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- Title
Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of S(g)
- Authors
Panyushev, Dmitri I.; Yakimova, Oksana S.
- Abstract
Let g be a semisimple Lie algebra, ϑ ∈ Aut (g) a finite order automorphism, and g 0 the subalgebra of fixed points of ϑ . Recently, we noticed that using ϑ one can construct a pencil of compatible Poisson brackets on S (g) , and thereby a ‘large’ Poisson-commutative subalgebra Z (g , ϑ) of S (g) g 0 . In this article, we study invariant-theoretic properties of (g , ϑ) that ensure good properties of Z (g , ϑ) . Associated with ϑ one has a natural Lie algebra contraction g (0) of g and the notion of a good generating system (=g.g.s.) in S (g) g . We prove that in many cases the equality ind g (0) = ind g holds and S (g) g has a g.g.s. According to V. G. Kac’s classification of finite order automorphisms (1969), ϑ can be represented by a Kac diagram, K (ϑ) , and our results often use this presentation. The most surprising observation is that g (0) depends only on the set of nodes in K (ϑ) with nonzero labels, and that if ϑ is inner and a certain label is nonzero, then g (0) is isomorphic to a parabolic contraction of g .
- Subjects
LIE algebras; POISSON brackets; AUTOMORPHISMS; COMMUTATIVE algebra; MATRIX pencils
- Publication
Mathematische Zeitschrift, 2023, Vol 303, Issue 2, p1
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-022-03199-x