Cohomology of symplectic reductions of generic coadjoint orbits

Let be a generic coadjoint orbit of a compact semi-simple Lie group . Weight varieties are the symplectic reductions of by the maximal torus in . We use a theorem of Tolman and Weitsman to compute the cohomology ring of these varieties. Our formula relies on a Schubert basis of the equivariant cohomology of , and it makes explicit the dependence on and a parameter in .

     

Matrix coefficients and coadjoint orbits of compact Lie groups

Let be a compact Lie group. We use Weyl functional calculus (Anderson, 1969) and symplectic convexity theorems to determine the support and singular support of the operator-valued Fourier transform of the product of the -function and the pull-back of an arbitrary unitary irreducible representation of to the Lie algebra, strengthening and generalizing the results of Cazzaniga, 1992. We obtain as a consequence a new demonstration of the Kirillov correspondence for compact Lie groups.

     

Complex adjoint orbits in Lie theory and geometry

This expository article is an introduction to the adjoint orbits of complex semisimple groups, primarily in the algebro-geometric and Lie-theoretic contexts, and with a pronounced emphasis on the properties of semisimple and nilpotent orbits. It is intended to build a foundation for more specialized settings in which adjoint orbits feature prominently (ex. hyperkähler geometry, Landau–Ginzburg models, and the theory of symplectic singularities). Also included are a few arguments and observations that, to the author’s knowledge, have not yet appeared in the research literature.     

Deformations of adjoint orbits for semisimple Lie algebras and Lagrangian submanifolds

We give a coadjoint orbit's diffeomorphic deformation between the classical semisimple case and the semi-direct product given by a Cartan decomposition. The two structures admit the Hermitian symplectic form defined in a semisimple complex Lie algebra. We provide some applications such as the constructions of Lagrangian submanifolds.     

The restricted Grassmannian, Banach Lie-Poisson spaces, and coadjoint orbits

We investigate some basic questions concerning the relationship between the restricted Grassmannian and the theory of Banach Lie-Poisson spaces. By using universal central extensions of Lie algebras, we find that the restricted Grassmannian is symplectomorphic to symplectic leaves in certain Banach Lie-Poisson spaces, and the underlying Banach space can be chosen to be even a Hilbert space. Smoothness of numerous adjoint and coadjoint orbits of the restricted unitary group is also established. Several pathological properties of the restricted algebra are pointed out.     

On the classification of the coadjoint orbits of the Sobolev Bott-Virasoro group

The purpose of this paper is to give the classification of the Bott-Virasoro coadjoint orbits, with nonzero central charge, in the functional analytic setting of smooth Hilbert manifolds. The central object of the paper is thus the completion of the Bott-Virasoro group with respect to a Sobolev topology, giving rise to a smooth Hilbert manifold and topological group, called the Sobolev Bott-Virasoro group. As a consequence of this approach, analytic and geometric properties of the coadjoint orbits are studied.     

一类图的伴随多项式的导数与积分

设Ｇ是不含三角形的图,本文给出了Ｇ的伴随多项式ｈ（Ｇ,ｘ）的导数和两个积分公式。     

Constructing an automorphic form from the orbit of a transformation

We consider the Fuchsian groups of linear-fractional transformations. We propose a new method for presenting automorphic forms as gap series over an appropriate subset of transformations of the group which is not a subgroup. Comparative analysis of the Poincaré theta-series and gap series demonstrates that the use of gap series requires less transformations and parameters that the summands of series depend on.     

The incidence class and the hierarchy of orbits

R. Rimányi defined the incidence class of two singularities η and ζ as [η]|ζ, the restriction of the Thom polynomial of η to ζ. He conjectured that (under mild conditions) [η]|ζ ≠ 0 ⇔ ζ ⊂ . Generalizing this notion we define the incidence class of two orbits η and ζ of a representation. We give a sufficient condition (positivity) for ζ to have the property that [η]|ζ ≠ 0 ⇔ ζ ⊂ for any other orbit η. We show that for many interesting cases, e.g. the quiver representations of Dynkin type positivity holds for all orbits. In other words in these cases the incidence classes completely determine the hierarchy of the orbits. We also study the case of singularities where positivity doesn’t hold for all orbits.      

Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems

We discuss bifurcation of periodic orbits in discontinuous planar systems with discontinuities on finitely many straight lines intersecting at the origin and the unperturbed system has either a limit cycle or an annulus of periodic orbits. Assume that the unperturbed periodic orbits cross every switching line transversally exactly once. For the first case we give a condition for the persistence of the limit cycle. For the second case, we obtain the expression of the first order Melnikov function and establish sufficient conditions on the number of limit cycles bifurcate from the periodic annulus. Then we generalize our results to systems with discontinuities on finitely many smooth curves. As an application, we present a piecewise cubic system with 4 switching lines and show that the maximum number of limit cycles bifurcate from the periodic annulus can be affected by the position of the switching lines.     

Multipliers for the symmetry groups of p-adic spacetime

The consequences for particle classification of the Volovich hypothesis that spacetime geometry is non-archimedean at the Planck scale are explored. The multiplier groups and universal topological central extensions of the p-adic Poincaré and Galilean groups are determined. The text was submitted by the author in English.     

Existence of periodic orbits and chaos in a class of three-dimensional piecewise linear systems with two virtual stable node-foci

In this paper, we consider a new class of piecewise linear (PWL) systems with two virtual stable node-foci (the meaning of “virtual” is from Bernardo et al. (2008)) which exhibits periodic orbits and chaos. This fact that PWL systems have no unstable equilibria but has chaos will unavoidably make the exploration of this chaos more complicated. Particular values for bifurcation diagram are provided. Based on mathematical analysis and Poincaré map, periodic orbits of this kind of system without unstable equilibrium points are derived, the corresponding existence theorems are given, and the obtained results are applied to specific examples.     

Finite cubic graphs admitting a cyclic group of automorphism with at most three orbits on vertices

The theory of voltage graphs has become a standard tool in the study of graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend the scope of this theory to graphs admitting a cyclic group of automorphisms that may not be semiregular. We use this new tool to classify all cubic graphs admitting a cyclic group of automorphisms with at most three vertex-orbits and we characterise vertex-transitivity for each of these classes. In particular, we show that a cubic vertex-transitive graph admitting a cyclic group of automorphisms with at most three orbits on vertices either belongs to one of 5 infinite families or is isomorphic to the well-known Tutte–Coxeter graph.     

A note on the periodic orbits and topological entropy of graph maps

This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits.

     

Deformations of vector fields and canonical coordinates on coadjoint orbits

We establish the concurrence of the existence of a canonical linear embedding, enabling us to pass to the Darboux coordinates on the coadjoint orbits, and the existence of a polarization of a certain linear functional. In a corollary to the main theorem we prove that every polarization is normal, which means that it satisfies the Pukansky condition.