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

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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.