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

JOURNAL OF GEOMETRY AND PHYSICS

卷:107

Article

Chang, Der-Chen^1,2,3  Markina, Irina⁴  Wang, Wei⁵ 

[1] Georgetown Univ, Dept Math, Washington, DC 20057 USA

[2] Georgetown Univ, Dept Comp Sci, Washington, DC 20057 USA

[3] Fu Jen Catholic Univ, Dept Math, Taipei 242, Taiwan

[4] Univ Bergen, Dept Math, NO-5008 Bergen, Norway

[5] Zhejiang Univ, Dept Math, Hangzhou 310027, Zhejiang, Peoples R China

关键词: The k-Cauchy-Fueter operator;    The k-Cauchy-Fueter complex;    Cohomology groups;    Regular boundary value problem;    Hodge-type decomposition;    Quaternionic analysis;   

DOI  :  10.1016/j.geomphys.2016.04.016

来源: Elsevier

【 摘 要 】

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 Omega in H is solvable if and only if f satisfies the compatibility condition and is orthogonal to the set 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. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

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