Algebra, Lie Theory and Geometry

Group leader
Adjunct Professor of Mathematics
Specific themes and goals
  • Structure and representations of infinite-dimensional locally finite Lie algebras: The structure and representation theory of semi-simple Lie algebras is a jewel of mathematics and has been developed by mathematical giants of the 20th century such as Hermann Weyl, Andre Weil, Bourbaki, Israel Gelfand, Harish-Chandra and others. A theory of infinite-dimensional locally semi-simple Lie algebras emerged in the late 1980s. By the mid-1990s, Prof. Penkov had developed a broad programme to understand these Lie algebras and their representations at the level of depth already acquired for finite dimensional semisimple Lie algebras. Since then, this programme has achieved several milestones, and the theory of locally finite Lie algebras has become a mainstream research area in algebraic representation theory. 
  • Tensor categories and representations of non-locally-finite infinitedimensional Lie algebras: This research direction arose in the last several years, driven by the possibility of describing universal non-rigid tensor categories as representation categories for “large” Lie algebras. 
  • Geometry of homogeneous infinite-dimensional spaces, more specifically geometry of homogeneous ind-varieties: Some of the deepest results in mathematics arise from the interaction of algebra and geometry. In the early 2000s, Penkov and colleagues started to build a geometric counterpart to the algebraic theory of locally finite Lie algebras. This brought the theory of ind-varieties of generalized flags to life.
Highlights and impact
  • In 2022, we published the joint monograph “Classical Lie Algebras at Infinity”, describing key results from our research during the last 20 years.
  • We completed a series of joint works with Aleksey Petukhov on primitive ideals of infinite-dimensional Lie algebras. One result is an algorithm which computes the annihilator in U(sl(∞)) of any simple highest weight sl(∞)-module Lb(λ). This algorithm is based on an infinite version of the Robinson-Schensted algorithm. Other papers were devoted to various aspects of the theory of highest-weight sl(∞)-modules. We discovered a new category of integrable sl(∞)-modules.
  • In one of our greatest research efforts, we provided a classification of all simple bounded weight modules over basic classical Lie superalgebras, and studied the categories of such modules.
  • We are also particularly proud of a publication in which we resolve a challenge which emerged a few years ago in our earlier work. Namely, when one builds general tensor categories over Mackey Lie algebras, the trivial representation is no longer injective as an object of those categories. In this research, we constructed an infinite-length injective hull of the trivial representation, which allowed us to understand general infinite-length injective hulls of tensor modules over Mackey Lie algebras.
  • We also developed a general theory of linear flag ind-varieties and showed that every such variety is GL(∞)-homogeneous, and is hence isomorphic to an ind-variety of generalized flags. This gives a purely algebraic-geometric construction of ind-varieties of generalized flags.
Selected publications
  • D. Grantcharov, I. Penkov, Simple bounded weight modules of sl(∞), o(∞), sp(∞), Transformation Groups 25(4) (2020), 1125-1160. 
  • A. Tikhomirov, I. Penkov, An algebraic-geometric construction of ind-varieties of generalized flags, Annali di Matematica Pura ed Applicata, doi:10.1007/s10231-022-01200-2. 
  • M. Ignatyev, I. Penkov, Automorphism groups of ind-varieties of generalized flags, Transformation groups, doi:10.1007/s00031-022-09703-1. 
  • C. Hoyt, V. Serganova, I. Penkov, Integrable sl(∞)-modules and category O for gl(m|n), Journal LMS 99 (2019), 403-427. 
  • V. Serganova, I. Penkov, Large annihilator category O for sl(∞), o(∞), sp(∞), Journal of Algebra 532 (2019), 249-279.