AI 中文总结
本文研究拟约化李超代数的正则元与平衡元，证明奇元素可分解为可交换的正则元与平衡元之和，将其应用于Duflo-Serganova函子的描述，并完成单拟约化李超代数 distinguished 奇元素的分类。
AI 中文摘要
设$G$为拟约化超群（其基础代数群$G_{\bar 0}$为约化群）。本文研究两类平凡相交的奇元素：正则元（neat elements）与平衡元（balanced elements）。正则元恒为伴随幂零的，且可嵌入同构于$\boldsymbol{\frak{osp}(1|2)}$的子代数中，$\frak{osp}(1|2)$是基础李代数为$\frak{sl}_2$的单李超代数；平衡奇元素则是自交换元（满足$[x,x]=0$的元素$x \text{Lie}(G)_{\bar 1}$）概念的自然推广，平衡元被用于在$G$的表示范畴上定义同调型函子。本文证明，任意元素$x \text{Lie}(G)_{\bar 1}$均可表示为彼此交换的正则奇元素与平衡奇元素之和，该定理具有范畴论应用：设$\frak{g}^{(1|1)}$为由$x \text{Lie}(G)_{\bar 1}$生成的$(1|1)$维李超代数，$\frak{g}^{(1|1)}$的有限维超表示范畴的半单化是函子$S: Rep(\frak{g}^{(1|1)}) \to Rep(SOSp(1|2))$；任意$x \text{Lie}(G)_{\bar 1}$诱导同态$i_x:\frak{g}^{(1|1)}\to \text{Lie}(G)$，令$\boldsymbol{\textit{Φ}_x}=S\boldsymbol{\textit{∘}}(-)\boldsymbol{\textit{↓}}_{i_x}$为限制函子$(-)\boldsymbol{\textit{↓}}_{i_x}$与函子$S$的复合，本文证明函子$\textit{Φ}_x$可通过上述分解中$x$的平衡部分对应的同调型函子$\boldsymbol{\textit{Φ}_{x_{bal}}}$明确描述，这类同调型函子称为Duflo-Serganova函子。最后，本文对单拟约化李超代数中的 distinguished 奇元素给出完全分类，证明除$\boldsymbol{\frak{spe}(n)}$外，所有此类元素均为平衡元或正则元。
英文摘要
Let $G$ be a quasi-reductive supergroup (so its underlying algebraic group $G_{\bar 0}$ is reductive). We consider two trivially intersecting classes of odd elements: neat elements and balanced elements. Neat elements are always $ad$-nilpotent and may be embedded into subalgebras that are isomorphic to $\mathfrak{osp}(1|2)$, a simple Lie superalgebra whose underlying Lie algebra is $\mathfrak{sl}_2$. Balanced odd elements, on the other hand, are a natural generalization of the notion of a self-commuting element (an element $x\in Lie(G)_{\bar 1}$ for which $[x,x]=0$). Balanced elements are used to define homology-type functors on the category of representations of $G$. We show that any element $x\in Lie(G)_{\bar 1}$ may be written as a sum of a neat and a balanced odd element which commute with each other. This theorem has a categorical application. Let $\mathfrak{g}^{(1|1)}$ be the $(1|1)$-dimensional Lie superalgebra generated by $x \in Lie(G)_{\bar 1}$. The semisimplification of the category of finite-dimensional super-representations of $\mathfrak{g}^{(1|1)}$ is a functor $S: Rep(\mathfrak{g}^{(1|1)}) \to Rep(SOSp(1|2))$. Any $x\in Lie(G)_{\bar 1}$ induces a homomorphism $ i_x:\mathfrak{g}^{(1|1)}\to Lie(G)$. Let $$Φ_x=S\circ (-)\downarrow_{i_x}:Rep(G)\to Rep(SOSp(1|2))$$ be the composition of the restriction functor $(-)\downarrow_{i_x}$ and the functor $S $. We show that the functor $Φ_x$ may be described explicitly using the homology-type functor $Φ_{x_{bal}}$ corresponding to the balanced part of $x$ in the above decomposition. These homology-type functors are known as Duflo-Serganova functors. Finally, we provide a full classification of distinguished odd elements in simple quasi-reductive Lie superalgebras and show that in all cases except $\mathfrak{spe}(n)$, such elements are either balanced or neat.
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