Multicomplex number
From Wikipedia, the free encyclopedia
In mathematics, the multicomplex numbers, 
, form an n dimensional algebra generated by one element e which satisfies 
. They are a vector space over the reals with a commutavite 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
with and 
real. For 
an exponential representation exists:
.
Two equivalent matrix representations of the algebra can be generated by choosing
where q is an ordinary complex nth root of -1, i.e. q = exp( − iπ / n).
[edit] Isomorphisms
For even n the multicomplex numbers can be expressed as direct sum
.
For odd n they are equivalent to
.
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 Zn: Applications en Théorie des Champs, Habilitation, Université Louis Pasteur, Strasbourg 1997 pp. 20-29 (in French).
