The Structure of Monogenic Functions Yakov Krasnov ABSTRACT We study the structure of monogenic functions using symmetries of the Dirac operator. Keywords: Monogenic functions, Clifford algebra, symmetries. 1 Preliminaries Since all the functions considered in this paper are supposed to take their values in a Clifford algebra, in this section we present some notation and de- finitions concerning the real Clifford algebra Am and infinitesimal analysis on this algebra. The real Clifford algebra Am is a real vector space with 2 m basis ele- ments eo, e1, ... ,e2m_1, defined by eo == eo = 1, el = el, ... ,em = em, e12 = ele2, e13 = ele3, ... ,em-I,m = em-l em,··· ,e12 ... m = ele2··· em, where el, ... ,em are standard hypercomplex numbers satisfying eiej + ejei = -2eOJij , (eiej)ek=ei(ejek), i,j,k=1, ... ,m. (1.1) As usual, we identify the canonical basis in JRm with m imaginary units generating Am and such that every element in Am can be (uniquely) represented in the form m U = L L UleI, UI E JR. (1.2) k=O III=k Hereafter, the summation index in the inner sum runs over all strictly increasing ordered k-tuples of elements from the set w = {1, 2, ... ,m}. In particular, I = ( i 1, i2, ... i k) in (1. 2) is defined in such a way that 1 S il < i2 < ... < ik S m. AMS Subject Classification: 3DG35, 31BD5, 41A1D. J. Ryan et al. (eds.), Clifford Algebras and their Applications in Mathematical Physics © Birkhäuser Boston 2000