Universal tensor categories, representations of infinite-dimensional Lie algebras, and infinite-dimensional geometry

这是无限维李表示理论及其相关的无限维几何一般领域中的一个广泛的建议。它建立在作者在这一领域的计划最新进展的基础上，可以追溯到长达20年的时间。具体的研究课题相当平均地分布在代数表示理论和几何表示理论之间。在代数方面，我们计划研究无限维Mackey-Lie代数的结构和表示理论。我们还建议研究一个有趣而又相当复杂的SL(∞)-模范畴。其次，我们提出的关于广义旗簇的自同构群的同构类的研究源于几何，但实际上主要是代数研究。此外，我们还打算追求两个几何研究方向。其一是研究对角in群的齐次ind空间，并利用广义标志的ind变种加以应用。二是粗旗品种的引种和后续研究，以适应∞的需要。后一种理论是由Kashiwara的Kac-Moody李代数的厚旗簇理论所推动的。

This is a broad proposal in the general field of infinite-dimensional Lie representation theory and the related infinite-dimensional geometry. It builds upon recent advances in the author’s program in this field going back for up to twenty years. The concrete topics of study are spread quite equally between algebraic and geometric representation theory. On the algebraic side, we plan to investigate the structure and representation theory of infinite-dimensional Mackey Lie algebras. We also propose to study an interesting and quite intricate category of sl(∞)-modules. Next, our proposed investigation of the isomorphism classes of automorphism groups of ind-varieties of generalized flags has its origin in geometry but is in fact a mostly algebraic study. In addition, we intend to pursue two geometric directions of research. One is the study of homogeneous ind-spaces of diagonal ind-groups, with applications of ind-varieties of generalized flags. The other is the introduction and subsequent study of thick flag varieties for the ind-group GL(∞). This latter theory is motivated by Kashiwara’s theory of thick flag varieties for a Kac- Moody Lie algebra.