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单词 Octonion
释义

octonion


Let be the quaternionsMathworldPlanetmath over the reals . Apply theCayley-Dickson construction to once, and we obtain an algebraMathworldPlanetmathPlanetmathPlanetmath,variously called Cayley algebra, the octonion algebra, or simply the octonions, over . Specifically the construction is carried out as follows:

  1. 1.

    Form the vector space 𝕆=𝐤; any element of𝕆 can be written as a+b𝐤, where a,b;

  2. 2.

    Define a binary operationMathworldPlanetmath on 𝕆 called the multiplication inO by

    (a+b𝐤)(c+d𝐤):=(ac-d¯b)+(da+bc¯)𝐤,

    where a,b,c,d, and c¯ is the quaternionic conjugationMathworldPlanetmathof c. When b=d=0, the multiplication is reduced themultiplication in . In addition, the multiplication rule aboveimply the following:

    a(d𝐤)=(da)𝐤(1)
    (b𝐤)c=(bc¯)𝐤(2)
    (b𝐤)(d𝐤)=-d¯b.(3)

    In particular, in the last equation, if b=d=1, 𝐤2=-1.

  3. 3.

    Define a unary operation on 𝕆 called the octonionicconjugation in O by

    a+b𝐤¯:=a¯-b𝐤,

    where a,b.Clearly, the octonionic conjugation is an involutionPlanetmathPlanetmath (http://planetmath.org/Involution2)(x¯¯=x).

  4. 4.

    Finally, define a unary operation N on 𝕆 called the norm inO by N(x):=xx¯, where x𝕆. Writex=a+b𝐤, then

    N(x)=(a+b𝐤)(a¯-b𝐤)=(aa¯+b¯b)+(-ba+ba¯¯)𝐤=aa¯+bb¯0.

    It is not hard to see that N(x)=0 iff x=0.

The above four (actually, only the first two suffice) steps makes𝕆 into an 8-dimensional algebra over such that isembedded as a subalgebraMathworldPlanetmathPlanetmathPlanetmath.

With the last two steps, one can define the inverseMathworldPlanetmathPlanetmathPlanetmath of a non-zeroelement x𝕆 by

x-1:=x¯N(x)

so thatxx-1=x-1x=1. Since x is arbitrary, 𝕆 has no zerodivisorsMathworldPlanetmath. Upon checking that x-1(xy)=y=(yx)x-1, the non-associative algebra 𝕆 is turned into a division algebraMathworldPlanetmath.

Since N(x)0 for any x𝕆, we can define a non-negativereal-valued function on 𝕆 by x=N(x). This is clearly well-defined and x=0 iffx=0. In addition, it is not hard to see that, for any rand x𝕆, rx=|r|x, and that satisfies the triangular inequality. This makes 𝕆 into a normeddivision algebra.

Since the multiplication in is noncommutative, 𝕆 isnoncommutative. In fact, if we write =𝐣,where are the complex numbers and 𝐣2=-1, thenB={1,𝐢,𝐣,𝐢𝐣} is a basisfor the vector space over . With the introductionof 𝐤𝕆, we quickly check that 𝐤anti-commute with the non-real basis elements in B:

𝐢𝐤=-𝐤𝐢,𝐣𝐤=-𝐤𝐣,(𝐢𝐣)𝐤=-𝐤(𝐢𝐣).

Furthermore, one checks that 𝐢(𝐣𝐤)=(𝐣𝐢)𝐤=-(𝐢𝐣)𝐤, so that𝕆 is not associative.

Since 𝕆=𝐤, the set {1,𝐢,𝐣,𝐢𝐣,𝐤,𝐢𝐤,𝐣𝐤,(𝐢𝐣)𝐤}(=BB𝐤) is abasis for 𝕆 over . A less messy way to represent these basiselements is done the following assignment:

Titleoctonion
Canonical nameOctonion
Date of creation2013-03-22 15:21:42
Last modified on2013-03-22 15:21:42
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id15
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 17A75
Classificationmsc 17D05
SynonymCayley algebra
Related topicTheoremsOnSumsOfSquares
Related topicDivisionAlgebra
Definesoctonion algebra
\\@unrecurse
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更新时间:2025/5/4 6:48:57