"Derivations, cohomology groups and second duals of Banach algebras"

“Banach 代数的导数、上同调群和第二对偶”

• 批准号: 36640-2012
• 负责人: Ghahramani, Fereidoun
• 金额: $ 1.53万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=593025
• 关键词: Derivations; cohomology; groups; second; duals

We learn in a high school (or University) first calculus course that if f and g are two functions whose derivatives D(f) and D(g) exist, then the product function fg also has a derivative D(fg), which satisfies D(fg) = D(f)g + fD(g). In higher Mathematics there are structures comprised of elements-often called vectors-which can be added or multiplied, but, unlike the multiplication of functions, the multiplication of vectors of these structures, in general, may fail to satisfy fg=gf. Let's think of differentiation as being a mapping that assigns to every differentiable function another function (its derivative D(f)) subject to differentiation rules. There are similar mappings D on abstract structures that assign to each vector f in that structure a vector D(f) in the same or another related structure, subject to preservation of addition and multiplication by numbers, and act on the product of two vectors in the same fashion that differentiation does, i.e. D(fg) = D(f)g + fD(g). These mappings are called derivations. One of the simplest derivations is given by the formula D(f) = fg - gf. Here g is a fixed vector and f varies among the vectors of the given structure that D acts upon. These derivations are called inner derivations. We propose to find formulas for derivations on certain structures and to study those abstract structures that have the property that all the derivations into certain related structures are inner or can be approximated by inner derivations. These structures occupy a significant place in research in an area of Mathematics called Banach Algebras. The study of derivations originates from Physics and my research (like every other pure mathematician's research) is theoretical and primarily geared at advancement of knowledge; it may later find its use in applied sciences or industry.

我们在高中（或大学）初学微积分课程中学到，如果f和g是两个函数，其导数D(f)和D(g)存在，那么乘积函数fg也有导数D(fg)，满足D(fg) = D(f)g + fD(g)。在高等数学中，存在由元素组成的结构（通常称为向量），它们可以进行加法或乘法，但与函数的乘法不同，这些结构的向量乘法通常可能无法满足 fg=gf。让我们将微分视为一种映射，为每个可微函数分配另一个遵循微分规则的函数（其导数 D(f)）。在抽象结构上也有类似的映射 D，为该结构中的每个向量 f 分配相同或另一个相关结构中的向量 D(f)，但要保留数字的加法和乘法，并以与微分相同的方式作用于两个向量的乘积，即 D(fg) = D(f)g + fD(g)。这些映射称为派生。最简单的推导之一由公式 D(f) = fg - gf 给出。这里 g 是一个固定向量，f 在 D 所作用的给定结构的向量之间变化。这些推导称为内推导。我们建议寻找某些结构的推导公式，并研究那些具有以下性质的抽象结构：所有对某些相关结构的推导都是内部的或可以通过内部推导来近似。这些结构在称为巴拿赫代数的数学领域的研究中占有重要地位。推导的研究源于物理学，而我的研究（就像所有其他纯数学家的研究一样）是理论性的，主要是为了知识的进步；它稍后可能会在应用科学或工业中得到应用。