Abstract
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).
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To the memory of our teacher Ernest Borisovich Vinberg
Dmitri I. Panyushev is the first author is funded by RFBR, project number 20-01-00515.
Oksana S. Yakimova is funded by the DFG (German Research Foundation) — project number 404144169. Received September 1, 2020. Accepted February 24, 2021.
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PANYUSHEV, D.I., YAKIMOVA, O.S. PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS. Transformation Groups 26, 641–670 (2021). https://doi.org/10.1007/s00031-021-09650-3
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DOI: https://doi.org/10.1007/s00031-021-09650-3