Abstract
An attempt to study the compatibility conditions, and the general free resolution, for the system associated with the Dirac operator in vector variables appeared already in Sabadini et al. (Math Z 239: 293-320, 2002), from the point of view of Clifford analysis, and in Sabadini et al. (Exp Math 12: 351-364, 2003) using the tool of megaforms. Other studies have been carried out in other papers, like Krump (Adv Appl Clifford Alg 19: 365-374, 2009), Krump and Souek (17: 537-548, 2007), Sala (The generalized Dolbeault complexes in Clifford analysis, Praha 2012), using methods of representation theory.
In this paper, we restrict our attention to the case of three variables and we describe the free resolution associate to the module from various different angles. The comparison has several noteworthy consequences.
In particular, it gives the explicit description of all the maps contained in the algebraic resolution and shows that they are invariant with respect to the action of. We also discuss how the methods used in this paper can be generalized to the case of Dirac operators.