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

这是代数无穷维李表示理论和相关几何领域的一个广泛的建议。我们建议从以下五个方面进行研究。麦基李代数上作为泛一元范畴的张量模的范畴。我们的计划是研究由两个对象$X$和$Y$生成的普遍加性线性对称单面范畴，它们具有配对$X\otimes Y \rightarrow \text{\textbf{1}}$，并具有长度为2的固定过滤：$X_0\subsetX$, $Y_0\subsetY$。我们将这个范畴视为麦基李代数$\mathfrak{gl}(V,V_*)$（李代数的定义$\mathfrak{gl}(V,V_*)$见第1.A节）上的一个模范畴。无限维李代数和李超代数上张量模的分类（Aleksandr Shevchenko博士项目）。我们计划将V. Serganova提出的范畴等价推广到有限李代数$\mathfrak{sl}(\infty)$, $\mathfrak{o}(\infty)$, $\mathfrak{sp}(\infty)$以及麦基李代数$\mathfrak{gl}(V,V_*)$上的其他模的范畴。这个等价将无限秩李代数上的模范畴与无限秩李超代数上的模范畴联系起来。经典李超代数无穷远处有界权模的范畴。我们建议对李超代数$\mathfrak{osp}(\infty|\infty)$上的简单有界权模进行分类和显式描述。进一步，我们打算研究有界权模的范畴，以及其他经典无限阶李超代数上的模的相应范畴。$G=GL(\infty)$, $O(\infty)$, $Sp(\infty)$的独立变种$G/P$的自同构群。我们系统地研究了广义标志的ind-变种的自同构群，并在ind-grassmannians的情况下给出了一个详细的方法。类别$\mathcal{O}$与大型本地湮灭器，$\mathcal{OLA}$（博士后项目博士Pablo Zadunaisky）。我们打算详细研究一个新的类似类别$\mathcal{O}$，其中包含最近研究的两个类别$\mathfrak{sl}(\infty)$ -模块：$\mathcal{OLA}$和$\mathbb{T}_{\mathfrak{g}, \mathfrak{k}}$。新范畴应为具有无限长标准模的最高权范畴，其可积的子范畴$\mathfrak{sl}(\infty)$ -模应与$\mathbb{T}_{\mathfrak{g}, \mathfrak{k}}$重合。

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 $\mathbb{T}_{\mathfrak{g}, \mathfrak{k}}$.