Explicit resolutions for the complex of several Fueter operators

Abstract

An analogue of the Dolbeault complex is introduced for regular functions of several quaternionic variables and studied by means of two different methods. The first one comes from algebraic analysis (for a thorough treatment see the book [F. Colombo, I. Sabadini, F. Sommen, D.C. Struppa, Analysis of Dirac systems and computational algebra, Progress in Mathematical Physics, Vol. 39, Birkhäuser, Boston, 2004]), while the other one relies on the symmetry of the equations and the methods of representation theory (see [F. Colombo, V. Souček, D.C. Struppa, Invariant resolutions for several Fueter operators, J. Geom. Phys. 56 (2006) 1175–1191; R.J. Baston, Quaternionic Complexes, J. Geom. Phys. 8 (1992) 29–52]). The comparison of the two results allows one to describe the operators appearing in the complex in an explicit form. This description leads to a duality theorem which is the generalization of the classical Martineau–Harvey theorem and which is related to hyperfunctions of several quaternionic variables.