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

The kk-Cauchy–Fueter operator D0(k) on one dimensional quaternionic space HH is the Euclidean version of spin k/2k/2 massless field operator on the Minkowski space in physics. The kk-Cauchy–Fueter equation for k≥2k≥2 is overdetermined and its compatibility condition is given by the kk-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 kk-Cauchy–Fueter equation D0(k)u=f on a smooth domain ΩΩ in HH is solvable if and only if ff 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 kk-Cauchy–Fueter complex over ΩΩ, which is finite dimensional, while the second cohomology group is always trivial.