View source for Non-associative algebra

{{about|a particular structure known as a non-associative algebra|non-associativity in general|Non-associativity}}
A '''non-associative algebra'''{{sfn|Schafer|1995|loc=Chapter 1}} (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-associative algebra over a [[field (mathematics)|field]] ''K'' if it is a [[vector space]] over ''K'' and is equipped with a ''K''-[[bilinear map|bilinear]] binary multiplication operation ''A'' × ''A'' → ''A'' which may or may not be associative. Examples include [[Lie algebra]]s, [[Jordan algebra]]s, the [[octonion]]s, and three-dimensional Euclidean space equipped with the [[cross product]] 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]]s.

An algebra is ''[[unital algebra|unital]]'' or ''unitary'' if it has an [[identity element]] ''I'' with ''Ix'' = ''x'' = ''xI'' for all ''x'' in the algebra. For example, the [[octonion]]s are unital, but [[Lie algebra]]s never are.

The nonassociative algebra structure of ''A'' may be studied by associating it with other associative algebras which are subalgebra 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 [[Module (mathematics)|''R''-module]] equipped with an ''R''-bilinear binary multiplication operation.{{sfn|Schafer|1995|loc=pp.1}} If a structure obeys all of the ring axioms apart from associativity (for example, any ''R''-algebra), then it is naturally a <math>\mathbb{Z}</math>-algebra, so some authors refer to non-associative <math>\mathbb{Z}</math>-algebras as '''non-associative rings'''.

{{Algebraic structures |Algebra}}

== 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 [[identity (mathematics)|identities]] which simplify multiplication somewhat. These include the following identities.

In the list, ''x'', ''y'' and ''z'' denote arbitrary elements of an algebra.
* [[Associativity|Associative]]:  (''xy'')''z'' = ''x''(''yz'').
* [[Commutativity|Commutative]]:  ''xy'' = ''yx''.
* [[Anticommutative]]:<ref name=Sch3>{{harvnb|Schafer|1995|p=3}}</ref> ''xy'' = −''yx''.<ref>This is always implied by the identity ''xx'' = 0 for all ''x'', and the converse holds for fields of characteristic other than two.</ref>
* [[Jacobi identity]]:<ref name=Sch3/><ref name=Oku12>{{harvnb|Okubo|2005|p=12}}</ref> (''xy'')''z'' + (''yz'')''x'' + (''zx'')''y'' = 0. 
* [[Jordan identity]]:<ref name=Sch91>{{harvnb|Schafer|1995|p=91}}</ref><ref name=Oku13>{{harvnb|Okubo|2005|p=13}}</ref> (''xy'')(''x''<sup>2</sup>) = ''x''(''y''(''x''<sup>2</sup>)).
* [[Power associative]]:<ref name=Sch30>{{harvnb|Schafer|1995|p=30}}</ref><ref name=Oku17>{{harvnb|Okubo|2005|p=17}}</ref><ref name=KMRT451/> For all ''x'', any three nonnegative powers of ''x'' associate. That is if ''a'', ''b'' and ''c'' are nonnegative powers of ''x'', then ''a''(''bc'') = (''ab'')''c''. This is equivalent to saying that ''x''<sup>''m''</sup> ''x''<sup>''n''</sup> = ''x''<sup>''n+m''</sup> for all non-negative integers ''m'' and ''n''.
* [[Alternative algebra|Alternative]]:<ref name=Sch5>{{harvnb|Schafer|1995|p=5}}</ref><ref name=Oku18>{{harvnb|Okubo|2005|p=18}}</ref><ref name=McC153>{{harvnb|McCrimmon|2004|p=153}}</ref> (''xx'')''y'' = ''x''(''xy'') and (''yx'')''x'' = ''y''(''xx'').
* [[Flexible algebra|Flexible]]:<ref name=Sch28>{{harvnb|Schafer|1995|p=28}}</ref><ref name=Oku16>{{harvnb|Okubo|2005|p=16}}</ref> ''x''(''yx'') = (''xy'')''x''.
* Elastic:<ref name=Ros91>{{cite book | last=Rosenfeld | first=Boris | title=Geometry of Lie groups | page=91 | zbl=0867.53002 | series=Mathematics and its Applications | location=Dordrecht | volume=393 | publisher=Kluwer Academic Publishers | year=1997 | isbn=0792343905 }}</ref> Flexible and (''xy'')(''xx'') = ''x''(''y''(''xx'')), ''x''(''xx'')''y'' = (''xx'')(''xy'').

These properties are related by 
# ''associative'' implies ''alternative'' implies ''power associative''; 
# ''associative'' implies ''Jordan identity'' implies ''power associative'';
# Each of the properties ''associative'', ''commutative'', ''anticommutative'', ''Jordan identity'', and ''Jacobi identity'' individually imply ''flexible''<ref name=Sch28/><ref name=Oku16/>, which implies ''power associative''.
# For a field with characteristic not two, being both commutative and anticommutative implies the algebra is just {0}.
# ''alternative'' implies ''flexible''. If the algebra is not commutative, any two out of the three following properties: ''left-alternative'', ''right-alternative'', ''flexible'', imply the third one.

===Associator===
{{main|Associator}}

The '''associator''' on ''A'' is the ''K''-[[multilinear map]] <math>[\cdot,\cdot,\cdot] : A \times A \times A \to A</math> given by

:<math>[x,y,z] = (xy)z - x(yz).\,</math>

It measures the degree of nonassociativity of <math>A</math>, and can be used to conveniently express some possible identities satisfied by ''A''.

* Associative: the associator is identically zero;
* Alternative: the associator is [[alternating form|alternating]], interchange of any two terms changes the sign;
* Flexible: <math>[x,y,x] = 0</math>;
* Jordan: <math>[x, y, x^2] = 0</math>.<ref name=Sch14/>

The '''nucleus''' is the set of elements that associate with all others:<ref name=McC56>{{harvnb|McCrimmon|2004|p=56}}</ref> that is, the ''n'' in ''A'' such that

:<math> [n,A,A] = [A,n,A] = [A,A,n] = \{0\} \ . </math>

The nucleus is an associative subring of ''A''.

===Center===

The '''center''' of a ''A'' is the set of elements that commute and associate with everything in ''A'', that is the intersection of 
:<math> C(A) = \{ n \in A \ | \ nr=rn \, \forall r \in A \, \} </math>
with the nucleus. It turns out that for elements of ''C(A)'' it is enough that two of the sets <math>([n,A,A], [A,n,A] , [A,A,n])</math> are <math>\{0\}</math> for the third to also be the zero set.

== Examples ==

* [[Euclidean space]] '''R'''<sup>3</sup> 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]]s are algebras satisfying anticommutativity and the Jacobi identity.
* Algebras of [[vector field]]s on a [[differentiable manifold]] (if ''K'' is '''R''' or the [[complex number]]s '''C''') or an [[algebraic variety]] (for general ''K'');
* [[Jordan algebra]]s are algebras which satisfy the commutative law and the Jordan identity.<ref name=Oku13/>
* 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 (algebra)|characteristic]] other than 2 gives rise to a Jordan algebra by defining a new multiplication ''x*y'' = (1/2)(''xy'' + ''yx''). In contrast to the Lie algebra case, not every Jordan algebra can be constructed this way. Those that can are called ''special''.
* [[Alternative algebra]]s 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 algebra]]s, are those algebras satisfying the power-associative identity. Examples include all associative algebras, all alternative algebras, Jordan algebras, and the [[sedenion]]s.
* The [[hyperbolic quaternion]] algebra over '''R''', which was an experimental algebra before the adoption of [[Minkowski space]] for [[special relativity]].

More classes of algebras:

* [[Graded algebra]]s. These include most of the algebras of interest to [[multilinear algebra]], such as the [[tensor algebra]], [[symmetric algebra]], and [[exterior algebra]] over a given [[vector space]]. Graded algebras can be generalized to [[filtered algebra]]s.
* [[Division algebra]]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]]s (dimension 1), the [[complex number]]s (dimension 2), the [[quaternion]]s (dimension 4), and the [[octonion]]s (dimension 8). The quaternions and octonions are not commutative. Of these algebras, all are associative except for the octonions.
* [[Quadratic algebra]]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 algebra]]s (where ''K'' is '''R'''), which begin with:
** '''C''' (a commutative and associative algebra);
** the [[quaternion]]s '''H''' (an associative algebra);
** the [[octonion]]s (an [[alternative algebra]]);
** the [[sedenion]]s, and the infinite sequence of Cayley-Dickson algebras ([[power-associative algebra]]s).
* The [[Poisson algebra]]s are considered in [[geometric quantization]]. They carry two multiplications, turning them into commutative algebras and Lie algebras in different ways.
*[[Genetic algebra]]s are non-associative algebras used in mathematical genetics.
*[[Triple system]]s
{{See also|list of algebras}}

== 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 [[sedenion]]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.<ref>{{cite book | title=Graduate Algebra: Noncommutative View | page=321 | series=Graduate studies in mathematics | first=Louis Halle | last=Rowen | publisher=[[American Mathematical Society]] | year=2008 | isbn=0-8218-8408-5 }}</ref>

Kurosh proved that every subalgebra of a free non-associative algebra is free.<ref>{{cite journal | last=Kurosh | first=A.G. | title=Non-associative algebras and free products of algebras | journal=Mat. Sbornik | volume=20 | issue = 62  | year=1947 | pages=237–262 | mr=20986 | zbl=0041.16803 }}</ref>

==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<sub>''K''</sub>(''A'') of ''K''-linear vector space endomorphism of ''A''.  We can associate to the algebra structure on ''A'' two subalgebras of End<sub>''K''</sub>(''A''), the '''derivation algebra''' and the '''(associative) enveloping algebra'''.

===Derivation algebra===
{{main|Derivation algebra}}
A ''[[Derivation (abstract algebra)|derivation]]'' on ''A'' is a map ''D'' with the property

:<math>D(x \cdot y) = D(x) \cdot y + x \cdot D(y) \ . </math>

The derivations on ''A'' form a subspace Der<sub>''K''</sub>(''A'') in End<sub>''K''</sub>(''A'').  The [[Commutator#Ring theory|commutator]] of two derivations is again a derivation, so that the [[Lie bracket]] gives Der<sub>''K''</sub>(''A'') a structure of [[Lie algebra]].<ref name=Sch4>{{harvnb|Schafer|1995|p=4}}</ref>

===Enveloping algebra===
There are linear maps ''L'' and ''R'' attached to each element ''a'' of an algebra ''A'':<ref name=Oku24>{{harvnb|Okubo|2005|p=24}}</ref>

:<math>L(a) : x \mapsto ax ; \ \  R(a) : x \mapsto xa \ . </math>

The ''associative enveloping algebra'' or ''multiplication algebra'' of ''A'' is the associative algebra generated by the left and right linear maps.<ref name=Sch14>{{harvnb|Schafer|1995|p=14}}</ref><ref>{{cite book | last=Albert | first=A. Adrian | authorlink=Abraham Adrian Albert | title=Structure of algebras | zbl=0023.19901 | series=American Mathematical Society Colloquium Publ. | volume=24 | location=New York | publisher=[[American Mathematical Society]] | year=2003 | origyear=1939 | isbn=0-8218-1024-3 | edition=Corrected reprint of the revised 1961 | page=113 }}</ref>  The ''centroid'' of ''A'' is the centraliser of the enveloping algebra in the endomorphism algebra End<sub>''K''</sub>(''A'').  An algebra is ''central'' if its centroid consists of the ''K''-scalar multiples of the identity.<ref name=KMRT451>{{harvnb|Knus|Merkurjev|Rost|Tignol|1998|p=451}}</ref>

Some of the possible identities satisfied by non-associative algebras may be conveniently expressed in terms of the linear maps:<ref name=McC57>{{harvnb|McCrimmon|2004|p=57}}</ref>
* 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''<sup>2</sup>);
* Alternative: every ''L''(''a'')<sup>2</sup> = ''L''(''a''<sup>2</sup>) and similarly for the right.

The ''quadratic representation'' ''Q'' is defined by<ref name=Koe57>{{harvnb|Koecher|1999|p=57}}</ref>

:<math>Q(a) : x \mapsto 2a \cdot (a \cdot x) - (a \cdot a) \cdot x \ </math>

or equivalently

:<math>Q(a) = 2 L^2(a) - L(a^2) \ . </math>

The article on [[universal enveloping algebra]]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]] 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

== Notes ==
{{reflist}}

== References ==
* {{citation | title=Some Aspects of Ring Theory: Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Varenna (Como), Italy, August 23-31, 1965 | volume=37 | series=C.I.M.E. Summer Schools | editor1-first=I. N. | editor1-last=Herstein | editor1-link=Israel Nathan Herstein | edition=reprint | publisher=[[Springer-Verlag]] | year=2011 | isbn=3642110363 | origyear=1965 }}
* {{citation | last1=Knus | first1=Max-Albert | last2=Merkurjev | first2=Alexander | author2-link=Alexander Merkurjev | last3=Rost | first3=Markus | author3-link=Markus Rost | last4=Tignol | first4=Jean-Pierre | title=The book of involutions | others=With a preface by J. Tits | zbl=0955.16001 | series=Colloquium Publications | publisher=[[American Mathematical Society]] | volume=44 | location=Providence, RI | year=1998 | isbn=0-8218-0904-0 |ref=harv}}
* {{citation | last=Koecher | first=Max | title=The Minnesota notes on Jordan algebras and their applications | editor1-first=Aloys | editor1-last=Krieg | editor2-first=Sebastian | editor2-last=Walcher | zbl=1072.17513 | series=Lecture Notes in Mathematics | volume=1710 | location=Berlin | publisher=[[Springer-Verlag]] | isbn=3-540-66360-6 | year=1999 |ref=harv|url=https://books.google.com/books?id=RHrdf06-vZ0C&printsec=frontcover&dq=isbn:3540663606&hl=en&sa=X&ved=0ahUKEwjbroOhxdDjAhXXZc0KHWUhAqAQ6AEIKjAA#v=onepage&q=%22non-associative%22&f=false}}
* {{Citation | last1=McCrimmon | first1=Kevin | title=A taste of Jordan algebras | url=https://books.google.com/books?isbn=0387954473 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Universitext | isbn=978-0-387-95447-9 | doi=10.1007/b97489 | id=[http://www.math.virginia.edu/Faculty/McCrimmon/ Errata] | year=2004 | mr=2014924 | zbl=1044.17001 }}
* {{citation |last=Okubo |first=Susumu |title=Introduction to Octonion and Other Non-Associative Algebras in Physics | year=2005 | origyear=1995 | publisher=[[Cambridge University Press]] | isbn=0-521-01792-0 | doi=10.1017/CBO9780511524479 | zbl=0841.17001 | series=Montroll Memorial Lecture Series in Mathematical Physics | volume=2 |ref=harv}}
* {{citation |first=Richard D. |last=Schafer |title=An Introduction to Nonassociative Algebras |year=1995 |origyear=1966 |publisher=Dover |isbn=0-486-68813-5 |url=http://www.gutenberg.org/ebooks/25156 | zbl=0145.25601 |ref=harv}}

[[Category:Non-associative algebras| ]]