We study the deformation of Courant pairs with a commutative algebra base. We consider the deformation cohomology bi-complex and describe a universal infinitesimal deformation. In a sequel, we formulate an extension of a given deformation of a Courant pair to another with extended base, which leads to describe the obstruction in extending a given deformation. We also discuss the construction of versal deformation of Courant pairs. As an application, we compute universal infinitesimal deformation of Poisson algebra structures on the three-dimensional complex Heisenberg Lie algebra. We compare the second deformation cohomology spaces of these Poisson algebra structures by considering them in the category of Leibniz pairs and Courant pairs, respectively.

中文翻译：

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Courant 对和泊松代数的变形

我们研究了具有交换代数基的 Courant 对的变形。我们考虑变形上同调双复并描述一个普遍的无穷小变形。在续集中，我们将 Courant 对的给定变形扩展到另一个具有扩展基础的变形，这导致描述了扩展给定变形的障碍。我们还讨论了 Courant 对的横向变形的构造。作为一个应用，我们计算泊松代数结构在三维复海森堡李代数上的普遍无穷小变形。我们通过将这些泊松代数结构的第二变形上同调空间分别考虑在莱布尼茨对和库朗对的类别中来比较它们。