flerkomplexa

Flerkomplexa, or multicomplex numbers, are a generalization of the ordinary complex numbers obtained by adjoining several imaginary units to the real numbers. A common construction fixes an integer m ≥ 1 and introduces imaginary units i1, i2, ..., im with each i_k^2 = -1 and with the units commuting with one another. The resulting algebra is a real vector space of dimension 2^m and is closed under addition and multiplication, forming a commutative, associative algebra over the reals. For m ≥ 2 the algebra typically contains zero divisors, meaning that nonzero elements can multiply to zero.

An element is a linear combination of the basis elements formed by all finite products of the i_k, so every flerkomplex number can be written as a sum over subsets S of {1,...,m} of a_S ∏_{k ∈ S} i_k, with real coefficients a_S. A useful special case is the bicomplex numbers (m = 2), which can be realized as the tensor product C ⊗ C and admit an idempotent decomposition that facilitates multiplication and analysis through projections.

Involutions (conjugations) in flerkomplexa can be obtained by negating arbitrary subsets of the imaginary units, yielding several conjugates and corresponding norms. A typical norm is built from a product of conjugates, producing a real-valued measure of size that can fail to be positive-definite due to zero divisors.

Analytical aspects extend concepts from complex analysis to multicomplex variables, giving rise to multicomplex analysis. Differentiability can be defined via generalized Cauchy–Riemann conditions; while some results parallel the one-variable theory, others differ because of zero divisors and the richer algebraic structure. Applications are mostly theoretical, serving as a framework in algebra, number theory and certain areas of physics and signal processing where multiple commuting imaginary directions are advantageous.