Abstract
Elliptic scator algebra is possible in 1+n dimensions, n∈N. It is isomorphic to complex algebra in 1 + 1 dimensions, when the real part and any one hypercomplex component are considered. It is endowed with two representations: an additive one, where the scator components are represented as a sum; and a polar representation, where the scator components are represented as products of exponentials. Within the scator framework, De Moivre’s formula is generalized to 1+n dimensions in the so called Victoria equation. This novel formula is then used to obtain compact expressions for the integer powers of scator elements. A scator in S1+n can be factored into a product of n scators that are geometrically represented as its projections onto n two dimensional planes. A geometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis is expounded. The powers of scators, when the ratio of their director components is a rational number, lie on closed curves. For 1 + 2 dimensional scators, twisted curves in a three dimensional space are obtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in 1 + 2 dimensions.
Highlights
De Moivre’s formula establishes a relationship between complex algebra and trigonometry through the evaluation of powers of numbers in modulus and angle variables. This type of relationship has been extended to higher dimensional algebras, notably quaternions [1], split quaternions [2], dual complex numbers [3], real and complex 2 × 2 matrices [4] and other Clifford algebras [5]
The exponential function is unique, between other things, because for the real and complex algebras, it maps the additive group onto the multiplicative group, the basic operations of the real and complex fields
The underlying philosophy of this algebra is to some extent, related to geometric algebras [8,9]
Summary
De Moivre’s formula establishes a relationship between complex algebra and trigonometry through the evaluation of powers of numbers in modulus and angle variables. Scator algebra has been successfully applied to other problems, such as a time–space description in deformed Lorentz metrics [12,13,14] and three dimensional fractal structures [15,16] This algebraic structure has two representations, an additive and a multiplicative representation that corresponds to the rectangular and polar versions of complex numbers. Just as in real and complex algebra, the components exponential function, reviewed maps scator addition onto scator multiplication. This result is used to prove the commutative group properties in the multiplicative representation and its relationship with non-associativity in the additive representation.
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