Representations and Cohomology of Algebras

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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.