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

This is a broad proposal in the general field of algebraic infinite-dimensional Lie representation theory and the related ind-geometry. We propose to do research in the following five directions.A. Categories of tensor modules over Mackey Lie algebras as universal monoidal categories. Our plan is to investigate the universal additive linear symmetric monoidal category generated by two objects $X$ and $Y$ with a paring $X\otimes Y \rightarrow \text{\textbf{1}}$, and with fixed filtrations of length two: $X_0\subsetX$, $Y_0\subsetY$. We realize this category as a module category over the Mackey Lie algebra $\mathfrak{gl}(V,V_*)$ (the definition of the Lie algebra $\mathfrak{gl}(V,V_*)$ see in Section 1.A). Categories of tensor modules over infinite-dimensional Lie algebras and Lie superalgebras (Ph.D project for Aleksandr Shevchenko). We plan to extend an equivalence of categories proposed by V. Serganova to other categories of modules over the finitary Lie algebras $\mathfrak{sl}(\infty)$, $\mathfrak{o}(\infty)$, $\mathfrak{sp}(\infty)$, as well over the Mackey Lie algebra $\mathfrak{gl}(V,V_*)$. This equivalence relates module categories over Lie algebras of infinite rank with module categories over Lie superalgebras of infinite rank. Categories of bounded weight modules for a classical Lie superalgebras at infinity.} We propose to classify and explicitly describe the simple bounded weight modules over the Lie superalgebra $\mathfrak{osp}(\infty|\infty)$. Further, we intend to study the category of bounded weight modules, as well as respective categories of modules over other classical Lie superalgebras of infinite rank. Automorphism groups of ind-varieties $G/P$ for $G=GL(\infty)$, $O(\infty)$, $Sp(\infty)$. We propose a systematic study of the automorphism groups of ind-varieties of generalized flags, and outline a detailed approach in the case of ind-grassmannians. Category $\mathcal{O}$ with large local annihilator, $\mathcal{OLA}$ (postdoc project for Dr. Pablo Zadunaisky). We intend to investigte in detail a new analogue of category $\mathcal{O}$ which contains two recently studied categories of $\mathfrak{sl}(\infty)$-modules: $\mathcal{OLA}$ and $\mathbb{T}_{\mathfrak{g}, \mathfrak{k}}$. The new category should be a highest weight category with standard modules of infinite length, and its subcategory of integrable $\mathfrak{sl}(\infty)$-modules should coincide with $\mathbb{T}_{\mathfrak{g}, \mathfrak{k}}$.

• DFG Programme: Research Grants