Representations and Cohomology of Algebras

摘要：代数的表示与上同调Witherspoon的工作涉及到各种类型的代数，如Hopf代数、量子群、群代数和交叉积。Witherspoon研究了这些代数在向量空间上作为线性变换的表示，以及它们的上同调，它测量了代数及其表示的性质。Witherspoon将计算由群作用在空间上产生的某些交叉积代数的Hochschild上同调。这与一些新的轨道上同调理论有联系，威瑟斯彭的研究将为许多数学家理解这些上同调理论的努力提供信息。同时，根据威瑟斯彭目前的工作，量子群理论家也会对这些交叉积代数的变形感兴趣，这些变形来自于这些代数上某些量子群的表示。威瑟斯彭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和有限量子群产生的三流形不变量。另一个项目涉及关于代数的表示如何与子代数的表示相关联的基本问题。Witherspoon将继续她的工作，将Clifford理论从群推广到某些类型的代数，目标是找到这些问题的建设性答案，并将应用这样的理论来回答关于代数表示的问题。代数是用方程或函数来表达物理对象。曲线或曲面是方程的图形，其物理性质可以直接由方程确定。更一般的代数系统，如许多函数的集合，称为代数，编码有关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，这种对称性就用相应的代数来表示。许多这样的例子都很容易理解，其中所涉及的数学被用于物理、化学和其他科学。然而，还有许多系统还没有被很好地理解，比如不到20年前在数学物理中出现的量子群。威瑟斯彭的工作涉及研究这些代数的性质及其作为物理对象的表示。威瑟斯彭作品中经常使用的一种技巧是上同源。威瑟斯彭将计算各种代数的上同调，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产生一系列关于代数的广泛主题的出版物，这些出版物也将影响代数以外的领域，如几何和数学物理。大多数活动都是针对许多数学家目前感兴趣的问题，而其他活动则是在发表后会找到感兴趣的读者的新想法。

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.