Multicomplex number

From Wikipedia, the free encyclopedia

In mathematics, the multicomplex numbers, ${\Bbb{MC}}_n$ , form an n dimensional algebra generated by one element e which satisfies $~e^n = -1$ . They are a vector space over the reals with a commutative and associative multiplication that distributes over addition. The term polynumber is used synonymously at times.

[edit] Representations

A multicomplex number x can be written as

$x = \sum_{i = 0}^{n-1} x_i e^i$

with $~e^n = -1$ and $~x_i$ real. For $|| x || \ne 0$ an exponential representation exists:

$x = \sum_{i=0}^{n-1} x_i e^i = \rho \exp ( \sum_{i=1}^{n-1} \Theta{}_i e^i )$ .

Two equivalent matrix representations of the algebra can be generated by choosing

$e \rightarrow E = \begin{pmatrix} q^{\frac{1}{2}} & 0 & 0 & ... & 0 \\ 0 & q^{\frac{3}{2}} & 0 & ... & 0 \\ \vdots & & \ddots & \ddots & \vdots \\ 0 & 0 & ... & q^{-\frac{3}{2}} & 0 \\ 0 & 0 & 0 & ... & q^{-\frac{1}{2}} \\ \end{pmatrix}, E^{\prime} = \begin{pmatrix} 0 & 1 & 0 & ... & 0 \\ 0 & 0 & 1 & ... & 0 \\ \vdots & & \ddots & \ddots & \vdots \\ 0 & 0 & ... & 0 & 1 \\ -1 & 0 & 0 & ... & 0 \\ \end{pmatrix}$

where q is an ordinary complex nth root of -1, i.e. $q = exp( - i π / n)$ .

[edit] Isomorphisms

For even n the multicomplex numbers can be expressed as direct sum

${\Bbb{MC}}_n \sim \oplus^{n/2} {\Bbb{C}}$ .

For odd n they are equivalent to

${\Bbb{MC}}_n \sim \oplus^{n} {\Bbb{R}}$ .

A special case of multicomplex numbers are the bicomplex numbers with n = 4, which are also isomorphic to the outer product C⊗C.

[edit] References

G. Baley Price, An Introduction to Multicomplex Spaces and Functions, Marcel Dekker Inc., New York, 1991
Michel Rausch de Traubenberg, Algèbres de Clifford, Supersymétrie et Symétries Z_n: Applications en Théorie des Champs, Habilitation, Université Louis Pasteur, Strasbourg 1997 pp. 20-29 (in French).