Some remarks on periodic gradings

2024/05/01 by Oksana Yakimova, Yakimova, Oksana · 1 citation
Mathematics · Computer Science · #Geometric Analysis and Curvature Flows #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2405.00599

Abstract

Let \mathfrak q be a finite-dimensional Lie algebra, ϑ∈ Aut(\mathfrak q) a finite order automorphism, and \mathfrak q₀ the subalgebra of fixed points of ϑ. Using ϑ one can construct a pencil \mathcal P of compatible Poisson brackets on S(\mathfrak q), and thereby a `large' Poisson-commutative subalgebra Z(\mathfrak q,ϑ) consisting of \mathfrak q₀-invariants in S(\mathfrak q). We study one particular bracket \ , \_∞∈\mathcal P and the related Poisson centre \mathcal Z_∞. It is shown that \mathcal Z_∞ is a polynomial ring, if \mathfrak q is reductive.

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• Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of \mathcal S(\mathfrak g)
• Periodic automorphisms, compatible Poisson brackets, and Gaudin\n subalgebras
• Poisson-commutative subalgebras of S(\mathfrak g) associated with involutions
• Invariants of twisted current algebras and related Poisson-commutative subalgebras
• Shifts of semi-invariants and complete commutative subalgebras in polynomial Poisson algebras