论文标题
$ \ Mathcal s（\ Mathfrak G）$的有限订单，定期收缩和泊松副总代理的自动形态学
Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of $\mathcal S(\mathfrak g)$
论文作者
论文摘要
令$ \ mathfrak g $为{\ sf aut}（\ mathfrak g）$有限的订单自动形态和$ \ mathfrak g_0 $ subalgebra的固定点的固定点$ \ \ vartheta $。最近，我们注意到，使用$ \ vartheta $一个人可以在$ \ Mathcal s（\ Mathfrak g）$上构建一支兼容的Poisson支架，从而构建一个“大型” poisson poisson-poisson-commitative subalgebra $ \ Mathcal z（\ Mathfrak G，\ vartheta）$ $ \ \ \ \ \ \ \ \ \ Mathak^\ ge） G_0} $。在本文中，我们研究了$（\ mathfrak g，\ vartheta）$的不变理论属性，以确保$ \ mathcal z（\ mathfrak g，\ vartheta）$的良好属性。与$ \ vartheta $相关联，一个自然的谎言代数收缩$ \ mathfrak g _ {（0）} $ $ \ mathfrak g $和$ \ MATHCAL S（= G.G.S。）的概念在$ \ Mathcal s（\ Mathfrak g）^\ Mathfrak g $中。我们证明，在许多情况下，平等$ {\ mathsf {ind \，}}} \ Mathfrak g _ {（0）} = {\ Mathsf {indsf {ind \，}} \ Mathfrak g $ holds和$ \ Mathcal S（\ Mathcal s（\ Mathfrak g.g $）根据V.G. KAC的有限订单自动形态学分类（1969），$ \ vartheta $可以用KAC图，$ \ Mathcal K（\ Vartheta）$表示，我们的结果经常使用此演示文稿。最令人惊讶的观察结果是，$ \ Mathfrak g _ {（0）} $仅取决于带有非零标签的$ \ Mathcal k（\ vartheta）$中的一组节点，并且如果$ \ \ vartheta $是内在的，并且某个标签是非零的，则是$ \ mathfrak g g _ _ ^ is as par par cottotion parection $ \ Mathfrak G $。
Let $\mathfrak g$ be a semisimple Lie algebra, $\vartheta\in {\sf Aut}(\mathfrak g)$ a finite order automorphism, and $\mathfrak g_0$ the subalgebra of fixed points of $\vartheta$. Recently, we noticed that using $\vartheta$ one can construct a pencil of compatible Poisson brackets on $\mathcal S(\mathfrak g)$, and thereby a `large' Poisson-commutative subalgebra $\mathcal Z(\mathfrak g,\vartheta)$ of $\mathcal S(\mathfrak g)^{\mathfrak g_0}$. In this article, we study invariant-theoretic properties of $(\mathfrak g,\vartheta)$ that ensure good properties of $\mathcal Z(\mathfrak g,\vartheta)$. Associated with $\vartheta$ one has a natural Lie algebra contraction $\mathfrak g_{(0)}$ of $\mathfrak g$ and the notion of a good generating system (=g.g.s.) in $\mathcal S(\mathfrak g)^\mathfrak g$. We prove that in many cases the equality ${\mathsf{ind\,}}\mathfrak g_{(0)}={\mathsf{ind\,}}\mathfrak g$ holds and $\mathcal S(\mathfrak g)^\mathfrak g$ has a g.g.s. According to V.G. Kac's classification of finite order automorphisms (1969), $\vartheta$ can be represented by a Kac diagram, $\mathcal K(\vartheta)$, and our results often use this presentation. The most surprising observation is that $\mathfrak g_{(0)}$ depends only on the set of nodes in $\mathcal K(\vartheta)$ with nonzero labels, and that if $\vartheta$ is inner and a certain label is nonzero, then $\mathfrak g_{(0)}$ is isomorphic to a parabolic contraction of $\mathfrak g$.
