Multicomplex number

In mathematics, the multicomplex numbers, ${\mathbb {MC}}_{n}$, 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.

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 $||x||\neq 0$ 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. $q=\exp(-i\pi /n)$.

Isomorphisms

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

${\mathbb {MC}}_{n}\sim \oplus ^{n/2}{\mathbb {C}}$.

For odd n they are equivalent to

${\mathbb {MC}}_{n}\sim \oplus ^{n}{\mathbb {R}}$.

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

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).