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

Abstract

AbstractLet $${\mathfrak g}$$ g be a semisimple Lie algebra, $$\vartheta \in \textsf{Aut}({\mathfrak g})$$ ϑ ∈ Aut ( g ) a finite order automorphism, and $${\mathfrak g}_0$$ g 0 the subalgebra of fixed points of $$\vartheta $$ ϑ . Recently, we noticed that using $$\vartheta $$ ϑ one can construct a pencil of compatible Poisson brackets on $${\mathcal {S}}({\mathfrak g})$$ S ( g ) , and thereby a ‘large’ Poisson-commutative subalgebra $${\mathcal {Z}}({\mathfrak g},\vartheta )$$ Z ( g , ϑ ) of $${\mathcal {S}}({\mathfrak g})^{{\mathfrak g}_0}$$ S ( g ) g 0 . In this article, we study invariant-theoretic properties of $$({\mathfrak g},\vartheta )$$ ( g , ϑ ) that ensure good properties of $${\mathcal {Z}}({\mathfrak g},\vartheta )$$ Z ( g , ϑ ) . Associated with $$\vartheta $$ ϑ one has a natural Lie algebra contraction $${\mathfrak g}_{(0)}$$ g ( 0 ) of $${\mathfrak g}$$ g and the notion of a good generating system (=g.g.s.) in $${\mathcal {S}}({\mathfrak g})^{\mathfrak g}$$ S ( g ) g . We prove that in many cases the equality $$\mathrm{ind\,}{\mathfrak g}_{(0)}=\mathrm{ind\,}{\mathfrak g}$$ ind g ( 0 ) = ind g holds and $${\mathcal {S}}({\mathfrak g})^{\mathfrak g}$$ S ( g ) g has a g.g.s. According to V. G. Kac’s classification of finite order automorphisms (1969), $$\vartheta $$ ϑ can be represented by a Kac diagram, $$\mathcal {K}(\vartheta )$$ K ( ϑ ) , and our results often use this presentation. The most surprising observation is that $${\mathfrak g}_{(0)}$$ g ( 0 ) depends only on the set of nodes in $$\mathcal {K}(\vartheta )$$ K ( ϑ ) with nonzero labels, and that if $$\vartheta $$ ϑ is inner and a certain label is nonzero, then $${\mathfrak g}_{(0)}$$ g ( 0 ) is isomorphic to a parabolic contraction of $${\mathfrak g}$$ g .