Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of $\mathcal S(\mathfrak g)$
| Autor: | Panyushev, Dmitri, Yakimova, Oksana |
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| Rok vydání: | 2022 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $\mathfrak g$. Comment: 32 pp., to appear in Math. Zeitschrift |
| Databáze: | arXiv |
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