Representations and Cohomology of Algebras

主要研究者：莎拉·威瑟斯彭求婚号码：0245560机构：阿默斯特学院摘要：代数的表示和上同调威瑟斯彭的工作涉及各种类型的代数，如霍普夫代数，量子群，群代数和交叉产品。 威瑟斯彭研究这些代数作为向量空间上的线性变换的表示，以及它们的上同调，它测量代数及其表示的性质。威瑟斯彭将计算Hochschild上同调的某些交叉产品代数所产生的群体行动的空间。有预期的联系，一些新的理论上同调的orbifolds，威瑟斯彭的研究将通知努力，许多数学家了解这些上同调理论。与此同时，它将感兴趣的量子群理论家作为威瑟斯彭预计，根据她目前的工作，这些交叉产品代数的变形来自表示某些量子群对这些代数。Witherspoon提出的相关工作将涉及Hopf代数和量子群的一些基本问题的进展，如上同调的有限生成，以及由有限量子群产生的三流形不变量。另一个项目涉及代数的表示如何与子代数的表示相关的基本问题。威瑟斯彭将继续她的工作在推广克利福德理论从团体到某些类型的代数，与目标找到建设性的答案，这些问题，并将应用这样一个理论来回答有关的问题表示代数。代数是表达的物理对象作为方程或功能。曲线或曲面是方程的图形，其物理性质可以直接从方程中确定。更一般的代数系统，如许多函数的集合，称为代数，编码关于更复杂的物理对象的信息。例如，如果一个物体（如晶体或分子）表现出对称性，那么这种对称性就可以用相应的代数来表示。许多这样的例子都很容易理解，所涉及的数学在物理学，化学和其他科学中得到了利用。然而，还有许多系统不太清楚，例如不到20年前在数学物理中出现的量子群。威瑟斯彭的工作包括研究这些代数的性质及其作为物理对象的表示。在威瑟斯彭的工作中经常使用的一种技术是上同调。威瑟斯彭将计算各种类型的代数的上同调，以及使用其他方法来研究它们及其表示。威瑟斯彭提出的活动将导致在广泛的出版物集合内的代数主题，也将影响领域以外的代数，如几何和数学物理。大多数活动解决的问题，其中许多数学家目前感兴趣的，而另一些是新的想法，将找到一个感兴趣的读者出版。

Principal Investigator: Sarah Witherspoon Proposal Number: 0245560Institution: Amherst CollegeAbstract: Representations and Cohomology of AlgebrasWitherspoon's work involves various types of algebras, such as Hopf algebras, quantum groups, group algebras, and crossed products. Witherspoon studies representations of these algebras as linear transformations on vector spaces, and their cohomology, which measures properties of the algebras and their representations. Witherspoon will compute the Hochschild cohomology of certain crossed product algebras arising from group actions on spaces. There are expected connections to some new theories of cohomology of orbifolds, and Witherspoon's research will inform the effort by many mathematicians to understand these cohomology theories. At the same time it will be of interest to quantum group theorists as Witherspoon expects, based on her current work, that deformations of these crossed product algebras come from representations of certain quantum groups on these algebras. Related work that Witherspoon proposes will involve progress on some basicquestions about Hopf algebras and quantum groups, such as finite generation of cohomology, and three-manifold invariants arising from finite quantum groups. Another project involves fundamental questions about how representations of an algebra are related to those of a subalgebra. Witherspoon will continue her work in generalizing Clifford theory from groups to certain types of algebras, with the goal of finding constructive answers to these questions, and will apply such a theory to answer questions about representations of algebras.Algebra is the expression of physical objects as equations or functions. A curve or surface is the graph of an equation, and its physical properties may be determined directly from the equation. More general algebraic systems such as collections of many functions, called algebras, encode information about more complicated physical objects. For example, if an object (such as a crystal or a molecule) exhibits symmetry, this symmetry is expressed in its corresponding algebra. Many such examples are well understood, and the mathematics involved is exploited in physics, chemistry, and other sciences. However there are many systems that are less well understood, such as the quantum groups that arose in mathematical physics less than two decades ago. Witherspoon's work involves the study of properties of such algebras and their representations as physical objects. One technique that is used frequently in Witherspoon's work is cohomology. Witherspoon will compute the cohomology of various types of algebras, as well as use other methods to study them and their representations. Witherspoon's proposed activities will result in a collection of publications on a wide range of topics within algebra that will also impact fields outside algebra such as geometry and mathematical physics. Most of the activities address questions, in which many mathematicians are currently interested, while others are new ideas that will find an interested audience upon publication.