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# Poincaré Lemma on Quaternion-Like Heisenberg Groups

Published:2017-07-10

• Der-Chen Chang,
Department of Mathematics and Statistics, Georgetown University, Washington D.C. 20057, USA
• Nanping Yang,
Department of Mathematics, Fu Jen Catholic University, Taipei 242, Taiwan, ROC
• Hsi-Chun Wu,
Graduate Institute of Applied Science and Engineering, Fu Jen Catholic University, Taipei 242, Taiwan, ROC
 Format: LaTeX MathJax PDF

## Abstract

For smooth functions $a_1, a_2, a_3, a_4$ on a quaternion Heisenberg group, we characterize the existence of solutions of the partial differential operator system $X_1f=a_1, X_2f=a_2, X_3f=a_3,$ and $X_4f=a_4$. In addition, a formula for the solution function $f$ is deduced provided the solvability of the system.
 Keywords: bracket generating property, quaternion Heisenberg group, curl, integrability condition, Poincaré lemma
 MSC Classifications: 93B05 - Controllability 49N99 - None of the above, but in this section

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