Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of {mathcal {S}}({mathfrak g})

Abstract

Let {mathfrak g} be a semisimple Lie algebra, vartheta in textsf{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 mathrm{ind,}{mathfrak g}_{(0)}=mathrm{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}.