# Hypercomplex number

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In mathematics, a hypercomplex number is a traditional term for an element of a unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory.

## History

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 1958 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.[2]

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[3] 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".[4] In 1973 Kantor and Solodovnikov published a textbook on hypercomplex numbers which was translated in 1989.[5][6]

Karen Parshall has written a detailed exposition of the heyday of hypercomplex numbers,[7] including the role of such luminaries as Theodor Molien[8] and Eduard Study.[9] For the transition to modern algebra, Bartel van der Waerden devotes thirty pages to hypercomplex numbers in his History of Algebra.[10]

## 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 ${\displaystyle (a_{0},\dots ,a_{n})}$ for a basis ${\displaystyle \{1,i_{1},\dots ,i_{n}\}}$. Where possible, it is conventional to choose the basis so that ${\displaystyle i_{k}^{2}\in \{-1,0,+1\}}$. A technical approach to hypercomplex numbers directs attention first to those of dimension two[why?].

## Two-dimensional real algebras

Theorem:[5]:14,15[11][12] 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 two dimensional, we can pick a basis {1, u}. Since the algebra is closed under squaring, the non-real basis element u squares to a linear combination of 1 and u:
${\displaystyle u^{2}=a_{0}+a_{1}u}$

for some 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

${\displaystyle u^{2}-a_{1}u+{\frac {a_{1}^{2}}{4}}=a_{0}+{\frac {a_{1}^{2}}{4}}.}$
${\displaystyle u^{2}-a_{1}u+{\frac {a_{1}^{2}}{4}}=\left(u-{\frac {a_{1}}{2}}\right)^{2}={\tilde {u}}^{2}}$  so that
${\displaystyle {\tilde {u}}^{2}~=a_{0}+{\frac {a_{1}^{2}}{4}}.}$

The three cases depend on this real value:

• If 4a0 = −a12, the above formula yields ũ2 = 0. Hence, ũ can directly be identified with the nilpotent element ${\displaystyle \epsilon }$ of the basis ${\displaystyle \{1,~\epsilon \}}$ of the dual numbers.
• If 4a0 > −a12, the above formula yields ũ2 > 0. This leads to the split-complex numbers which have normalized basis ${\displaystyle \{1,~j\}}$ with ${\displaystyle j^{2}=+1}$. To obtain j from ũ, the latter must be divided by the positive real number ${\displaystyle a:={\sqrt {a_{0}+{\frac {a_{1}^{2}}{4}}}}}$ which has the same square as ũ has.
• If 4a0 < −a12, the above formula yields ũ2 < 0. This leads to the complex numbers which have normalized basis ${\displaystyle \{1,~i\}}$ with ${\displaystyle i^{2}=-1}$. To yield i from ũ, the latter has to be divided by a positive real number ${\displaystyle a:={\sqrt {{\frac {a_{1}^{2}}{4}}-a_{0}}}}$ 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 ${\displaystyle {\tfrac {1}{2}}(1\pm j)}$ and zero divisors ${\displaystyle (1+j)(1-j)=0}$, 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".[13] The idea of cross-ratio of four complex numbers can be extended to the two-dimensional real algebras.[14]

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

### Clifford algebras

A 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, uv = ½(uv + vu) that can be used to orthogonalise the quadratic form, to give a set of bases {e1, ..., ek} such that:

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

Imposing closure under multiplication 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 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 ei2 = 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 Cp,q(R), indicating that the algebra is constructed from p simple basis elements with ei2 = +1, q with ei2 = −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 C0,1(R), split-complex numbers C1,0(R), quaternions C0,2(R), split-biquaternions C0,3(R), split-quaternions C1,1(R) ≈ C2,0(R) (the natural algebra of two-dimensional space); C3,0(R) (the natural algebra of three-dimensional space, and the algebra of the Pauli matrices); and the spacetime algebra C1,3(R).

The elements of the algebra Cp,q(R) form an even subalgebra C0q+1,p(R) of the algebra Cq+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 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 1995 Ian R. Porteous wrote on "The recognition of subalgebras" in his book on Clifford algebras. His Proposition 11.4 summarizes the hypercomplex cases:[15]

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 ${\displaystyle e_{0}^{2}=e_{1}^{2}=-1}$ 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 ${\displaystyle e_{0}^{2}=e_{1}^{2}=1}$ is isomorphic to M2(R) (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 ${\displaystyle e_{0}^{2}=e_{1}^{2}=e_{2}^{2}=-1}$ 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 ${\displaystyle e_{0}^{2}=e_{1}^{2}=e_{2}^{2}=1}$ is isomorphic to M2(C) (biquaternions, Pauli algebra, 2 × 2 complex matrices).

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

### Cayley–Dickson construction

All of the Clifford algebras Cp,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 = 2, 3, 4, ..., with bases ${\displaystyle \{1,i_{1},\dots ,i_{2^{n}-1}\}}$, where all the non-real basis elements anti-commute and satisfy ${\displaystyle i_{m}^{2}=-1}$. In 8 or more dimensions (n ≥ 3) these algebras are non-associative. In 16 or more dimensions (n ≥ 4) these algebras also have 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 the "split algebras" in the collection of composition algebras instead of the division algebras:

split-complex numbers with basis ${\displaystyle \{1,i_{1}\}}$ satisfying ${\displaystyle \ i_{1}^{2}=+1}$,
split-quaternions with basis ${\displaystyle \{1,i_{1},i_{2},i_{3}\}}$ satisfying ${\displaystyle \ i_{1}^{2}=-1,i_{2}^{2}=i_{3}^{2}=+1}$, and
split-octonions with basis ${\displaystyle \{1,i_{1},\dots ,i_{7}\}}$ satisfying ${\displaystyle \ i_{1}^{2}=i_{2}^{2}=i_{3}^{2}=-1}$, ${\displaystyle \ i_{4}^{2}=\cdots =i_{7}^{2}=+1.}$

Unlike the complex numbers, the split-complex numbers are not algebraically closed, and further contain zero divisors and non-trivial idempotents. As with the 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 ${\displaystyle \mathbb {C} \otimes _{\mathbb {R} }\mathbb {C} }$, eight-dimensional biquaternions ${\displaystyle \mathbb {C} \otimes _{\mathbb {R} }\mathbb {H} }$, and 16-dimensional complex octonions ${\displaystyle \mathbb {C} \otimes _{\mathbb {R} }\mathbb {O} }$.

## References

1. ^ Linear Associative Algebra (1881) American Journal of Mathematics 4(1):221–6
2. ^ Adams, J. F. (July 1960). "On the Non-Existence of Elements of Hopf Invariant One". Annals of Mathematics. 72 (1): 20–104. doi:10.2307/1970147. JSTOR 1970147.
3. ^ Thomas Hawkins (1972) "Hypercomplex numbers, Lie groups, and the creation of group representation theory", Archive for History of Exact Sciences 8:243–87
4. ^ 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
5. ^ a b Kantor, I.L., Solodownikow (1978), Hyperkomplexe Zahlen, BSB B.G. Teubner Verlagsgesellschaft, Leipzig
6. ^ Kantor, I. L.; Solodovnikov, A. S. (1989), Hypercomplex numbers, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96980-0, MR 996029
7. ^ Karen Parshall (1985) "Wedderburn and the Structure of Algebras" Archive for History of Exact Sciences 32:223–349
8. ^ Theodor Molien (1893) "Über Systeme höher complexen Zahlen", Mathematische Annalen 41:83–156
9. ^ Eduard Study (1898) "Theorie der gemeinen und höhern komplexen Grössen", Encyclopädie der mathematischen Wissenschaften I A 4 147–83
10. ^ * 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
11. ^ Isaak Yaglom (1968) Complex Numbers in Geometry, pages 10 to 14
12. ^ John H. Ewing editor (1991) Numbers, page 237, Springer, ISBN 3-540-97497-0
13. ^ Anthony A. Harkin & Joseph B. Harkin (2004) Geometry of Generalized Complex Numbers, Mathematics Magazine 77(2):118–29
14. ^ Sky Brewer (2013) "Projective Cross-ratio on Hypercomplex Numbers", Advances in Applied Clifford Algebras 23(1):1–14
15. ^ Ian R. Porteous (1995) Clifford Algebras and the Classical Groups, pages 88–89, Cambridge University Press ISBN 0-521-55177-3