论文标题
在步骤2 Carnot代数上的水平仿射功能
Horizontally affine functions on step-2 Carnot algebras
论文作者
论文摘要
在本文中，我们简要介绍了水平仿射，H型的概念，并在STEP-2 CARNOT代数上提供了此类功能的完整描述。我们表明，Haffine函数的矢量空间在免费的step-2等级上 - $ n $ carnot代数对$ \ mathbb {r}^n $的外部代数同构。使用每个Carnot代数可以将其写成免费的Carnot代数的商，我们将从自由案例中推论出对任意步骤2 Carnot代数的H型功能的描述，以及这些步骤-2 Carnot代数的几个表征，而H-affine功能在矢量空间的本质意义上是亲密的。我们对H型功能的兴趣源于它们与最近在文献中引入的一类称为“精确单调”的集合的关系，以及它们与最小的Hypersurfaces的关系。
In this paper we introduce the notion of horizontally affine, h-affine in short, function and give a complete description of such functions on step-2 Carnot algebras. We show that the vector space of h-affine functions on the free step-2 rank-$n$ Carnot algebra is isomorphic to the exterior algebra of $\mathbb{R}^n$. Using that every Carnot algebra can be written as a quotient of a free Carnot algebra, we shall deduce from the free case a description of h-affine functions on arbitrary step-2 Carnot algebras, together with several characterizations of those step-2 Carnot algebras where h-affine functions are affine in the usual sense of vector spaces. Our interest for h-affine functions stems from their relationship with a class of sets called precisely monotone, recently introduced in the literature, as well as from their relationship with minimal hypersurfaces.
