Finitary Lie algebras: representations, primitive ideals, and related geometry

这是代数无限维李表示理论和相关的ind-几何的一般领域中的一个广泛的建议。它建立在单有限复李代数sl（∞），so（∞），sp（∞）的表示理论的最新进展的基础上。其中一个进展是最近对包络代数U（g）中的本原理想的研究，这导致了U（sl（∞））的本原理想的分类。本文的目的是通过给出U（so（∞））和U（sp（∞））的本原理想的明确分类，以及计算任意单最高权g-模的U（g）中的零化子的算法，来完善U（g）的本原理想理论。这个项目的动机是双重的：一方面，为了进一步发展有限李代数g=sl（∞），so（∞），sp（∞）的表示理论，需要对U（g）中的本原理想有很好的了解，另一方面，最终的答案肯定与有限维情况大不相同，突出了这些无限维无限秩李代数的表示理论的新特征，我们还提出了与李代数g=sl（∞），so（∞），sp（∞）和相应的ind-群G=SL（∞），SO（∞），SP（∞）.其中一个工作方向旨在理解至少三个不同的g-模范畴，其中两个是O范畴的类似物，第三个是可积模范畴。另一个项目涉及有界（非最高权重）权重模块的分类。最后，我们提出了对任意分裂抛物ind-子群PG的ind-簇G/P上的Q-轨道的几何研究，以及对G/P上的Matsuki对偶的理解（最近的工作考虑了P=B的情况）。学生（Aleksandr Fadeev），而所有其他研究方向应是主要研究者及其合作者的共同工作。

This is a broad proposal in the general field of algebraic infinite-dimensional Lie representation theory and the related ind-geometry. It builds up on essential recent advances in the representation theory of the simple finitary complex Lie algebras sl(∞), so(∞), sp(∞). One such advance has been the recent study of primitive ideals in the enveloping algebra U(g), which has led to a classification of primitive ideals of U(sl(∞)). It is an objective of this proposal to complete the theory of primitive ideals of U(g) by providing explicit classifications of the primitive ideals also for U(so(∞)) and U(sp(∞)), as well as algorithms for computing the annihilator in U(g) of any simple highest weight g-module. The motivation for this project is twofold: on the one hand, one needs good knowledge of the primitive ideals in U(g) in order to further develop the representation theory of the finitary Lie algebras g=sl(∞), so(∞), sp(∞), and, on the other hand, the final answer promises to be very different from the finite-dimensional case, highlighting new features in the representation theory of these infinite-dimensional Lie algebras of infinite rank.We propose to work also on several other directions related to the representation theory of the Lie algebras g=sl(∞), so(∞), sp(∞) and the corresponding ind-groups G=SL(∞), SO(∞), SP(∞). One of these work directions aims at understanding at least three different categories of g-modules, two of which are analogs of category O, and the third is a category of integrable modules. Another project concerns the classification of bounded (non-highest-weight) weight modules. This is a step in a broader program of investigating categories of g-modules whose simple objects are not highest weight modules.Finally, we propose a geometric study of the Q-orbits on ind-varieties G/P for arbitrary splitting parabolic ind-subgroups P G, as well as understanding Matsuki duality on G/P (the case P=B was considered in recent work).The study of primitive ideals should be carried out mostly by a Ph.D. student (Aleksandr Fadeev) while all other directions of study should be joint work of the principal investigator and his collaborators.