Complete integrability from Poisson-Nijenhuis structures on compact hermitian symmetric spaces
2015 (English)Article in journal (Refereed) Submitted
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.
Place, publisher, year, edition, pages
Research subject Theoretical Physics
IdentifiersURN: urn:nbn:se:uu:diva-287825OAI: oai:DiVA.org:uu-287825DiVA: diva2:923504