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Hypercomplex number

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

History

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Alattice of some hypercomplex number systems and the corresponding lattice ofgroups generated by the basis elements

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 reals⁠R{\displaystyle \mathbb {R} }⁠, the complexes⁠C{\displaystyle \mathbb {C} }⁠, the quaternions⁠H{\displaystyle \mathbb {H} }⁠, and the octonions⁠O{\displaystyle \mathbb {O} }⁠, and theFrobenius theorem says the only realassociative division algebras are⁠R{\displaystyle \mathbb {R} }⁠,⁠C{\displaystyle \mathbb {C} }⁠, and⁠H{\displaystyle \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 × 2real matrices were foundisomorphic 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

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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(a0,…,an){\displaystyle (a_{0},\dots ,a_{n})} for a basis⁠{1,i1,…,in}{\displaystyle \{1,i_{1},\dots ,i_{n}\}}⁠. Where possible, it is conventional to choose the basis so that⁠ik2∈{−1,0,+1}{\displaystyle i_{k}^{2}\in \{-1,0,+1\}}⁠. A technical approach to hypercomplex numbers directs attention first to those ofdimension two.

Two-dimensional real algebras

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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:

u2=a0+a1u{\displaystyle 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

u2−a1u+14a12=a0+14a12.{\displaystyle u^{2}-a_{1}u+{\frac {1}{4}}a_{1}^{2}=a_{0}+{\frac {1}{4}}a_{1}^{2}.}

Rewriting the left half of the above equation as(u−12a1)2{\textstyle \left(u-{\frac {1}{2}}a_{1}\right)^{2}} and then substituting in new elementũ where⁠u~=u−12a1{\displaystyle {\tilde {u}}=u-{\tfrac {1}{2}}a_{1}}⁠, one obtains

u~2=a0+14a12.{\displaystyle {\tilde {u}}^{2}=a_{0}+{\frac {1}{4}}a_{1}^{2}.}

The three cases depend on this real value:

The complex numbers are the onlyquadratic algebra that is afield.Split algebras such as the split-complex numbers that include non-real roots of 1 also containidempotents12(1±j){\textstyle {\frac {1}{2}}(1\pm j)} andzero divisors⁠(1+j)(1−j)=0{\displaystyle (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)

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Clifford algebras

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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, ...,ek} such that:12(eiej+ejei)={−1,0,+1i=j,0i≠j.{\displaystyle {\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 2k elements, {1,e1,e2,e3, ...,e1e2, ...,e1e2e3, ...}. These can be interpreted as the basis of a hypercomplex number system. Unlike the basis {e1, ...,ek}, 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 elementei such thatei2 = 0 (i.e. directions in the original space over which the quadratic form wasdegenerate), the remaining Clifford algebras can be identified by the label Clp,q(⁠R{\displaystyle \mathbb {R} }⁠), indicating that the algebra is constructed fromp simple basis elements withei2 = +1,q withei2 = −1, and whereR{\displaystyle \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 Cl0,1(⁠R{\displaystyle \mathbb {R} }⁠),split-complex numbers Cl1,0(⁠R{\displaystyle \mathbb {R} }⁠),quaternions Cl0,2(⁠R{\displaystyle \mathbb {R} }⁠),split-biquaternions Cl0,3(⁠R{\displaystyle \mathbb {R} }⁠),split-quaternionsCl1,1(⁠R{\displaystyle \mathbb {R} }⁠) ≈ Cl2,0(⁠R{\displaystyle \mathbb {R} }⁠) (the natural algebra of two-dimensional space); Cl3,0(⁠R{\displaystyle \mathbb {R} }⁠) (the natural algebra of three-dimensional space, and the algebra of thePauli matrices); and thespacetime algebra Cl1,3(⁠R{\displaystyle \mathbb {R} }⁠).

The elements of the algebra Clp,q(⁠R{\displaystyle \mathbb {R} }⁠) form an even subalgebra Cl[0]
q+1,p(⁠R{\displaystyle \mathbb {R} }⁠) of the algebra Clq+1,p(⁠R{\displaystyle \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
For extension beyond the classical algebras, seeClassification of Clifford algebras.

Cayley–Dickson construction

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Further information:Cayley–Dickson construction
CayleyQ8 graph of quaternion multiplication showing cycles of multiplication ofi (red),j (green) andk (blue). After clicking on the diagram, hover over or click a path to highlight it.

All of the Clifford algebras Clp,q(⁠R{\displaystyle \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{1,i1,…,i2n−1}{\displaystyle \left\{1,i_{1},\dots ,i_{2^{n}-1}\right\}}⁠, where all the non-real basis elements anti-commute and satisfy⁠im2=−1{\displaystyle {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 have nonzerozero-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]

NameNo. of
dimensions		Dimensions
(2n)		Symbol
real numbers120R{\displaystyle \mathbb {R} }
complex numbers221C{\displaystyle \mathbb {C} }
quaternions422H{\displaystyle \mathbb {H} }
octonions823O{\displaystyle \mathbb {O} }
sedenions1624S{\displaystyle \mathbb {S} }
trigintaduonions3225T{\displaystyle \mathbb {T} }
sexagintaquatronions6426
centumduodetrigintanions12827
ducentiquinquagintasexions25628

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,i1}{\displaystyle \{1,\,i_{1}\}} satisfying⁠i12=+1{\displaystyle i_{1}^{2}=+1}⁠,
split-quaternions with basis{1,i1,i2,i3}{\displaystyle \{1,\,i_{1},\,i_{2},\,i_{3}\}} satisfying⁠i12=−1,i22=i32=+1{\displaystyle i_{1}^{2}=-1,\,i_{2}^{2}=i_{3}^{2}=+1}⁠, and
split-octonions with basis{1,i1,…,i7}{\displaystyle \{1,\,i_{1},\,\dots ,\,i_{7}\}} satisfying⁠i12=i22=i32=−1{\displaystyle i_{1}^{2}=i_{2}^{2}=i_{3}^{2}=-1}⁠,⁠i42=i52=i62=i72=+1{\displaystyle 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

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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 numbersC⊗RC{\displaystyle \mathbb {C} \otimes _{\mathbb {R} }\mathbb {C} } (isomorphic to tessarines⁠C⊗RD{\displaystyle \mathbb {C} \otimes _{\mathbb {R} }D}⁠), eight-dimensionalbiquaternions⁠C⊗RH{\displaystyle \mathbb {C} \otimes _{\mathbb {R} }\mathbb {H} }⁠, and 16-dimensionalcomplex octonions⁠C⊗RO{\displaystyle \mathbb {C} \otimes _{\mathbb {R} }\mathbb {O} }⁠.

Another example is the tensor product of two quaternion algebrasH⊗2=H⊗RH{\displaystyle \mathbb {H} ^{\otimes 2}=\mathbb {H} \otimes _{\mathbb {R} }\mathbb {H} } (isomorphic to the Clifford algebraCl3,1(R){\displaystyle Cl_{3,1}\mathbb {(R)} } and to4×4{\displaystyle 4\times 4} real matrices⁠M(4,R){\displaystyle M(4,\mathbb {R} )}⁠) leading to applications in relativistic physics.[19][20]

More generally, one definesH⊗m{\displaystyle \mathbb {H} ^{\otimes m}} (and its subalgebras) as ”hyperquaternion algebras”. In particular,H⊗3=M(4,H){\displaystyle \mathbb {H} ^{\otimes 3}=M(4,\mathbb {H} )} yields aquaternionic matrix and its even subalgebraH⊗2⊗RC{\displaystyle \mathbb {H} ^{\otimes 2}\otimes _{\mathbb {R} }\mathbb {C} } (Dirac algebra).[21][22]

Further examples

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See also

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References

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  1. ^Peirce, Benjamin (1881),"Linear Associative Algebra",American Journal of Mathematics,4 (1):221–6,doi:10.2307/2369153,JSTOR 2369153
  2. ^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
  3. ^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
  4. ^Emil Artin later generalized Wedderburn's result so it is known as theArtin–Wedderburn theorem
  5. ^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
  6. ^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
  7. ^abKantor, I.L., Solodownikow (1978),Hyperkomplexe Zahlen, BSB B.G. Teubner Verlagsgesellschaft, Leipzig
  8. ^Kantor, I. L.; Solodovnikov, A. S. (1989),Hypercomplex numbers, Berlin, New York:Springer-Verlag,ISBN 978-0-387-96980-0,MR 0996029
  9. ^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
  10. ^Molien, Theodor (1893),"Ueber Systeme höherer complexer Zahlen",Mathematische Annalen,41 (1):83–156,doi:10.1007/BF01443450,S2CID 122333076
  11. ^Study, Eduard (1898), "Theorie der gemeinen und höhern komplexen Grössen",Encyclopädie der mathematischen Wissenschaften, vol. I A, pp. 147–183
  12. ^van der Waerden, B.L. (1985), "10. The discovery of algebras, 11. Structure of algebras",A History of Algebra, Springer,ISBN 3-540-13610X
  13. ^Yaglom, Isaak (1968),Complex Numbers in Geometry, pp. 10–14
  14. ^Ewing, John H., ed. (1991),Numbers, Springer, p. 237,ISBN 3-540-97497-0
  15. ^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
  16. ^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
  17. ^Porteous, Ian R. (1995),Clifford Algebras and the Classical Groups,Cambridge University Press, pp. 88–89,ISBN 0-521-55177-3
  18. ^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.
  19. ^Girard, P. R. (2007).Quaternions, Clifford algebras and relativistic physics. Springer Science & Business Media.
  20. ^Gürlebeck, K.; Sprössig, W. (1997).Quaternionic and Clifford calculus for physicists and engineers. New York: John Wiley & Sons.
  21. ^Girard, P.R.; Clarysse, P.; Pujol, R.; Delachartre, P. (2025). "Hyperquaternionic unitary symplectic groups: A unifying tool for physics".Advances in Applied Clifford Algebras.35 (40). Springer.doi:10.1007/s00006-025-01402-w.
  22. ^Sprössig, W. (2020). "Some new aspects in hypercomplex analysis".Advancements in Complex Analysis: From Theory to Practice. Springer. pp. 497–518.

Further reading

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External links

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