Hypercomplex number

In mathematics, a hypercomplex number is a traditional term for an element of an algebra over the field of real numbers. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, added to the real and complex 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 when Benjamin Peirce first published his Linear Associative Algebra, and was carried forward by his son Charles Sanders Peirce.^[1] Most significantly, they identified the nilpotent and the idempotent elements as useful hypercomplex numbers for classifications. The Cayley–Dickson construction used involutions 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 (normed division algebras), and Frobenius theorem (associative division algebras). Finally in 1956 J. Frank Adams used topological methods to prove that there exist only four finite-dimensional real division algebras: the reals ℝ, the complexes ℂ, the quaternions ℍ, and the octonions 𝕆.

It was matrix algebra that harnessed the hypercomplex systems. First, matrices contributed new hypercomplex numbers like 2 × 2 real matrices. Soon the matrix paradigm began to explain the others as they became represented by matrices and their operations. In 1907 Joseph Wedderburn showed that associative hypercomplex systems could be represented by matrices, or direct sums of systems of matrices. From that date the preferred term for a hypercomplex system became associative algebra as seen in the title of Wedderburn’s thesis at University of Edinburgh. Note however, that non-associative systems like octonions and hyperbolic quaternions represent another type of hypercomplex number.

As Hawkins (1972) explains, the hypercomplex numbers are stepping stones to learning about Lie groups and group representation theory. For instance, in 1929 Emmy Noether wrote on "hypercomplex quantities and representation theory". Review of the historic particulars gives body to the generalities of modern theory. In 1973 Kantor and Solodovnikov published a textbook on hypercomplex numbers which was translated in 1989; a reviewer says it has a "highly classical flavour". See Karen Parshall (1985) for a detailed exposition of the heyday of hypercomplex numbers, including the role of such luminaries as Theodor Molien and Eduard Study. For the transition to modern algebra, Bartel van der Waerden devotes thirty pages to hypercomplex numbers in his History of Algebra (1985).

Definition

A definition of a hypercomplex number is given by Kantor & Solodovnikov (1989) as an element of a finite-dimensional algebra over the real numbers that is unital and distributive (but not necessarily associative). 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 of dimension two. Higher dimensions are configured as Cliffordian or algebraic sums of other algebras.

Two-dimensional real algebras

Theorem:^[2]^[3]^[4] Up to isomorphism, there are exactly three 2-dimensional unital algebras over the reals: the ordinary complex numbers, the split-complex numbers, and the dual numbers.

proof: Since the algebra is closed under squaring, and it has but two dimensions, the non-real basis element u squares to an arbitrary linear combination of 1 and u:

u^2=a_0+a_1u

with arbitrary real numbers a₀ and a₁. Using the common method of completing the square by subtracting a₁u and adding the quadratic complement a₁²/4 to both sides yields

u^2-a_1u+\frac{a_1^2}{4}=a_0+\frac{a_1^2}{4} .

u^2-a_1u+\frac{a_1^2}{4}=\left(u-\frac{a_1}{2}\right)^2= \tilde{u}^2

so that

\tilde{u}^2~=a_0+\frac{a_1^2}{4} .

The three cases depend on this real value:

If 4a₀ = −a 2
1 , the above formula yields ũ² = 0. Hence, ũ can directly be identified with the nilpotent element $\epsilon$ of the Dual numbers' basis $\{ 1, ~\epsilon \}$ .
If 4a₀ > −a 2
1 , the above formula yields ũ² > 0. This leads to the split-complex numbers which have normalized basis $\{ 1 , ~j \}$ with $j^2 = +1$ . To obtain j from ũ, the latter must be divided by the positive real number $a:=\sqrt{a_0+\frac{a_1^2}{4}}$ which has the same square as ũ.
If 4a₀ < −a 2
1 , the above formula yields ũ² < 0. This leads to the complex numbers which have normalized basis $\{ 1 , ~i \}$ with $i^2 = -1$ . To yield i from ũ, the latter has to be divided by a positive real number $a:=\sqrt{\frac{a_1^2}{4}-a_0}$ which squares to the negative of ũ².

The complex numbers are the only two-dimensional hypercomplex algebra that is a field. Algebras such as the split-complex numbers that include non-real roots of 1 also contain idempotents $\tfrac{1}{2} (1 \pm j)$ and zero divisors $(1 + j)(1 - j) = 0$ , so such algebras cannot be division algebras. However, these properties can turn out to be very meaningful, for instance in describing the Lorentz transformations of special relativity.

In a 2004 edition of Mathematics Magazine the two-dimensional real algebras have been styled the "generalized complex numbers".^[5] The idea of cross-ratio of four complex numbers can be extended to the two-dimensional real algebras.^[6]

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

Clifford algebras

Clifford algebra is the unital associative algebra generated over an underlying vector space equipped with a quadratic form. Over the real numbers this is equivalent to being able to define a symmetric scalar product, u⋅v = ½(uv + vu) that can be used to orthogonalise the quadratic form, to give a set of bases {e₁, ..., e_k} such that:

\tfrac{1}{2} (e_i e_j + e_j e_i) = \Bigg\{ \begin{matrix} -1, 0, +1 & i=j, \\ 0 & i \not = j. \end{matrix}

Imposing closure under multiplication now generates a multivector space spanned by 2^k bases, {1, e₁, e₂, e₃, ..., e₁e₂, ..., e₁e₂e₃, ...}. These can be interpreted as the bases of a hypercomplex number system. Unlike the bases {e₁, ..., e_k}, the remaining bases may or may not anti-commute, depending on how many simple exchanges must be carried out to swap the two factors. So e₁e₂ = −e₂e₁; but e₁(e₂e₃) = +(e₂e₃)e₁.

Putting aside the bases for which e_i² = 0 (i.e. directions in the original space over which the quadratic form was degenerate), the remaining Clifford algebras can be identified by the label Cℓ_p,q(R) indicating that the algebra is constructed from p simple bases with e_i² = +1, q with e_i² = −1, and where 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, called geometric algebras, form a systematic set which turn out to be very useful in physics problems which involve rotations, phases, or spins, notably in classical and quantum mechanics, electromagnetic theory and relativity.

Examples include: the complex numbers Cℓ_0,1(R); split-complex numbers Cℓ_1,0(R); quaternions Cℓ_0,2(R); split-biquaternions Cℓ_0,3(R); coquaternions Cℓ_1,1(R) ≈ Cℓ_2,0(R) (the natural algebra of 2d space); Cℓ_3,0(R) (the natural algebra of 3d space, and the algebra of the Pauli matrices); and Cℓ_1,3(R) the spacetime algebra.

The elements of the algebra Cℓ_p,q(R) form an even subalgebra Cℓ⁰_q+1,p(R) of the algebra Cℓ_q+1,p(R), which can be used to parametrise rotations in the larger algebra. There is thus a close connection between complex numbers and rotations in 2D space; between quaternions and rotations in 3D space; between split-complex numbers and (hyperbolic) rotations (Lorentz transformations) in 1+1 D space, and so on.

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

In 1995 Ian R. Porteous wrote on "The recognition of subalgebras" in his book on Clifford algebras. His Proposition 11.4 summarizes the hypercomplex cases:^[7]

Let A be a real associative algebra with unit element 1. Then

1 generates R (algebra of real numbers),
any two-dimensional subalgebra generated by an element e₀ of A such that e₀² = −1 is isomorphic to C (algebra of complex numbers),
any two-dimensional subalgebra generated by an element e₀ of A such that e₀² = 1 is isomorphic to ²R (algebra of split-complex numbers),
any four-dimensional subalgebra generated by a set {e₀, e₁} of mutually anti-commuting elements of A such that $e_0 ^2 = e_1 ^2 = -1$ is isomorphic to H (algebra of quaternions),
any four-dimensional subalgebra generated by a set {e₀, e₁} of mutually anti-commuting elements of A such that $e_0 ^2 = e_1 ^2 = 1$ is isomorphic to R(2) (2 × 2 real matrices, coquaternions),
any eight-dimensional subalgebra generated by a set {e₀, e₁, e₂} of mutually anti-commuting elements of A such that $e_0 ^2 = e_1 ^2 = e_2 ^2 = -1$ is isomorphic to ²H (split-biquaternions),
any eight-dimensional subalgebra generated by a set {e₀, e₁, e₂} of mutually anti-commuting elements of A such that $e_0 ^2 = e_1 ^2 = e_2 ^2 = 1$ is isomorphic to C(2) (biquaternions, Pauli algebra, 2 × 2 complex matrices).

For extension beyond the classical algebras, see Classification of Clifford algebras.

Cayley–Dickson construction

For more details on this topic, see Cayley–Dickson construction.

All of the Clifford algebras Cℓ_p,q(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 the Cayley–Dickson construction. This generates number systems of dimension 2ⁿ, n in {2, 3, 4, ...}, with bases $\{1, i_1, \dots, i_{2^n-1}\}$ , 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 have zero-divisors.

The first algebras in this sequence are the four-dimensional quaternions, eight-dimensional octonions, and 16-dimensional sedenions. An algebraic symmetry is lost with each increase in dimensionality: quaternion multiplication is not commutative, octonion multiplication is non-associative, and the norm of sedenions is not multiplicative.

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

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 = \cdots = i_7^2 = +1 .

As with quaternions, split-quaternions are not commutative, but further contain nilpotents; they are isomorphic to the 2 × 2 real matrices. Split-octonions are non-associative and contain nilpotents.

Tensor products

The tensor 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-dimensional tessarines $\mathbb C\otimes_\mathbb{R}\mathbb C$ , eight-dimensional biquaternions $\mathbb C\otimes_\mathbb{R}\mathbb H$ , and 16-dimensional complex octonions $\mathbb C\otimes_\mathbb{R}\mathbb O$ .

Further examples

bicomplex numbers: a 4D vector space over the reals, or 2D over the complex numbers
multicomplex numbers: 2ⁿ⁻¹-dimensional vector spaces over the complex numbers
composition algebra: algebras with a quadratic form that composes with the product

Notes and references

↑ Linear Associative Algebra (1881) American Journal of Mathematics 4(1):221–6
↑ Isaak Yaglom (1968) Complex Numbers in Geometry, pages 10 to 14
↑ John H. Ewing editor (1991) Numbers, page 237, Springer, ISBN 3-540-97497-0
↑ Kantor & Solodovnikov (1978) 14,15
↑ Anthony A. Harkin & Joseph B. Harkin (2004) Geometry of Generalized Complex Numbers, Mathematics Magazine 77(2):118–29
↑ Sky Brewer (2013) "Projective Cross-ratio on Hypercomplex Numbers", Advances in Applied Clifford Algebras 23(1):1–14
↑ Ian R. Porteous (1995) Clifford Algebras and the Classical Groups, pages 88 & 89, Cambridge University Press ISBN 0-521-55177-3

Daniel Alfsmann (2006) On families of 2^N dimensional hypercomplex algebras suitable for digital signal processing, 14th European Signal Processing Conference, Florence, Italy.
Emil Artin (1928) "Zur Theorie der hyperkomplexen Zahlen" and "Zur Arithmetik hyperkomplexer Zahlen", in The Collected Papers of Emil Artin, Serge Lang and John T. Tate editors, pp 301–45, Addison-Wesley, 1965.
Baez, John (2002), "The Octonions", Bulletin of the American Mathematical Society 39: 145–205, doi:10.1090/S0273-0979-01-00934-X, ISSN 0002-9904
Elie Cartan (1908) "Les systems de nombres complex et les groupes de transformations", Encyclopédie des sciences mathématiques pures et appliquées I 1. and Ouvres Completes T.2 pt. 1, pp 107–246.
Thomas Hawkins (1972) "Hypercomplex numbers, Lie groups, and the creation of group representation theory", Archive for History of Exact Sciences 8:243–87.
Kantor, 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 996029
Jeanne La Duke (1983) "The study of linear associative algebras in the United States, 1870–1927", see pp. 147–159 of Emmy Noether in Bryn Mawr Bhama Srinivasan & Judith Sally editors, Springer Verlag.
Theodor Molien (1893) "Über Systeme höher complexen Zahlen", Mathematische Annalen 41:83–156.
Silviu Olariu (2002) Complex Numbers in N Dimensions, North-Holland Mathematics Studies #190, Elsevier ISBN 0-444-51123-7 .
Karen Parshall (1985) "Wedderburn and the Structure of Algebras" Archive for History of Exact Sciences 32:223–349.
Irene Sabadini, Michael Shapiro & Frank Sommen, editors (2009) Hypercomplex Analysis and Applications Birkhauser ISBN 978-3-7643-9892-7 .
Eduard Study (1898) "Theorie der gemeinen und höhern komplexen Grössen", Encyclopädie der mathematischen Wissenschaften I A 4 147–83.
Henry Taber (1904) "On Hypercomplex Number Systems", Transactions of the American Mathematical Society 5:509.
B.L. van der Waerden (1985) A History of Algebra, Chapter 10: The discovery of algebras, Chapter 11: Structure of algebras, Springer, ISBN 3-540-13610X .
Joseph Wedderburn (1908) "On Hypercomplex Numbers", Proceedings of the London Mathematical Society 6:77–118.

External links

Hazewinkel, Michiel, ed. (2001), "Hypercomplex number", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
History of the Hypercomplexes on hyperjeff.com
Hypercomplex.info
Weisstein, Eric W., "Hypercomplex number", MathWorld.
E. Study, "On systems of complex numbers and their application to the theory of transformation groups" (English translation)
G. Frobenius, "Theory of hypercomplex quantities" (English translation)

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