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

TL;DR

This paper establishes a Hodge-type decomposition and analyzes the cohomology groups of the $k$-Cauchy-Fueter complex over domains in quaternionic space, revealing conditions for solvability of related boundary value problems.

Contribution

It proves the regularity of a boundary value problem for the $k$-Cauchy-Fueter complex and characterizes the solvability conditions via cohomology groups, extending quaternionic analysis analogous to complex Dolbeault theory.

Findings

Hodge-type orthogonal decomposition established

Non-homogeneous equation solvability characterized by cohomology

First cohomology group is finite-dimensional, second is trivial

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 $k\geq 2$ 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 $\mathscr H^1_{ (k) }(\Omega)$ 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.