Invariants of Twisted Current Algebras and Related Poisson-Commutative Subalgebras

Abstract Let ${\mathfrak q}$ be a finite-dimensional Lie algebra and $\vartheta $ an automorphism of order $m$ of ${\mathfrak q}$. We extend $\vartheta $ to an automorphism of the loop algebra ${\widehat{{\mathfrak q}}}={\mathfrak q}[t,t^{-1}]$ and consider the fixed-point subalgebra ${\widehat{{\mathfrak q}}}^\vartheta $. Using the splitting ${\widehat{{\mathfrak q}}}^\vartheta ={\mathfrak q}[t]^\vartheta \oplus (t^{-1}{\mathfrak q}[t^{-1}])^\vartheta $, we construct subalgebras of the symmetric algebra ${\mathcal Z}_{1}\subset{\mathcal S}({\mathfrak q}[t]^\vartheta )$ and ${\mathcal Z}_{2}\subset{\mathcal S}((t^{-1}{\mathfrak q}[t^{-1}])^\vartheta )$. We prove that if ${\mathfrak q}$ is reductive, then both ${\mathcal Z}_{1}$ and ${\mathcal Z}_{2}$ are Poisson-commutative. Moreover, ${\mathcal Z}_{1}$ is always a polynomial ring with infinitely many generators, which are explicitly described. The algebras ${\mathcal Z}_{1}$ and ${\mathcal Z}_{2}$ are $\vartheta $-twisted Poisson versions of the universal Gaudin subalgebra introduced by Ilin and Rybnikov in 2021 and the Feigin–Frenkel centre, respectively. Unlike the untwisted case there is a significant difference between ${\mathcal Z}_{1}$ and ${\mathcal Z}_{2}$. For the natural Lie algebras homomorphism $\psi \!: {\widehat{{\mathfrak q}}}^\vartheta \to{\mathfrak q}[t,t^{-1}]^\vartheta /(t^{m}-1) \cong{\mathfrak q}$, it is shown that $\psi ({\mathcal Z}_{1})$ and $\psi ({\mathcal Z}_{2})$ are closely related to the Poisson-commutative subalgebras of ${\mathcal S}({\mathfrak q})$ constructed by Panyushev–Yakimova in 2021. An explicit description of $\psi ({\mathcal Z}_{2})$ is obtained under some assumptions on $\vartheta $, while $\psi ({\mathcal Z}_{1})$ is described for all $\vartheta $ and all reductive ${\mathfrak q}$.