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

• Final Report Year: 2018

Final Report Abstract

This DFG project consisted of five different directions of research A)-D). In three of these directions my collaborators and I have been able to carry out the proposed work in full. In two proposed work directions our approach deviated somewhat from the proposal but also yielded ample fruit. In addition, we have established results which fit perfectly in the area of the project but could not have been anticipated in 2015. More precisely, we established an equivalence of categories which was conjectural at the time of proposal. In the topic C) Categories of tensor modules for diagonal Lie algebras our studies deviated a bit from the proposal. We studied categories of representations of Mackey Lie algebras glM instead of a diagonal Lie algebras. Both these types of Lie algebras are natural generalizatons of the Lie algebra sl(∞). We obtained extensive results about certain categories of tensor representations, and in particular showed that these categories are universal tensor categories. In the topic E) Geometry of homogeneous ind-varieties G/P for G = SL(∞), O(∞), Sp(∞) L. Fresse and I were able to establish and analogue of Matsuki duality for any homogeneous ind-variety G/B where G = SL(∞), O(∞), Sp(∞). This result surpasses our expectations form 2015. In addition, we establish results which fit the area of the project but not were not mentioned in the proposal.

Publications

• On categories O for root-reductive Lie algebras
  T. Nampaisarn
  
• Orbit duality in ind-varieties of maximal generalized flags, Transactions of the Moscow Mathematical Society, 2017, 131-160
  I. Penkov, L. Fresse
  (See online at https://doi.org/10.1090/mosc/266)
• Representation categories of Mackey Lie algebras as universal monoidal categories, Pure and Applied Mathematics Quarterly 13 (2017), 77-121
  A. Chirvasitu, I. Penkov
  (See online at https://doi.org/10.4310/PAMQ.2017.v13.n1.a3)
• On the categories of admissible g, sl(2) - modules, Transformation Groups 23(2) (2018), 463-489
  I. Penkov, V. Serganova, G. Zuckerman
  (See online at https://doi.org/10.1007/s00031-017-9458-1)
• Ordered tensor categories and representations of the Mackey Lie algebra of infinite matrices, Algebra and Representation Theory (2018)
  A. Chirvasitu, I. Penkov
  (See online at https://doi.org/10.1007/s10468-018-9765-9)
• Primitive ideals of U (sl(∞)) and the Robinson-Schensted algorithm at infinity. To appear in: Representation of Lie Algebraic Systems and Nilpotent Orbits: in Honour of the 75th Birthday of Tony Joseph, Birkhauser, 2019
  A. Petukhov, I. Penkov