Cohomology, Support Varieties, and Representation THeory

本课题研究超群、李超代数、有限群及相关代数的表示理论和上同调。主要研究者对这些物体的表示理论、几何和代数组合之间的相互作用特别感兴趣。例如，研究者工作的一个重点将是继续发展复李超代数的支持变体。这些几何对象提供了这些代数的表示理论、上同调和组合学之间的联系。此外，他打算研究涉及对称群的模表示和相关代数对象的问题。这个项目是在数学领域称为表示理论。诸如群和李代数之类的代数结构在自然界中是作为某些物体的对称性而出现的。因此，研究这些对象如何与其他数学对象相互作用是很自然的。这是表征理论的观点。由于它提供了关于对象的底层结构和对称性的洞察力，因此表示理论在数学、物理、化学、生物学、密码学、网络、计算机图形学和艺术等其他领域被证明是有价值的，这并不奇怪。这个提议的重点是这些代数对象的“超级”版本；也就是说，它包含了对称和反对称。受其在数学物理中的实用性的启发，数学家们发现超级代数对象扮演着与经典对象相似的重要角色。研究者预计，随着“超级”表征理论的新应用被发现，这一趋势将继续下去。

This project addresses questions in the representation theory and cohomology of supergroups, Lie superalgebras, finite groups, and related algebras. The principal investigator is particularily interested in the interplay between the representation theory, geometry, and algebraic combinatorics of these objects. For example, one focus of the investigator's work will be the continued development of support varieties for complex Lie superalgebras. These geometric objects provide links between the representation theory, cohomology, and combinatorics of these algebras. Additionally, he intends to investigate questions involving the modular representations of the symmetric group and related algebraic objects.This project is in the area of mathematics known as representation theory. Algebraic structures such as groups and Lie algebras arise in nature as the symmetries of some object. Thus it is natural to study these objects in terms of how they interact with other mathematical objects. This is the point of view of representation theory. Because of the insight it provides regarding the underlying structure and symmetries of an object, it is not surprising that representation theory has proven valuable in other areas of mathematics, physics, chemistry, biology, cryptography, networking, computer graphics, and art. This proposal focuses on ``super'' versions of these algebraic objects; that is, ones which involve both symmetries and anti-symmetries. Inspired by their usefulness in mathematical physics, mathematicians have found that super algebraic objects play an important role similar to their classical counterparts. The investigator expects this trend to continue as new applications of ``super'' representation theory are discovered.