2020/12/07 by Dmitri I. Panyushev, Panyushev, Dmitri I., Oksana Yakimova +1
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) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2012.04014
openalex publication_date 2020/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathfrak g be a semisimple Lie algebra, \mathfrak h⊂\mathfrak g a reductive subalgebra such that \mathfrak h^⊥ is a complementary \mathfrak h-submodule of \mathfrak g. In 1983, Bogoyavlenski claimed that one obtains a Poisson commutative subalgebra of the symmetric algebra \mathcal S(\mathfrak g) by taking the subalgebra \mathcal Z generated by the bi-homogeneous components of all H∈\mathcal S(\mathfrak g)\mathfrak g. But this is false, and we present a counterexample. We also provide a criterion for the Poisson commutativity of such subalgebras \mathcal Z. As a by-product, we prove that \mathcal Z is Poisson commutative if \mathfrak h is abelian and describe \mathcal Z in the special case when \mathfrak h is a Cartan subalgebra. In this case, \mathcal Z appears to be polynomial and has the maximal transcendence degree (dim \mathfrak g+rk \mathfrak g)/2.