Hypercomplex number

For the unrelated use of this term in non-standard analysis, see hyperreal number.

In mathematics, hypercomplex number is an element of a hypercomplex system, which is an archaic term for a finite dimensional algebra over the real numbers (possibly non-associative), such as the quaternions or octonions. Hypercomplex numbers have had a long lineage of devotees including Hermann Hankel, Georg Frobenius, Eduard Study, and Élie Cartan.

None of these extensions other than the real or complex numbers form a field, because the field of complex numbers is algebraically closed by the fundamental theorem of algebra.

Because of changes in algebraic terminology around the beginning of the 20th century, the term "hypercomplex number" is no longer used much in mathematics, except in translations from Russian where the term is still in use.

1 Definitions
2 Two-dimensional examples (one non-real axis)
- 2.1 Split-complex numbers
- 2.2 Dual numbers
3 Higher dimensional examples (more than one non-real axis)
4 References
5 External links

[ Definitions

A definition of hypercomplex number is given by Kantor & Solodovnikov (1989) as finite dimensional algebras over the reals that are unital and distributive (but not necessarily associative). Elements are generated through real number coefficients $(a_0 ,~... , a_n)$ for bases $\{ 1,i_1, \dots, i_n \}$ . It is conventional to normalize the basis so that $i_k^2 \in \{ -1, 0, +1 \}$ if possible.

[ Two-dimensional examples (one non-real axis)

There are up to isomorphism exactly 3 2-dimensional algebras over the reals: the complex numbers, the split complex numbers, and the dual numbers.

[ Split-complex numbers

Split-complex numbers have a base $\{ 1 , ~j \}$ with $j 2 = + 1$ . The algebra of split complex numbers is isomorphic to the sum of 2 copies of the real numbers.

Algebras that include such non-real roots of 1 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.

[ Dual numbers

Dual numbers have a base ${1,ε}$ with nilpotent $ε 2 = 0$ .

[ 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 (ie 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 - ie 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 space-time 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.

[ 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 complex numbers and the quaternions contain non-real elements j 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 bases anti-commute and satisfy $i_m^2 = -1$ . These are non-associative in dimension at least 8.

The first algebras in this sequence are the four-dimensional quaternions, eight-dimensional octonions, and 16-dimensional sedenions. However, satisfying these requirements comes at a price: Each increase in dimensionality introduces new algebraic complications. Quaternion multiplication is not commutative anymore, octonion multiplication is non-associative, and the normed of sedenions is not multiplicative.

The Cayley-Dickson construction can be modified by inserting an extra sign at some stages. This leads to coquaternions (split-quaternions; e.g. to bases $\{ 1,~i_1, i_2, i_3 \}$ with $i_1^2 = -1, i_2^2 = i_3^2 = +1$ , ) and split-octonions (e.g. to bases $\{ 1,~i_1, \dots , i_7 \}$ with $i_1^2 = i_2^2 = i_3^2 = -1$ , $i_4^2 = \cdots = i_7^2 = +1$ ). The coquaternions contain nilpotents, have a non-commutative multiplication, and are isomorphic to the 2 × 2 real matrices. Split-octonions are non-associative.

[ 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 leads to four-dimensional tessarines $\mathbb C\otimes\mathbb C$ , eight-dimensional biquaternions $\mathbb C\otimes\mathbb H$ , and 16-dimensional complex octonions $\mathbb C\otimes\mathbb O$ .

Tessarines offer a commutative and associative multiplication, biquaternions are associative but not commutative, and conic sedenions are not associative and not commutative. They all contain idempotents and zero-divisors, but offer a multiplicative modulus. Biquaternions contain nilpotents, conic sedenions are also not power associative.

[ Further examples

There are many hyperalgebras found and named in the 19th and early 20th centuries. Most of them turned out to be isomorphic to sums of matrix algebras over the real numbers, complex numbers or quaternions, so their names are no longer in use. Examples include split complex numbers, split quaternions, tessarines, biquaternions, multicomplex numbers, bicomplex numbers, and so on.

The hyperbolic quaternions (after Alexander MacFarlane) have a non-associative and non-commutative multiplication. All basis elements are roots of 1, i.e. $i_n^2 = +1$ for $n \in \{ 1, 2, 3 \}$ .

[ References

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, http://math.ucr.edu/home/baez/octonions/octonions.html
Jeanne La Duke "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 1983.
Kantor, I. L.; Solodovnikov, A. S. (1989), Hypercomplex numbers, Berlin, New York: Springer-Verlag, MR 996029, ISBN 978-0-387-96980-0
Vil'yams, N.N. (2001), "Hypercomplex number", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/H/h048390.htm
Weisstein, Eric W., "Hypercomplex number" from MathWorld.

[ External links

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