On the Hodge-type decomposition and cohomology groups of k-Cauchy–Fueter complexes over domains in the quaternionic space

Abstract

The k -Cauchy–Fueter operator D 0 ( k ) on one dimensional quaternionic space H is the Euclidean version of spin k / 2 massless field operator on the Minkowski space in physics. The k -Cauchy–Fueter equation for k ≥ 2 is overdetermined and its compatibility condition is given by the k -Cauchy–Fueter complex. In quaternionic analysis, these complexes play the role of Dolbeault complex in several complex variables. We prove that a natural boundary value problem associated to this complex is regular. Then by using the theory of regular boundary value problems, we show the Hodge-type orthogonal decomposition, and the fact that the non-homogeneous k -Cauchy–Fueter equation D 0 ( k ) u = f on a smooth domain Ω in H is solvable if and only if f satisfies the compatibility condition and is orthogonal to the set ℋ ( k ) 1 ( Ω ) of Hodge-type elements. This set is isomorphic to the first cohomology group of the k -Cauchy–Fueter complex over Ω , which is finite dimensional, while the second cohomology group is always trivial.