On the Hodge-type decomposition and cohomolgy 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 $\mathbb{H}$ is the Euclidean version of helicity $\frac{k}{2}$ massless field operator on the Minkowski space in physics. The $k$-Cauchy-Fueter equation for $k\ge 2$ 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 $\Omega$ in $\mathbb{H}$ is solvable if and only if $f$ satisfies the compatibility condition and is orthogonal to the set ${\mathcal{H}}_{(k)}^1(\Omega )$ of Hodge-type elements. This set is isomorphic to the first cohomology group of the $k$-Cauchy-Fueter complex over $\Omega$, which is finite dimensional, while the second cohomology group is always trivial.

24 pages

Cited by in corpus (4)

• The Neumann problem for the $k$-Cauchy-Fueter complexes over $k$-pseudoconvex domains in ${\mathbb{R}}^4$ and the $L^2$ estimate
• On quaternionic complexes over unimodular quaternionic manifolds
• The resolution of Euclidean massless field operators of higher spins on ${\mathbb{R}}^6$ and the $L^2$ method
• On the weighted $L^2$ estimate for the $k$-Cauchy-Fueter operator and the weighted $k$-Bergman kernel