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

这是代数无限维李表示理论及其相关几何一般领域中的一个广泛的建议。本文主要从以下五个方面进行研究：a.作为泛么半群范畴的Mackey Lie代数上的张量模范畴。我们的计划是研究由两个对象$X$和$Y$生成的泛可加线性对称么半群范畴，其中两个对象的长度为2：$X_0\subsetX$，$Y_0\subsetY$，其中$X\o乘Y\right tarrow\Text{\extbf{1}}$.我们将这个范畴实现为Mackey李代数$\mathfrak{gl}(V，V_*)$(李代数$\mathfrak{gl}(V，V_*)$的定义见第1.A节)上的一个模范畴。无限维李代数和李超代数上的张量模范畴(亚历山大·舍甫琴科博士项目)。我们计划将V.Serganova提出的范畴的一个等价推广到有限李代数$\mathfrak{sl}(\infty)$，$\mathfrak{o}(\infty)$，$\mathfrak{sp}(\infty)$上的其他模范畴，以及Mackey李代数$\mathfrak{gl}(V，V_*)$上。这种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。经典李超代数在无穷远处的有界权重模的范畴。}我们建议对李超代数$\mathfrak{osp}(\inty|\inty)$上的单有界重模进行分类和显式刻画。进一步，我们打算研究有界权模范畴，以及其他经典无限秩李超代数上的相应模范畴。当$G=GL(\INFTY)$，$O(\INFTY)$，$Sp(\INFTY)$时，自同构群的自同构群。我们对广义旗帜变种的自同构群进行了系统的研究，并以广义旗帜变种为例给出了一个详细的研究方法。具有大型局部湮灭装置的$\Mathcal{O}$类别，$\Mathcal{Ola}$(Pablo Zadunaisky博士博士后项目)。我们打算详细地研究一个新的范畴$\mathcal{O}$，它包含两个最近研究的范畴$\mathfrak{sl}(\inty)$-模：$\mathcal{Ola}$和$\mathbb{T}_{\mathfrak{g}，\mathfrak{k}}$。新范畴应该是具有无限长标准模的最高权范畴，其可积$\mathfrak{sl}(\infty)$-模的子范畴应与$\mathbb{T}_{\mathfrak{g}，\mathfrak{k}}$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 $\mathbb{T}_{\mathfrak{g}, \mathfrak{k}}$.