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Abstract
Much attention has recently been paid to a study of analogues of the Dolbeault complex for the Dirac equation in several vector variables. In this article, we study these questions in dimension 4, in several quaternionic variables. Instead of the Fueter equation and quaternionic (or spinor) valued functions, we consider invariant first-order equations for functions with values in higher spin representations. We present a classification of conformally invariant equations on with the property that the corresponding equation in n variables is invariant with respect to the symmetry group of the projective quaternionic space. We get two series of equations. For each of them, we construct complexes starting from these equations and we relate them to complexes constructed earlier on quaternionic manifolds by R. Baston and S. Salamon (see Baston, R.J., 1992, Quaternionic Complexes, J. Geom. Phys., 8(1–4), 29–52, and Salamon, S.M., 1986, Differential geometry of quaternionic manifolds, Ann. Scient. Éc. Norm. Sup., 4
o
-serie, 19, pp. 31–55, respectively).
§Dedicated to Richard Delanghe on the occasion of his 65th birthday.
Acknowledgements
We would like to mention that the main ideas of this study (supported by grants GA UK 447/2004 and MSM0021620839) were developed during a recent visit of the authors to Ghent University. It was one of the many fruitful contacts between the authors and the Ghent Clifford Research Group, which was created and inspired for decades by Richard Delanghe. His strong influence was felt not only on the level of mathematics and the authors would like to express their gratitude to him and his family for many pleasant days spent with them in Ghent.
Notes
§Dedicated to Richard Delanghe on the occasion of his 65th birthday.