论文标题
功能代数的非线性同态是由厚的形态诱导的
Non-linear homomorphisms of algebras of functions are induced by thick morphisms
论文作者
论文摘要
2014年，沃诺夫（Voronov）引入了（超级）歧管的厚实形态的概念，作为构建$ l _ {\ infty} $的工具 - 同型泊松代数的形态。厚实的形态概括了普通的平滑地图，但不是地图本身。但是，它们在$ c^{\ infty} $ function上诱导背带。这些撤回后背是在所谓的“非线性同态”的函数代数之间的一般非线性图。从定义上讲，这意味着它们的差异是代数同构的。制定了以下猜想：平滑功能代数代数的任意非线性同态是由某些厚实的形态产生的。我们在这里证明了正式功能类别中的这种猜想。通过这种方式，我们扩展了$ c^{\ infty} $函数的歧管和代数同态的平滑地图的众所周知的结果，更一般而言，在非线性环境中提供了经典的“函数 - 代数双重性”的类似物。
In 2014, Voronov introduced the notion of thick morphisms of (super)manifolds as a tool for constructing $L_{\infty}$-morphisms of homotopy Poisson algebras. Thick morphisms generalise ordinary smooth maps, but are not maps themselves. Nevertheless, they induce pull-backs on $C^{\infty}$ functions. These pull-backs are in general non-linear maps between the algebras of functions which are so-called 'non-linear homomorphisms'. By definition, this means that their differentials are algebra homomorphisms in the usual sense. The following conjecture was formulated: an arbitrary non-linear homomorphism of algebras of smooth functions is generated by some thick morphism. We prove here this conjecture in the class of formal functionals. In this way, we extend the well-known result for smooth maps of manifolds and algebra homomorphisms of $C^{\infty}$ functions and, more generally, provide an analog of classical 'functional-algebraic duality' in the non-linear setting.
