Categories of Lie algebra representations, primitive ideals, and geometry of homogeneous ind-spaces

• Applicant: Professor Dr. Ivan Penkov
• Subject Area: Mathematics
• Term: from 2015 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 280374544

This is a broad proposal in the field of Lie representation theory and the related geometry. It builds upon recent advances in the author's program in this field going back for up to 20 years. A unifying feature of the proposal is the study of various categories of representations introduced in part by the author and his collaborators. Our first topic of study is the category of (g, sl(2))-modules for a semisimple complex Lie algebra g and any sl(2)-subalgebra. We plan to work on a major conjecture which we made a few years ago. We hope also to make essential progress in the study of certain analogues of the category O for g = sl(infty), o(infty), sp(infty). This latter topic is center to the entire application as it is the Ph. D. project which we propose to be funded.In addition we are launching a program to study tensor categories of tensor representations of more general diagonal Lie algebras, as well as certain extensions of the categories of tensor representations of o(infty) and sp(infty). This is motivated by our recent success in using the category of tensor representations of sl(infty) to categorify the boson-fermion correspondence. Another direction of study is the theory of primitive ideals of U(o(infty)) and U(sp(infty)), which we propose to advance by using methods developed recently for U(sl(infty)) and via a striking new isomorphism of the lattices of ideals in U(o(infty)) and U(sp(infty)). Finally, we propose to study the automorphism groups of the homogeneous ind-spaces G/P for G=SL(infty), O(infty), SP(infty) where P is a splitting parabolic subgroup of G, and to launch a study of the homogeneous ind-spaces G/P for general (nonsplitting) parabolic subgroups P subset of G.

• DFG Programme: Research Grants
• International Connection: France, United Kingdom, USA
• Cooperation Partners: Professor Dr. Lucas Fresse; Dr. Johanna Hennig; Dr. Alexey Vladimirovich Petukhov; Professorin Vera Serganova, Ph.D.; Professor Gregg J. Zuckerman, Ph.D.
• Co-Investigator: Thanasin V. Nampaisarn