Double and dual numbers. SU(2) groups, two-component spinors and generating functions

G. F. Torres del Castillo, K. C. Gutiérrez-Herrera


We explicitly show that the groups of $2 \times 2$ unitary matrices with determinant equal to 1 whose entries are double or dual numbers are homomorphic to ${\rm SO}(2,1)$ or to the group of rigid motions of the Euclidean plane, respectively, and we introduce the corresponding two-component spinors. We show that with the aid of the double numbers we can find generating functions for separable solutions of the Laplace equation in the $(2 + 1)$ Minkowski space, which contain special functions that also appear in the solution of the Laplace equation in the three-dimensional Euclidean space, in spheroidal and toroidal coordinates.


Double numbers; dual numbers; unitary groups; spinors; Minkowski $(2 + 1)$ space; Laplace's equation; spheroidal coordinates; toroidal coordinates.

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