Representation categories of infinite-dimensional Lie algebras and superalgebras, and automorphisms of homogeneous ind-spaces

• Applicant: Professor Dr. Ivan Penkov
• Subject Area: Mathematics
• Term: from 2020 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 448324667

• Final Report Year: 2024

Final Report Abstract

The first cluster of results concerns the study of several categories of representations of Mackey Lie algebras (certain infinite-dimensional analogues of the classical simple finitedimensional complex Lie algebras), and in particular establishing the universality of these categories as tensor categories. The seed for this research is the category Tsl(∞) of tensor modules over the Lie algebra sl(∞) of finitary infinite matrices. This category was introduced and thoroughly studied almost a decade ago by A. Sam, V. Serganova, A. Snowden, the author, and others. The Koszulity and the universality properties of Tsl(∞) as a tensor category are far reaching generalizations of Weyl’s semisimplicity theorem for finite-dimensional representations of the Lie algebra sl(n). It turned out however that the above results for Tsl(∞) are only the “tip of an iceberg” and that there exists a variety of more complex categories of representations of Mackey Lie algebras with more interesting universality properties. More precisely, in my joint works with A. Chirvasitu and V. Tsanov we construct a C-linear abelian symmetric tensor category I Tt generated by two objects X and Y with socle filtrations of length t + 2, and show that this category has a natural universality property. We realize this universal tensor category as the category of tensor representations of a Mackey Lie algebra of dimension 2ℵt , compute all Ext-groups between simples in I Tt , and establish the Koszulity of the category. In a further series of two joint papers with F. Esposito we introduce and study two tensor categories of topological tensor modules. The first one is the category of continuous tensor modules over the Lie algebra gl(V ), where V is a countable-dimensional discrete vector space and gl(V ) is endowed with ind-linearly compact topology. The second category is the category of continuous tensor representations of the Lie algebra gl(V ⊕ V ∗ ), the “semiinfinite” Lie algebra. We prove that both these tensor categories are anti-equivalent to the universal category Tsl(∞) . Finally, we also prove Lie superalgebra versions of the above results. In another direction of research, jointly with D. Grantcharov and V. Serganova, we have investigated the categories of bounded weight modules over the infinite-dimensional Lie superalgebras osp(∞|∞) and sl(∞|∞). These categories are “almost orthogonal” to the respective categories of tensor modules. In our work we classify the simple bounded weight modules of osp(∞|∞) and sl(∞|∞), and make some essential steps towards understanding the structure of the respective categories of bounded weight modules. As a first stage of the project, we classify the simple weight modules over the Weyl and Clifford superalgebras, D(∞|∞) and Cl(∞|∞), and over certain their subalgebras. Then we show that the classification of simple bounded weight osp(∞|∞)- and sl(∞|∞)-modules reduces to a classification of weight modules of D(∞|∞) and Cl(∞|∞) via certain natural homomorphisms from U (osp(∞|∞)) and U (sl(∞|∞)) to D(∞|∞) and Cl(∞|∞). Finally, a third main direction of our studies is concerned with the automorphism groups of ind-varieties of generalized flags. These ind-varieties are direct limits of usual flag varieties and are homogeneous ind-spaces for the finitary ind-group GL(∞). There are also ind-varieties of isotropic generalized flags which are homogeneous ind-spaces for the ind-groups O(∞) and Sp(∞). In our joint work with M. Ignatiev, we describe the automorphism group of an arbitrary ind-variety of generalized flags (also in the isotropic case) explicitly and in full generality. In particular, we discover that a right framework for such a description is a class of groups which we refer to as Mackey groups. In a related development, L. Fresse and I have completed a work in which we study homogeneous ind-spaces of diagonal ind-groups. As an application, we characterize all ind-varieties of generalized flags which are homogeneous spaces for a given pure diagonal ind-group.

Publications

• Universal Tensor Categories Generated by Dual Pairs. Applied Categorical Structures, 29(5), 915-950.
  Chirvasitu, Alexandru & Penkov, Ivan
  
• Automorphism Groups of ind-Varieties of Generalized Flags. Transformation Groups, 29(2), 743-772.
  Ignatev, Mikhail & Penkov, Ivan
  
• Bounded weight modules for basic classical Lie superalgebras at infinity. European Journal of Mathematics, 10(1).
  Grantcharov, Dimitar; Penkov, Ivan & Serganova, Vera
  
• On Homogeneous Spaces for Diagonal Ind-Groups. Transformation Groups, 30(4), 1723-1749.
  Fresse, Lucas & Penkov, Ivan
  
• Topological semiinfinite tensor (super)modules. Journal of Algebra, 642, 237-255.
  Esposito, Francesco & Penkov, Ivan
  
• Topological tensor representations of $\mathfrak{gl}(V)$ for a space $V$ of countable dimension. Algebraic Geometry and Physics, 2(1), 125-149.
  Esposito, Francesco & Penkov, Ivan