Philip Foth

Polygons in Minkowski space and the Gelfand–Tsetlin method for pseudo-unitary groups

Abstract We study the symplectic geometry of the moduli spaces of polygons in the Minkowski 3-space. These spaces naturally carry completely integrable systems with periodic flows. We extend the Gelfand–Tsetlin method to pseudo-unitary groups and show that the action variables are given by the Minkowski lengths of non-intersecting diagonals.

Poisson structures on complex flag manifolds associated with real forms

For a complex semisimple Lie group G and a real form Go we define a Poisson structure on the variety of Borel subgroups of G with the property that all Go-orbits in X as well as all Bruhat cells (for a suitable choice of a Borel subgroup of G) are Poisson submanifolds. In particular, we show that every non-empty intersection of a Go-orbit and a Bruhat cell is a regular Poisson manifold, and we compute the dimension of its symplectic leaves.

Bruhat Poisson structure on CPn and integrable systems

In this note we present a simple formula for the Bruhat Poisson structure on complex projective spaces in terms of the momentum coordinates. We also give a simple description of a family of functions in involution on compact Hermitian symmetric spaces obtained via the bi-Hamiltonian approach using the Bruhat Poisson structure and an invariant symplectic structure. We compute these functions explicitly on CPn and relate them to the Gelfand–Tsetlin coordinates. We also show how the Lenard scheme can be applied.

EIGENVALUES OF SUMS OF PSEUDO-HERMITIAN MATRICES ∗

We study analogues of classical inequalities for the eigenvalues of sums of Hermitian matrices for the cone of admissible elements in the pseudo-Hermitian case. In particular, we obtain analogues of the Lidskii-Wielandt inequalities.

The Poisson geometry of SU(1,1)

We study the natural Poisson structure on the group SU(1,1) and related questions. In particular, we give an explicit description of the Ginzburg–Weinstein isomorphism for the sets of admissible elements. We also establish an analog of Thompson’s conjecture for this group.

Generalized Kostant convexity theorems

We give a simple proof of the equality of the spectra for projections to Levi factors in the linear and non-linear cases, generalizing a classical theorem of Kostant.

Tetraplectic structures, tri-momentum maps, and quaternionic flag manifolds

Abstract The purpose of this note is to define tri-momentum maps for certain manifolds with an Sp(1)n-action. We exhibit many interesting examples of such spaces using quaternions. We show how these maps can be used to reduce such manifolds to ones with fewer symmetries. The images of such maps for quaternionic flag manifolds, which are defined using the Dieudonne determinant, resemble the polytopes from the complex case.

On sums of admissible coadjoint orbits

Given a quasi-Hermitian semisimple Lie algebra, we describe possible spectra of the sum of two admissible elements from its dual vector space.