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| {{about|a particular structure known as a non-associative algebra|non-associativity in general|Non-associativity}}
| | The writer's title is Christy Brookins. I've usually cherished living in Kentucky but now I'm contemplating other choices. The favorite hobby for him and his kids is to play lacross and he'll be beginning something else alongside with it. I am an invoicing officer and I'll be promoted quickly.<br><br>My web blog: [http://Www.taehyuna.net/xe/?document_srl=78721 free online tarot card readings] |
| {{mergefrom|Example of a non-associative algebra|date=February 2013|discuss=Talk:Non-associative algebra#Merge example}}
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| {{one source|date=April 2012}}
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| A '''non-associative [[Algebra over a field|algebra]]'''{{sfn|Schafer|1966|loc=Chapter 1}} (or '''distributive algebra''') over a field (or a commutative ring) ''K'' is a ''K''-vector space (or more generally a [[Module (mathematics)|module]]{{sfn|Schafer|1966|loc=pp.1}}) ''A'' equipped with a ''K''-[[bilinear]]{{dn|date=December 2013}} map ''A'' × ''A'' → ''A'' which establishes a binary multiplication operation on ''A''. 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.
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| 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.
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| Multiplication by elements of ''A'' on the left or on the right give rise to left and right ''K''-[[linear transformation]]s of ''A'' given by
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| <math>L_a : x \mapsto ax</math> and <math>R_a : x \mapsto xa</math>. The '''enveloping algebra''' of a non-associative algebra ''A'' is the subalgebra of the full algebra of ''K''-[[endomorphisms]] of ''A'' which is generated by the left and right multiplication maps of ''A''.{{sfn|Schafer|1966|loc=pp.14-15}} This enveloping algebra is necessarily associative, even though ''A'' may be non-associative. In a sense this makes the enveloping algebra "the smallest associative algebra containing ''A''".
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| 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.
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| {{Algebraic structures |Algebra}}
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| == Algebras satisfying identities ==
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| 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.
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| In the list, ''x'', ''y'' and ''z'' denote arbitrary elements of an algebra.
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| * [[Associativity|Associative]]: (''xy'')''z'' = ''x''(''yz'').
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| * [[Commutativity|Commutative]]: ''xy'' = ''yx''.
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| * [[Anticommutative]]: ''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>
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| * [[Jacobi identity]]: (''xy'')''z'' + (''yz'')''x'' + (''zx'')''y'' = 0.
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| * [[Jordan identity]]: (''xy'')''x''<sup>2</sup> = ''x''(''yx''<sup>2</sup>).
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| * [[Power associative]]: For all ''x'', and 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''.
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| * [[Alternative algebra|Alternative]]: (''xx'')''y'' = ''x''(''xy'') and (''yx'')''x'' = ''y''(''xx'').
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| * [[Flexible algebra|Flexible]]:{{sfn|Okubo|1995|loc=p. 16}} ''x''(''yx'') = (''xy'')''x''.
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| These properties are related by
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| # ''associative'' implies ''alternative'' implies ''power associative'';
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| # ''associative'' implies ''Jordan identity'' implies ''power associative'';
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| # Each of the properties ''associative'', ''commutative'', ''anticommutative'', ''Jordan identity'', and ''Jacobi identity'' individually imply ''flexible''.{{sfn|Okubo|1995|loc=p. 16}}
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| # For a field with characteristic not two, being both commutative and anticommutative implies the algebra is just {0}.
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| == Examples ==
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| * [[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.
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| * [[Lie algebra]]s are algebras satisfying anticommutativity and the Jacobi identity.
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| * 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'');
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| * [[Jordan algebra]]s are algebras which satisfy the commutative law and the Jordan identity.
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| * 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.
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| * 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''.
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| * [[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.
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| * [[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.
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| * The [[hyperbolic quaternion]] algebra over '''R''', which was an experimental algebra before the adoption of [[Minkowski space]] for [[special relativity]].
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| More classes of algebras:
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| * [[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.
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| * [[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.
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| * [[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.
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| * The [[Cayley–Dickson algebra]]s (where ''K'' is '''R'''), which begin with:
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| ** '''C''' (a commutative and associative algebra);
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| ** the [[quaternion]]s '''H''' (an associative algebra);
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| ** the [[octonion]]s (an [[alternative algebra]]);
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| ** the [[sedenion]]s (a [[power-associative algebra]], like all of the Cayley-Dickson algebras).
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| * The [[Poisson algebra]]s are considered in [[geometric quantization]]. They carry two multiplications, turning them into commutative algebras and Lie algebras in different ways.
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| *[[Genetic algebra]]s are non-associative algebras used in mathematical genetics.
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| == See also ==
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| *[[List of algebras]]
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| == Notes ==
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| {{reflist}}
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| == References ==
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| * {{citation |last=Okubo |first=Susumu |title=Introduction to Octonion and Other Non-Associative Algebras in Physics |year=1995 |publisher=Cambridge University Press |ISBN=978-0-521-47215-9 |DOI=10.1017/CBO9780511524479 }}
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| * {{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}}
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| [[Category:Non-associative algebras]]
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The writer's title is Christy Brookins. I've usually cherished living in Kentucky but now I'm contemplating other choices. The favorite hobby for him and his kids is to play lacross and he'll be beginning something else alongside with it. I am an invoicing officer and I'll be promoted quickly.
My web blog: free online tarot card readings