Interplay of Algebraic Groups, Finite Groups, Lie Algebras, Quantum Groups, and their Representations and Cohomology

特征除群阶的域上有限群的上同调簇反映了群的许多重要的上同调性质和拓扑性质。 但与限制李代数的上同调变种相比，这些上同调变种并不是自然实现的，限制李代数的上同调变种被实现为李代数直到有限映射的幂零锥。 利用李代数的上同调簇构造了有限李型群的上同调簇的一种实现。 利用代数群在限制李代数的上同调簇上的作用，可以计算有限群的模的支撑簇。 一个重要的任务是建立有限群和限制李代数之间的关系。其他研究领域包括从上同调理论的角度研究代数群、李代数、有限群和量子群的表示理论之间的关系。群、李代数和量子群都是数学对象，它们作为对称、场和算子从几何学和物理学中产生。它们与数学的其他领域和数学以外的其他学科有着广泛的联系。 研究者的特别兴趣是融合这些对象的表示和上同调理论，并研究它们之间的相互关系。

The cohomological varieties for finite groups over fields ofcharacteristic dividing the order of the groups reflect many importantcohomological and topological properties of the groups. But thesecohomological varieties are not naturally realized in contrast to thecohomological varieties of restricted Lie algebras, which are realized as the nilpotent cones of the Lie algebras up to finite maps. The proposed research will construct a realization of the cohomological varieties for finite groups of Lie type in terms of the cohomological varieties of the corresponding Lie algebras. Using the actions of algebraic groups on the cohomological varieties of restricted Lie algebras, the support varieties of modules for finite groups can be computed. An important task is to establish the relation between finite groups and restricted Lie algebras. Other areas of research includes studying relations among representation theories of algebraic groups, Lie algebras, finite groups, and quantum groups from the aspects of cohomology theory. Groups, Lie algebras, and quantum groups are mathematical objects thatarise from geometry and physics as symmetries, fields, and operators. They have broad connections to other areas of mathematics and other disciplines outside mathematics. The investigator's special interest is to meld the representation and cohomological theories of these objects and to study their inter-relations.