论文标题
雅各布森·莫罗佐夫（Jacobson-Morozov）的引理代数超组
Jacobson-Morozov Lemma for Algebraic Supergroups
论文作者
论文摘要
给定一个准还原代数超级组$ g $，我们使用对称单体类别的半密码理论来定义一个对称的单体函数$φ_x：\ rep（g）\ to \ rep（osp（osp（1 | 2））与任何给定元素$ x \ in \ in \ in \ bar 1 rm y lie lie y lie y lie ply y lie peffect（1 | 2）$。对于nilpotent Elements $ x $，我们表明可以使用与$ x $相关的deligne过滤来定义函子$φ_x$。 我们使用这种方法证明了代数超组的Jacobson-Morozov引理的类似物。即，我们在\ mathrm {lie}（g）_ {\ bar 1} $上给出了奇数nilpotent元素$ x \的必要条件，它定义了supergroups $ osp（1 | 2）\ g $ to g $的嵌入，以便$ x $在相应的lie algebra homomormorphism中均为$ x $。
Given a quasi-reductive algebraic supergroup $G$, we use the theory of semisimplifications of symmetric monoidal categories to define a symmetric monoidal functor $Φ_x: \Rep(G) \to \Rep(OSp(1|2))$ associated to any given element $x \in \mathrm{Lie}(G)_{\bar 1}$. For nilpotent elements $x$, we show that the functor $Φ_x$ can be defined using the Deligne filtration associated to $x$. We use this approach to prove an analogue of the Jacobson-Morozov Lemma for algebraic supergroups. Namely, we give a necessary and sufficient condition on odd nilpotent elements $x\in \mathrm{Lie}(G)_{\bar 1}$ which define an embedding of supergroups $OSp(1|2)\to G$ so that $x$ lies in the image of the corresponding Lie algebra homomorphism.
