Let 𝔮 be a finite-dimensional Lie algebra. The symmetric algebra (𝔮) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on (𝔮) to any finite order automorphism ϑ of 𝔮. We study related Poisson-commutative subalgebras (𝔮; ϑ) of 𝒮(𝔮) and associated Lie algebra contractions of 𝔮. To obtain substantial results, we have to assume that 𝔮 = 𝔤 is semisimple. Then we can use Vinberg’s theory of ϑ-groups and the machinery of Invariant Theory.

If 𝔤 = 𝔥⊕⋯⊕𝔥 (sum of k copies), where 𝔥 is simple, and ϑ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra (𝔮; ϑ) is polynomial and maximal. Furthermore, we quantise this (𝔤; ϑ) using a Gaudin subalgebra in the enveloping algebra 𝒰(𝔤).

令𝔮为有限维李代数。对称代数（𝔮）配备有标准的Lie-Poisson括号。在本文中，我们详细解释了一个令人惊讶的发现，即自然地将（𝔮）上的第二个兼容Poisson括号与any的任何有限阶自同构associate关联。我们研究𝒮（𝔮）的相关Poisson可交换子代数（𝔮; ϑ）和associated的相关李代数压缩。为了获得实质性的结果，我们必须假设𝔮=𝔤是半简单的。然后我们可以使用Vinberg'-群理论和不变理论的机制。

如果𝔤=𝔥⊕⋯⊕𝔥（k个副本的总和），其中𝔥是简单的，并且per是循环置换，那么我们证明相应的Poisson可交换子代数（𝔮; ϑ）是多项式且是最大的。此外，我们使用包络代数𝒰（𝔤）中的高丁子代数对此（𝔤; ϑ）进行量化。