2021/02/19 by Dmitri I. Panyushev, Panyushev, Dmitri I., Oksana Yakimova +1 · 1 citation
Mathematics · #14L30 #17B08 #17B20 #17B63 #22E46 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2102.10065
openalex publication_date 2021/02/19 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Let mathfrak g be a finite-dimensional Lie algebra. The symmetric algebra\n mathcal S( mathfrak g) is equipped with the standard Lie-Poisson bracket. In\nthis paper, we elaborate on a surprising observation that one naturally\nassociates the second compatible Poisson bracket on mathcal S( mathfrak g)\nto any finite order automorphism \θ of mathfrak g. We study related\nPoisson-commutative subalgebras mathcal C of mathcal S( mathfrak g) and\nassociated Lie algebra contractions of mathfrak g. To obtain substantial\nresults, we have to assume that mathfrak g is semisimple. Then we can use\nVinberg's theory of \θ-groups and the machinery of Invariant Theory.\n If mathfrak g= mathfrak h\⊕\… \⊕ mathfrak h (sum of k\ncopies), where mathfrak h is simple, and \θ is the cyclic permutation,\nthen we prove that the corresponding Poisson-commutative subalgebra mathcal\nC is polynomial and maximal. Furthermore, we quantise this mathcal C using\na Gaudin subalgebra in the enveloping algebra mathcal U( mathfrak g).\n