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 addition ...
where the binary multiplication operation is not assumed to be associative. 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 of ...
''A'' is a non-associative algebra over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''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 can ...
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 Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s,
Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan alg ...
s, 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, 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 is ...
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 su ...
''e'' with ''ex'' = ''x'' = ''xe'' for all ''x'' in the algebra. For example, 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 are unital, but
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
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 In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a grou ...
of ''A'' as a ''K''-vector space. Two such are the
derivation algebra In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A natu ...
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 In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
''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: . *
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 In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...
: . *
Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
: or depending on authors. *
Jordan identity In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan al ...
: or depending on authors. *
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 ...
: (left alternative) and (right alternative). *
Flexible Flexible may refer to: Science and technology * Power cord, a flexible electrical cable. ** Flexible cable, an Electrical cable as used on electrical appliances * Flexible electronics * Flexible response * Flexible-fuel vehicle * Flexible rake re ...
: . * th power associative with : for all integers so that . ** Third power associative: . ** Fourth power associative: (compare with ''fourth power commutative'' below). *
Power associative In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity. Definition An algebra (or more generally a magma) is said to be power-associative if the subalgebra ...
: 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 cla ...
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 Nil may refer to: * nil (the number 0, zero) Acronyms * NIL (programming language), an implementation of the Lisp programming language * Student athlete compensation, Name, Image and Likeness, a set of rules in the American National Collegiate At ...
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, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
R3 with multiplication given by the
vector 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 is ...
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 Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
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 form ...
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. Mo ...
(for general ''K''); *
Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan alg ...
s are algebras which satisfy the commutative law and the Jordan identity. * Every associative algebra gives rise to a Lie algebra by using the commutator 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 algebra In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only 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 is o ...
s are algebras satisfying the alternative property. The most important examples of alternative algebras are the
octonions 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 have ...
(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 algebra In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity. Definition An algebra over a field, algebra (or more generally a magma (algebra), magma) is said to be ...
s, 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
sedenion In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers; they are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic to ...
s. * 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 eleme ...
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 inerti ...
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 laws o ...
. 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 se ...
s. These include most of the algebras of interest to
multilinear algebra Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of ''p' ...
, 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 being ...
,
symmetric algebra In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
, 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 a ...
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 can ...
. Graded algebras can be generalized to
filtered algebra In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory. A filtered algebra over the field k is an alge ...
s. *
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 fie ...
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 measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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 form ...
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 quatern ...
s (dimension 4), and 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 (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 i ...
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 quatern ...
s H (an associative algebra); ** 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 (an
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 *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the algebra. Every associative algebra is o ...
); ** the
sedenion In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers; they are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic to ...
s, and the infinite sequence of Cayley-Dickson algebras (
power-associative algebra In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity. Definition An algebra over a field, algebra (or more generally a magma (algebra), magma) is said to be ...
s). *
Hypercomplex algebra In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group represen ...
s 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 central ...
s are considered in
geometric quantization In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a wa ...
. 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 system In algebra, a triple system (or ternar) is a vector space ''V'' over a field F together with a F-trilinear map : (\cdot,\cdot,\cdot) \colon V\times V \times V\to V. The most important examples are Lie triple systems and Jordan triple systems. The ...
s


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 In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zer ...
. For example, all non-zero elements of the
sedenion In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers; they are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic to ...
s 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 Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
'' 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 of two derivations is again a derivation, so that the
Lie bracket 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 identi ...
gives Der''K''(''A'') a structure of
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
.


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 represent ...
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, an exceptional
Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan alg ...
that is not enveloped by the canonical construction of the enveloping algebra for Jordan algebras.


See also

*
List of algebras {{short description, None This is a list of possibly nonassociative algebras. An algebra is a module, wherein you can also multiply two module elements. (The multiplication in the module is compatible with multiplication-by-scalars from the base ri ...
*
Commutative non-associative magmas In mathematics, there exist magma (algebra), magmas that are commutative but not associative. A simple example of such a magma may be derived from the children's game of rock, paper, scissors. Such magmas give rise to non-associative algebras. A m ...
, which give rise to non-associative algebras


Citations


Notes


References

* * * * * * * * * * * * * * * * * {{Authority control