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

• Applicant: Professor Dr. Ivan Penkov
• Subject Area: Mathematics
• Term: from 2018 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 404426225

• Final Report Year: 2020

Final Report Abstract

This DFG project consisted of four different directions of research, A)-D). Specifically, in topic A) A. Fadeev has completed his dissertation and has provided a complete description of the primitive ideals of U (o∞ ) and U (sp∞ ). In topic B) two joint papers have been published and a third one is in progress. The proposed work in direction C) has been fully done and published. In topic D) one of the proposed lines of inquiry has led to final results in the work, and another line of inquiry is still in progress. Our work on the topics A)-C) went more or less along expectations, but in topic D) we were surprised by the fact that we could obtain the final classification of all k-tuples of splitting parabolic subgroups P1 , P2 , . . . , Pk of G = SL(∞), O(∞), Sp(∞) such that G/P1 × . . . × G/Pk has finitely many G-orbits. The same problem in the finite-dimensional case has been solved in a series of a highly non-trivial works. For the duration of the grant the principal investigator has published several joint papers on topics not strictly following to the grant proposal but related to it. Moreover, in a further work we establish results for which we had no hope at the time of submisson of the grant proposal.

Publications

• Large annihilator category O for sl∞ , o∞ , sp∞ , Journal of Algebra 532 (2019), 249-279
  I. Penkov, V. Serganova
  (See online at https://doi.org/10.1016/j.jalgebra.2019.05.020)
• Multiple flag ind-varieties with finitely many orbits, Transformation Groups 27, 833–865 (2022)
  L. Fresse, I. Penkov
  (See online at https://doi.org/10.1007/s00031-021-09653-0)
• On an infinite limit of BGG categories O, Moscow Mathematical Journal 19 (2019), 655–693
  K. Coulembier, I. Penkov
  (See online at https://doi.org/10.17323/1609-4514-2019-19-4-655-693)
• Simple bounded weight modules of sl∞ , o∞ , sp∞ , Transformation Groups, 25 (2020), 1125–1160
  D. Grantcharov, I. Penkov
  (See online at https://doi.org/10.1007/s00031-020-09571-7)