Symmetries in quaternionic analysis

Abstract

This survey-type paper deals with the symmetries related to quaternionic analysis. The main goal is to formulate an SU(2) invariant version of the theory. First, we consider the classical Lie groups related to the algebra of quaternions. After that, we recall the classical Spin(4) invariant case, that is Cauchy–Riemann operators, and recall their basic properties. We define the SU(2) invariant operators called the Coifman– Weiss operators. Then we study their relations with the classical Cauchy–Riemann operators and consider the factorization of the Laplace operator. Using SU(2) invariant harmonic polynomials, we obtain the Fourier series representations for quaternionic valued functions studying in detail the matrix coefficients.