Movatterモバイル変換

[0]ホーム

Jump to content

Hypercomplex number

Edit links

From Wikipedia, the free encyclopedia

Element of a unital algebra over the field of real numbers

Not to be confused withsurcomplex number.

"Hypernumber" redirects here. For the extension of the real numbers used innon-standard analysis, seeHyperreal number.

Inmathematics,hypercomplex number is a traditional term for anelement of a finite-dimensionalunital algebra over thefield ofreal numbers. The study of hypercomplex numbers in the late 19th century forms the basis of moderngroup representation theory.

History

[edit]

In the nineteenth century,number systems calledquaternions,tessarines,coquaternions,biquaternions, andoctonions became established concepts in mathematical literature, extending the real andcomplex numbers. The concept of a hypercomplex number covered them all, and called for a discipline to explain and classify them.

The cataloguing project began in 1872 whenBenjamin Peirce first published hisLinear Associative Algebra, and was carried forward by his sonCharles Sanders Peirce.^[1] Most significantly, they identified thenilpotent and theidempotent elements as useful hypercomplex numbers for classifications. TheCayley–Dickson construction usedinvolutions to generate complex numbers, quaternions, and octonions out of the real number system. Hurwitz and Frobenius proved theorems that put limits on hypercomplexity:Hurwitz's theorem says finite-dimensional realcomposition algebras are the reals $\mathbb {R}$ , the complexes $\mathbb {C}$ , the quaternions $\mathbb {H}$ , and the octonions $\mathbb {O}$ , and theFrobenius theorem says the only realassociative division algebras are $\mathbb {R}$ , $\mathbb {C}$ , and $\mathbb {H}$ . In 1958J. Frank Adams published a further generalization in terms of Hopf invariants onH-spaces which still limits the dimension to 1, 2, 4, or 8.^[2]

It wasmatrix algebra that harnessed the hypercomplex systems. For instance, 2 x 2real matrices were found isomorphic tocoquaternions. Soon the matrix paradigm began to explain several others as they were represented by matrices and their operations. In 1907Joseph Wedderburn showed that associative hypercomplex systems could be represented bysquare matrices, ordirect products of algebras of square matrices.^[3]^[4] From that date the preferred term for ahypercomplex system becameassociative algebra, as seen in the title of Wedderburn's thesis atUniversity of Edinburgh. Note however, that non-associative systems like octonions andhyperbolic quaternions represent another type of hypercomplex number.

AsThomas Hawkins^[5] explains, the hypercomplex numbers are stepping stones to learning aboutLie groups andgroup representation theory. For instance, in 1929Emmy Noether wrote on "hypercomplex quantities and representation theory".^[6] In 1973Kantor and Solodovnikov published a textbook on hypercomplex numbers which was translated in 1989.^[7]^[8]

Karen Parshall has written a detailed exposition of the heyday of hypercomplex numbers,^[9] including the role of mathematicians includingTheodor Molien^[10] andEduard Study.^[11] For the transition tomodern algebra,Bartel van der Waerden devotes thirty pages to hypercomplex numbers in hisHistory of Algebra.^[12]

Definition

[edit]

A definition of ahypercomplex number is given byKantor & Solodovnikov (1989) as an element of aunital, but not necessarilyassociative orcommutative, finite-dimensional algebra over the real numbers. Elements are generated with real number coefficients $(a_{0},\dots ,a_{n})$ for a basis $\{1,i_{1},\dots ,i_{n}\}$ . Where possible, it is conventional to choose the basis so that $i_{k}^{2}\in \{-1,0,+1\}$ . A technical approach to hypercomplex numbers directs attention first to those ofdimension two.

Two-dimensional real algebras

[edit]

Theorem:^[7]^: 14, 15^[13]^[14] Up to isomorphism, there are exactly three 2-dimensional unital algebras over the reals: the ordinarycomplex numbers, thesplit-complex numbers, and thedual numbers. In particular, every 2-dimensional unital algebra over the reals is associative and commutative.

Proof: Since the algebra is 2-dimensional, we can pick a basis{1,u}. Since the algebra isclosed under squaring, the non-real basis elementu squares to a linear combination of 1 andu:

u^{2}=a_{0}+a_{1}u

for some real numbersa₀ anda₁.

Using the common method ofcompleting the square by subtractinga₁u and adding the quadratic complementa²
₁ / 4 to both sides yields

u^{2}-a_{1}u+{\frac {1}{4}}a_{1}^{2}=a_{0}+{\frac {1}{4}}a_{1}^{2}.

Thus ${\textstyle \left(u-{\frac {1}{2}}a_{1}\right)^{2}={\tilde {u}}^{2}}$ where ${\textstyle {\tilde {u}}^{2}~=a_{0}+{\frac {1}{4}}a_{1}^{2}.}$ The three cases depend on this real value:

If4a₀ = −a₁², the above formula yieldsũ² = 0. Hence,ũ can directly be identified with thenilpotent element $\varepsilon$ of the basis $\{1,~\varepsilon \}$ of the dual numbers.
If4a₀ > −a₁², the above formula yieldsũ² > 0. This leads to the split-complex numbers which have normalized basis $\{1,~j\}$ with $j^{2}=+1$ . To obtainj fromũ, the latter must be divided by the positive real number ${\textstyle a\mathrel {:=} {\sqrt {a_{0}+{\frac {1}{4}}a_{1}^{2}}}}$ which has the same square asũ has.
If4a₀ < −a₁², the above formula yieldsũ² < 0. This leads to the complex numbers which have normalized basis $\{1,~i\}$ with $i^{2}=-1$ . To yieldi fromũ, the latter has to be divided by a positive real number ${\textstyle a\mathrel {:=} {\sqrt {{\frac {1}{4}}a_{1}^{2}-a_{0}}}}$ which squares to the negative ofũ².

The complex numbers are the only 2-dimensional hypercomplex algebra that is afield.Split algebras such as the split-complex numbers that include non-real roots of 1 also containidempotents ${\textstyle {\frac {1}{2}}(1\pm j)}$ andzero divisors $(1+j)(1-j)=0$ , so such algebras cannot bedivision algebras. However, these properties can turn out to be very meaningful, for instance in representing alight cone with anull cone.

In a 2004 edition ofMathematics Magazine the 2-dimensional real algebras have been styled the "generalized complex numbers".^[15] The idea ofcross-ratio of four complex numbers can be extended to the 2-dimensional real algebras.^[16]

Higher-dimensional examples (more than one non-real axis)

[edit]

Clifford algebras

[edit]

AClifford algebra is the unital associative algebra generated over an underlying vector space equipped with aquadratic form. Over the real numbers this is equivalent to being able to define a symmetric scalar product,u ⋅v =⁠1/2⁠(uv +vu) that can be used toorthogonalise the quadratic form, to give a basis{e₁, ...,e_k} such that: ${\frac {1}{2}}\left(e_{i}e_{j}+e_{j}e_{i}\right)={\begin{cases}-1,0,+1&i=j,\\0&i\not =j.\end{cases}}$

Imposing closure under multiplication generates a multivector space spanned by a basis of 2^k elements, {1,e₁,e₂,e₃, ...,e₁e₂, ...,e₁e₂e₃, ...}. These can be interpreted as the basis of a hypercomplex number system. Unlike the basis {e₁, ...,e_k}, the remaining basis elements need notanti-commute, depending on how many simple exchanges must be carried out to swap the two factors. Soe₁e₂ = −e₂e₁, bute₁(e₂e₃) = +(e₂e₃)e₁.

Putting aside the bases which contain an elemente_i such thate_i² = 0 (i.e. directions in the original space over which the quadratic form wasdegenerate), the remaining Clifford algebras can be identified by the label Cl_p,q( $\mathbb {R}$ ), indicating that the algebra is constructed fromp simple basis elements withe_i² = +1,q withe_i² = −1, and where $\mathbb {R}$ indicates that this is to be a Clifford algebra over the reals—i.e. coefficients of elements of the algebra are to be real numbers.

These algebras, calledgeometric algebras, form a systematic set, which turn out to be very useful in physics problems which involverotations,phases, orspins, notably inclassical andquantum mechanics,electromagnetic theory andrelativity.

Examples include: thecomplex numbers Cl_0,1( $\mathbb {R}$ ),split-complex numbers Cl_1,0( $\mathbb {R}$ ),quaternions Cl_0,2( $\mathbb {R}$ ),split-biquaternions Cl_0,3( $\mathbb {R}$ ),split-quaternionsCl_1,1( $\mathbb {R}$ ) ≈ Cl_2,0( $\mathbb {R}$ ) (the natural algebra of two-dimensional space); Cl_3,0( $\mathbb {R}$ ) (the natural algebra of three-dimensional space, and the algebra of thePauli matrices); and thespacetime algebra Cl_1,3( $\mathbb {R}$ ).

The elements of the algebra Cl_p,q( $\mathbb {R}$ ) form an even subalgebra Cl^[0]
_q+1,p( $\mathbb {R}$ ) of the algebra Cl_q+1,p( $\mathbb {R}$ ), which can be used to parametrise rotations in the larger algebra. There is thus a close connection between complex numbers and rotations in two-dimensional space; between quaternions and rotations in three-dimensional space; between split-complex numbers and (hyperbolic) rotations (Lorentz transformations) in 1+1-dimensional space, and so on.

Whereas Cayley–Dickson and split-complex constructs with eight or more dimensions are not associative with respect to multiplication, Clifford algebras retain associativity at any number of dimensions.

In 1995Ian R. Porteous wrote on "The recognition of subalgebras" in his book on Clifford algebras. His Proposition 11.4 summarizes the hypercomplex cases:^[17]

LetA be a real associative algebra with unit element 1. Then

1 generates $\mathbb {R}$ (algebra of real numbers),
any two-dimensional subalgebra generated by an elemente₀ ofA such thate₀² = −1 is isomorphic to $\mathbb {C}$ (algebra of complex numbers),
any two-dimensional subalgebra generated by an elemente₀ ofA such thate₀² = 1 is isomorphic to $\mathbb {R}$ ² (pairs of real numbers with component-wise product, isomorphic to thealgebra of split-complex numbers),
any four-dimensional subalgebra generated by a set {e₀,e₁} of mutually anti-commuting elements ofA such that $e_{0}^{2}=e_{1}^{2}=-1$ is isomorphic to $\mathbb {H}$ (algebra of quaternions),
any four-dimensional subalgebra generated by a set {e₀,e₁} of mutually anti-commuting elements ofA such that $e_{0}^{2}=e_{1}^{2}=1$ is isomorphic to M₂( $\mathbb {R}$ ) (2 × 2real matrices,coquaternions),
any eight-dimensional subalgebra generated by a set {e₀,e₁,e₂} of mutually anti-commuting elements ofA such that $e_{0}^{2}=e_{1}^{2}=e_{2}^{2}=-1$ is isomorphic to² $\mathbb {H}$ (split-biquaternions),
any eight-dimensional subalgebra generated by a set {e₀,e₁,e₂} of mutually anti-commuting elements ofA such that $e_{0}^{2}=e_{1}^{2}=e_{2}^{2}=1$ is isomorphic to M₂( $\mathbb {C}$ ) (2 × 2 complex matrices,biquaternions,Pauli algebra).

For extension beyond the classical algebras, seeClassification of Clifford algebras.

Cayley–Dickson construction

[edit]

Further information:Cayley–Dickson construction

Cayley Q8 graph of quaternion multiplication showing cycles of multiplication ofi (red),j (green) andk (blue). Inthe SVG file, hover over or click a path to highlight it.

All of the Clifford algebras Cl_p,q( $\mathbb {R}$ ) apart from the real numbers, complex numbers and the quaternions contain non-real elements that square to +1; and so cannot be division algebras. A different approach to extending the complex numbers is taken by theCayley–Dickson construction. This generates number systems of dimension 2ⁿ,n = 2, 3, 4, ..., with bases $\left\{1,i_{1},\dots ,i_{2^{n}-1}\right\}$ , where all the non-real basis elements anti-commute and satisfy $i_{m}^{2}=-1$ . In 8 or more dimensions (n ≥ 3) these algebras are non-associative. In 16 or more dimensions (n ≥ 4) these algebras also havezero-divisors.

The first algebras in this sequence include the 4-dimensionalquaternions, 8-dimensionaloctonions, and 16-dimensionalsedenions. An algebraic symmetry is lost with each increase in dimensionality: quaternion multiplication is notcommutative, octonion multiplication is non-associative, and thenorm ofsedenions is not multiplicative. After the sedenions are the 32-dimensionaltrigintaduonions (or 32-nions), the 64-dimensional sexagintaquatronions (or 64-nions), the 128-dimensional centumduodetrigintanions (or 128-nions), the 256-dimensional ducentiquinquagintasexions (or 256-nions), andad infinitum, as summarized in the table below.^[18]

Name	No. of dimensions	Dimensions (2ⁿ)	Symbol
real numbers	1	2⁰	$\mathbb {R}$
complex numbers	2	2¹	$\mathbb {C}$
quaternions	4	2²	$\mathbb {H}$
octonions	8	2³	$\mathbb {O}$
sedenions	16	2⁴	$\mathbb {S}$
trigintaduonions	32	2⁵	$\mathbb {T}$
sexagintaquatronions	64	2⁶
centumduodetrigintanions	128	2⁷
ducentiquinquagintasexions	256	2⁸

The Cayley–Dickson construction can be modified by inserting an extra sign at some stages. It then generates the "split algebras" in the collection ofcomposition algebras instead of the division algebras:

split-complex numbers with basis

\{1,\,i_{1}\}

satisfying

\ i_{1}^{2}=+1

split-quaternions with basis

\{1,\,i_{1},\,i_{2},\,i_{3}\}

satisfying

\ i_{1}^{2}=-1,\,i_{2}^{2}=i_{3}^{2}=+1

, and

split-octonions with basis

\{1,\,i_{1},\,\dots ,\,i_{7}\}

satisfying

\ i_{1}^{2}=i_{2}^{2}=i_{3}^{2}=-1

\ i_{4}^{2}=i_{5}^{2}=i_{6}^{2}=i_{7}^{2}=+1.

Unlike the complex numbers, the split-complex numbers are notalgebraically closed, and further contain nontrivialzero divisors and nontrivialidempotents. As with the quaternions, split-quaternions are not commutative, but further containnilpotents; they are isomorphic to thesquare matrices of dimension two. Split-octonions are non-associative and contain nilpotents.

Tensor products

[edit]

Thetensor product of any two algebras is another algebra, which can be used to produce many more examples of hypercomplex number systems.

In particular taking tensor products with the complex numbers (considered as algebras over the reals) leads to four-dimensionalbicomplex numbers $\mathbb {C} \otimes _{\mathbb {R} }\mathbb {C}$ (isomorphic to tessarines $\mathbb {C} \otimes _{\mathbb {R} }D$ ), eight-dimensionalbiquaternions $\mathbb {C} \otimes _{\mathbb {R} }\mathbb {H}$ , and 16-dimensionalcomplex octonions $\mathbb {C} \otimes _{\mathbb {R} }\mathbb {O}$ .

Further examples

[edit]

bicomplex numbers: a 4-dimensional vector space over the reals, 2-dimensional over the complex numbers, isomorphic to tessarines.
multicomplex numbers: 2ⁿ-dimensional vector spaces over the reals, 2ⁿ⁻¹-dimensional over the complex numbers
composition algebra: algebra with aquadratic form that composes with the product

References

[edit]

^Peirce, Benjamin (1881),"Linear Associative Algebra",American Journal of Mathematics,4 (1):221–6,doi:10.2307/2369153,JSTOR 2369153
^Adams, J. F. (July 1960),"On the Non-Existence of Elements of Hopf Invariant One"(PDF),Annals of Mathematics,72 (1):20–104,CiteSeerX 10.1.1.299.4490,doi:10.2307/1970147,JSTOR 1970147
^J.H.M. Wedderburn (1908),"On Hypercomplex Numbers",Proceedings of the London Mathematical Society,6:77–118,doi:10.1112/plms/s2-6.1.77
^Emil Artin later generalized Wedderburn's result so it is known as theArtin–Wedderburn theorem
^Hawkins, Thomas (1972), "Hypercomplex numbers, Lie groups, and the creation of group representation theory",Archive for History of Exact Sciences,8 (4):243–287,doi:10.1007/BF00328434,S2CID 120562272
^Noether, Emmy (1929),"Hyperkomplexe Größen und Darstellungstheorie" [Hypercomplex Quantities and the Theory of Representations],Mathematische Annalen (in German),30:641–92,doi:10.1007/BF01187794,S2CID 120464373, archived fromthe original on 2016-03-29, retrieved2016-01-14
^^a ^bKantor, I.L., Solodownikow (1978),Hyperkomplexe Zahlen, BSB B.G. Teubner Verlagsgesellschaft, Leipzig
^Kantor, I. L.; Solodovnikov, A. S. (1989),Hypercomplex numbers, Berlin, New York:Springer-Verlag,ISBN 978-0-387-96980-0,MR 0996029
^Parshall, Karen (1985), "Joseph H. M. Wedderburn and the structure theory of algebras",Archive for History of Exact Sciences,32 (3–4):223–349,doi:10.1007/BF00348450,S2CID 119888377
^Molien, Theodor (1893),"Ueber Systeme höherer complexer Zahlen",Mathematische Annalen,41 (1):83–156,doi:10.1007/BF01443450,S2CID 122333076
^Study, Eduard (1898), "Theorie der gemeinen und höhern komplexen Grössen",Encyclopädie der mathematischen Wissenschaften, vol. I A, pp. 147–183
^van der Waerden, B.L. (1985), "10. The discovery of algebras, 11. Structure of algebras",A History of Algebra, Springer,ISBN 3-540-13610X
^Yaglom, Isaak (1968),Complex Numbers in Geometry, pp. 10–14
^Ewing, John H., ed. (1991),Numbers, Springer, p. 237,ISBN 3-540-97497-0
^Harkin, Anthony A.; Harkin, Joseph B. (2004),"Geometry of Generalized Complex Numbers"(PDF),Mathematics Magazine,77 (2):118–129,doi:10.1080/0025570X.2004.11953236,S2CID 7837108
^Brewer, Sky (2013), "Projective Cross-ratio on Hypercomplex Numbers",Advances in Applied Clifford Algebras,23 (1):1–14,arXiv:1203.2554,doi:10.1007/s00006-012-0335-7,S2CID 119623082
^Porteous, Ian R. (1995),Clifford Algebras and the Classical Groups,Cambridge University Press, pp. 88–89,ISBN 0-521-55177-3
^Cariow, Aleksandr (2015). "An unified approach for developing rationalized algorithms for hypercomplex number multiplication".Przegląd Elektrotechniczny.1 (2). Wydawnictwo SIGMA-NOT:38–41.doi:10.15199/48.2015.02.09.ISSN 0033-2097.

External links

[edit]

The WikibookAbstract Algebra has a page on the topic of:Hypercomplex numbers

"Hypercomplex number",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
Weisstein, Eric W."Hypercomplex number".MathWorld.
Study, E.,On systems of complex numbers and their application to the theory of transformation groups(PDF) (English translation)
Frobenius, G.,Theory of hypercomplex quantities(PDF) (English translation)

v t e Number systems
Sets ofdefinable numbers	Natural numbers ( $\mathbb {N}$ ) Integers ( $\mathbb {Z}$ ) Rational numbers ( $\mathbb {Q}$ ) Constructible numbers Algebraic numbers ( $\mathbb {A}$ ) Closed-form numbers Periods ( ${\mathcal {P}}$ ) Computable numbers Arithmetical numbers Set-theoretically definable numbers Gaussian integers Gaussian rationals Eisenstein integers
Composition algebras	Division algebras:Real numbers ( $\mathbb {R}$ ) Complex numbers ( $\mathbb {C}$ ) Quaternions ( $\mathbb {H}$ ) Octonions ( $\mathbb {O}$ )
Split types	Over $\mathbb {R}$ : Split-complex numbers Split-quaternions Split-octonions Over $\mathbb {C}$ : Bicomplex numbers Biquaternions Bioctonions
Otherhypercomplex	Dual numbers Dual quaternions Dual-complex numbers Hyperbolic quaternions Sedenions ( $\mathbb {S}$ ) Trigintaduonions ( $\mathbb {T}$ ) Split-biquaternions Multicomplex numbers Geometric algebra/Clifford algebra Algebra of physical space Spacetime algebra Plane-based geometric algebra
Infinities andinfinitesimals	Cardinal numbers Extended natural numbers Extended real numbers Projective Extended complex numbers Hyperreal numbers Levi-Civita field Ordinal numbers Supernatural numbers Surreal numbers Superreal numbers
Other types	Irrational numbers Fuzzy numbers Transcendental numbers p-adic numbers (p-adic solenoids) Profinite integers Normal numbers
Classification List

v t e Dimension
Dimensional spaces	Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime
Other dimensions	Krull Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom
Polytopes andshapes	Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Number systems	Hypercomplex numbers Cayley–Dickson construction
Dimensions by number	Zero One Two Three Four Five Six Seven Eight n-dimensions
See also	Hyperspace Codimension
Category