关于具有固定平行度的修正扩张图
On modified extension graphs of a fixed atypicality
摘要：
本文研究了经典李超代数$\mathfrak{gl}(m|n), \mathfrak{osp}(M|2n)$和$\mathfrak{q}_m$上有限维单模之间的扩张.我们考虑一个简化版本的扩展图，这是从$Ext^1$ -图通过确定表示奇偶校验的变化和删除的循环。我们给出了一对顶点连通的一个必要条件，并证明了这个条件在大多数情况下是充分的。这个条件意味着有限维单模在Duflo-Serganova函子下的像有不可分解的同构分量。这产生了$\mathcal{F}in(\mathfrak{gl}(m|n))$和$\mathcal{F}in(\mathfrak{osp}(M|2n))$的Duflo-Serganova函子的半简单性。
英文摘要：
In this paper we study extensions between finite-dimensional simple modules over classical Lie superalgebras $\mathfrak{gl}(m|n), \mathfrak{osp}(M|2n)$ and $\mathfrak{q}_m$. We consider a simplified version of the extension graph which is produced from the $Ext^1$-graph by identifying representations obtained by parity change and removal of the loops. We give a necessary condition for a pair of vertices to be connected and show that this condition is sufficient in most of the cases. This condition implies that the image of a finite-dimensional simple module under the Duflo-Serganova functor has indecomposable isotypical components. This yields semisimplicity of Duflo-Serganova functor for $\mathcal{F}in(\mathfrak{gl}(m|n))$ and for $\mathcal{F}in(\mathfrak{osp}(M|2n))$.
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