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

点击翻译按钮获取中文摘要

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.