2015/03/25 by Francesco Bonechi, Bonechi, Francesco, Jian Qiu +4
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Symplectic Geometry (math.SG) #hep-th #math-ph #math.DG #math.MP #math.QA #math.SG
paper · pdf · doi:10.48550/arxiv.1503.07339
40 pages, 1 figure, 17 references
arxiv created 2015/03/25 · openalex publication_date 2015/03/25 · arxiv updated 2015/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a class of Poisson-Nijenhuis systems defined on compact hermitian symmetric spaces, where the Nijenhuis tensor is defined as the composition of Kirillov-Konstant-Souriau symplectic form with the so called Bruhat-Poisson structure. We determine its spectrum. In the case of Grassmannians the eigenvalues are the Gelfand-Tsetlin variables. We introduce the abelian algebra of collective hamiltonians defined by a chain of nested subalgebras and prove complete integrability. By construction, these models are integrable with respect to both Poisson structures. The eigenvalues of the Nijenhuis tensor are a choice of action variables. Our proof relies on an explicit formula for the contravariant connection defined on vector bundles that are Poisson with respect to the Bruhat-Poisson structure.