Cohomological Methods in the Representation Theory of Algebraic Groups, Quantum Groups and Superalgebras

首席研究员(PI)将研究涉及上同调方法的问题，并将其应用于代数群、量子群和李超代数的表示理论。更具体地说，PI旨在扩展我们对代数群、量子群和李代数的重要上同调计算的知识。通过一系列涉及李代数和Frobenius核上同调的步骤，研究FLAG簇的线丛上同调将在这些计算中发挥重要作用。PI建议利用几何方法，通过使用支持变种来研究表示理论。对于李超代数，这个方法提供了一个漂亮的同调解释，充分研究了缺陷和非典型性的组合概念。表示论在大约100年前随着Frobenius和Schur的开创性工作而出现，并因其与组合学、代数几何、数论和物理应用的联系而成为数学的中心领域。上同调理论是由拓扑学家在整个20世纪发展起来的，用来构造用于流形和拓扑空间研究的代数不变量。上同调也被定义为代数结构，如群和李代数，以确定它们的表示可以粘合在一起的方式。更令人惊讶的是，代数结构的上同调如何被用来将在表示理论层面上看不到的潜在几何引入画面。国际数学家协会一直在积极推广表示理论中上同调方法的工作知识，最近在该领域组织了几次会议，重点是发展初级数学家。PI还联合领导了佐治亚大学的一个代数学研究小组。该小组通过使用计算机代数工具包和出版研究成果，为博士后研究员和学生提供表示理论方面的实践培训。

The principal investigator (PI) will investigate problems involving cohomological methods with applications to the representation theory of algebraic groups, quantum groups, and Lie superalgebras. More specifically, the PI aims to expand our knowledge of important cohomological calculations for algebraic groups, quantum groups and Lie algebras. Studying line bundle cohomology for the flag variety will play a prominent role in these calculations through a series of steps involving Lie algebra and Frobenius kernel cohomology. The PI proposes to utilize geometric methods to study the representation theory through the use of support varieties. For Lie superalgebras, this approach provides a beautiful homological interpretation of the well-studied combinatorial notions of defect and atypicality. Representation theory emerged about 100 years ago with the pioneering work of Frobenius and Schur, and has become a central area of mathematics because of its connections to combinatorics, algebraic geometry, number theory, and applications to physics. Cohomology theories were developed throughout the 20th century by topologists to construct algebraic invariants for the investigation of manifolds and topological spaces. Cohomology was also defined for algebraic structures like groups and Lie algebras to determine ways in which their representations can be glued together. Even more striking is how the cohomology of algebraic structures can be used to introduce the underlying geometry, which is not seen at the representation theoretic level, into the picture. The PI has been actively promoting the working knowledge of cohomological methods in representation theory and has recently organized several conferences in the area with an emphasis toward the development of junior mathematicians. The PI also co-directs a research group in algebra at the University of Georgia. This group provides practical training in representation theory for postdoctoral fellows and students through the use of computer algebra packages and the publication of research.