras (where ''K'' is R), which begin with: ** C (a commutative and associative algebra); ** the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
s H (an associative algebra); ** the octonions (an alternative algebra); ** the sedenions, and the infinite sequence of Cayley-Dickson algebras ( power-associative algebras). * Hypercomplex algebras are all finite-dimensional unital R-algebras, they thus include Cayley-Dickson algebras and many more. * The
Poisson algebra In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also centr ...
s are considered in geometric quantization. They carry two multiplications, turning them into commutative algebras and Lie algebras in different ways. *
Genetic algebra In mathematical genetics, a genetic algebra is a (possibly non-associative) algebra used to model inheritance in genetics. Some variations of these algebras are called train algebras, special train algebras, gametic algebras, Bernstein algebras, c ...
s are non-associative algebras used in mathematical genetics. * Triple systems


Properties

There are several properties that may be familiar from ring theory, or from associative algebras, which are not always true for non-associative algebras. Unlike the associative case, elements with a (two-sided) multiplicative inverse might also be a zero divisor. For example, all non-zero elements of the sedenions have a two-sided inverse, but some of them are also zero divisors.


Free non-associative algebra

The free non-associative algebra on a set ''X'' over a field ''K'' is defined as the algebra with basis consisting of all non-associative monomials, finite formal products of elements of ''X'' retaining parentheses. The product of monomials ''u'', ''v'' is just (''u'')(''v''). The algebra is unital if one takes the empty product as a monomial. Kurosh proved that every subalgebra of a free non-associative algebra is free.


Associated algebras

An algebra ''A'' over a field ''K'' is in particular a ''K''-vector space and so one can consider the associative algebra End''K''(''A'') of ''K''-linear vector space endomorphism of ''A''. We can associate to the algebra structure on ''A'' two subalgebras of End''K''(''A''), the derivation algebra and the (associative) enveloping algebra.


Derivation algebra

A '' derivation'' on ''A'' is a map ''D'' with the property :D(x \cdot y) = D(x) \cdot y + x \cdot D(y) \ . The derivations on ''A'' form a subspace Der''K''(''A'') in End''K''(''A''). The
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
of two derivations is again a derivation, so that the Lie bracket gives Der''K''(''A'') a structure of
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
.


Enveloping algebra

There are linear maps ''L'' and ''R'' attached to each element ''a'' of an algebra ''A'': :L(a) : x \mapsto ax ; \ \ R(a) : x \mapsto xa \ . The ''associative enveloping algebra'' or ''multiplication algebra'' of ''A'' is the associative algebra generated by the left and right linear maps. The ''centroid'' of ''A'' is the centraliser of the enveloping algebra in the endomorphism algebra End''K''(''A''). An algebra is ''central'' if its centroid consists of the ''K''-scalar multiples of the identity. Some of the possible identities satisfied by non-associative algebras may be conveniently expressed in terms of the linear maps: * Commutative: each ''L''(''a'') is equal to the corresponding ''R''(''a''); * Associative: any ''L'' commutes with any ''R''; * Flexible: every ''L''(''a'') commutes with the corresponding ''R''(''a''); * Jordan: every ''L''(''a'') commutes with ''R''(''a''2); * Alternative: every ''L''(''a'')2 = ''L''(''a''2) and similarly for the right. The ''quadratic representation'' ''Q'' is defined by: :Q(a) : x \mapsto 2a \cdot (a \cdot x) - (a \cdot a) \cdot x \ or equivalently :Q(a) = 2 L^2(a) - L(a^2) \ . The article on
universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representa ...
s describes the canonical construction of enveloping algebras, as well as the PBW-type theorems for them. For Lie algebras, such enveloping algebras have a universal property, which does not hold, in general, for non-associative algebras. The best-known example is, perhaps the
Albert algebra In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra. They are named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers. Over the real numbers, there a ...
, an exceptional Jordan algebra that is not enveloped by the canonical construction of the enveloping algebra for Jordan algebras.


See also

* List of algebras * Commutative non-associative magmas, which give rise to non-associative algebras


Citations


Notes


References

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