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arXiv · 1810.08848

Lax formalism for Gelfand-Tsetlin integrable systems

Eder M. Correa·Lino Grama
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Abstract

In the present work, we study Hamiltonian systems on (co)adjoint orbits and propose a Lax pair formalism for Gelfand-Tsetlin integrable systems defined on (co)adjoint orbits of the compact Lie groups ${\rm{U}}(n)$ and ${\rm{SO}}(n)$. In the particular setting of (co)adjoint orbits of ${\rm{U}}(n)$, by means of the associated Lax matrix we construct a family of algebraic curves which encodes the Gelfand-Tsetlin integrable systems as branch points. This family of algebraic curves enables us to explore some new insights into the relationship between the topology of singular Gelfand-Tsetlin fibers, singular algebraic curves and vanishing cycles. Further, we provide a new description for Guillemin and Sternberg's action coordinates in terms of hyperelliptic integrals.

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Eder M. Correa, Lino Grama. 2018-10-20. Lax formalism for Gelfand-Tsetlin integrable systems. https://arxiv.org/abs/1810.08848

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