We give a possible extension of definition of shears and overshears in the case of two non commutative (quaternionic) variables in relation with the associated vector fields and flows. We define the divergence operator and determine the vector fields with divergence. Given the non‐existence of quaternionic volume form on H², we define automorphisms with volume to be time‐one maps of vector fields with divergence and volume preserving automorphisms to be time‐one maps of vector fields with divergence 0. To these two classes the Andersen–Lempert theory applies. Finally, we exhibit an example of a quaternionic automorphism, which is not in the closure of the set of finite compositions of volume preserving quaternionic shears even though its restriction to the complex subspace $\mathbf{C}\times \mathbf{C}$ is in the closure of the set of finite compositions of complex shears.

中文翻译：

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关于H2中的一类自同构，它类似于保留体积的性质

在与相关矢量场和流有关的两个非交换（四元）变量的情况下，我们给出了剪切和超剪的定义的可能扩展。我们定义了散度算子，并确定了具有散度的向量场。给定H ²上不存在四元体体积形式，我们将体积的自同构定义为具有散度的矢量场的时间一图，而体积保留的自同构是具有散度为0的矢量场的时间一图。对于这两个类别Andersen-Lempert理论适用。最后，我们展示了一个四元自同构的例子，尽管它限制了复杂的子空间，但它并不在保留四元剪应力的有限成分集的闭合中$\mathbf{C}\times \mathbf{C}$ 在复杂剪力有限组的闭合中。