Hamiltonian Group Research Articles

Some relations for the lengths of orbits on k -sets and ( k - 1)-sets

Let G be a permutation group on a finite set \(\mit\Omega \) of size n. The paper is devoted to investigation of the relationship between the lengths of orbits on k-sets and (k - 1)-sets. The main results are the following:¶¶Theorem A. Let G be an Abelian or Hamiltonian group acting transitively on the finite set $\mit\Omega $ . Let $ \mit\Sigma \subseteq \mit\Omega $ be a k-set and $ \mit\Delta $ be a (k - 1)-subset of $\mit\Sigma $ . Then $ \mit\Sigma^G $ or $ \mit\Delta^G $ must be of the maximal length.¶¶Theorem B. Let G be a permutation group on the set $\mit\Omega $ and let $\mit\Sigma \subseteq \mit\Omega $ be a k-set, k > 2. Then there is a (k - 1)-set $\mit\Delta \subset \mit\Sigma $ such that¶¶\( \left |\mit\Delta ^G\right |\ge {2 \over k^2 }\cdot {\left |\mit\Sigma ^G\right |}^{{k-1} \over {k}} \).¶¶Thus length is in some sense a hereditary property of orbits.

Hidden symmetry in molecular graphs

The concept of the Hamiltonian group of a graph is introduced to specify the full symmetry properties of the adjacency matrix (Hamiltonian operator). Three group theoretical rules and a new method are presented which allow the hidden symmetry of a graph to be systematically and rigorously investigated. Based on one subgroup of the Hamiltonian group, the hidden symmetry problem in an alternant molecular graph sharing doubly degenerate eigenvalues x=0 is solved in general.

Contractor renormalization group technology and exact Hamiltonian real-space renormalization group transformations.

The COntractor REnormalization group (CORE) method, a new approach to solving Hamiltonian lattice systems, is presented. The method defines a systematic and nonperturbative means of implementing Kadanoff-Wilson real-space renormalization group transformations using cluster expansion and contraction techniques. We illustrate the approach and demonstrate its effectiveness using scalar field theory, the Heisenberg antiferromagnetic chain, and the anisotropic Ising chain. Future applications to the Hubbard and t-J models and lattice gauge theory are discussed.

The classification of transversal multiplicity-free group actions

Multiplicity-free Hamiltonian group actions are the symplectic analogs of multiplicity-free representations, that is, representations in which each irreducible appears at most once. The most well-known examples are toric varieties. The purpose of this paper is to show that under certain assumptions multiplicity-free actions whose moment maps are transversal to a Cartan subalgebra are in one-to-one correspondence with a certain collection of convex polytopes. This result generalizes a theorem of Delzant concerning torus actions.

A compact symmetric symplectic non-Kaehler manifold

In this paper I construct, using off the shelf components, a compact symplectic manifold with a non-trivial Hamiltonian circle action that admits no Kaehler structure. The non-triviality of the action is guaranteed by the existence of an isolated fixed point. The motivation for this work comes from the program of classification of Hamiltonian group actions. The Audin-Ahara-Hattori-Karshon classification of Hamiltonian circle actions on compact symplectic 4-manifolds showed that all of such manifolds are Kaehler. Delzant's classification of $2n$-dimensional symplectic manifolds with Hamiltonian action of $n$-dimensional tori showed that all such manifolds are projective toric varieties, hence Kaehler. An example in this paper show that not all compact symplectic manifolds that admit Hamiltonian torus actions are Kaehler. Similar technique allows us to construct a compact symplectic manifold with a Hamiltonian circle action that admits no invariant complex structures, no invariant polarizations, etc.

Cobordism theory and localization formulas for Hamiltonian group actions

Cobordism theory and localization formulas for Hamiltonian group actions Get access Viktor Ginzburg, Viktor Ginzburg Search for other works by this author on: Oxford Academic Google Scholar Victor Guillemin, Victor Guillemin Search for other works by this author on: Oxford Academic Google Scholar Yael Karshon Yael Karshon Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 1996, Issue 5, 1996, Pages 221–234, https://doi.org/10.1155/S1073792896000165 Published: 01 January 1996 Article history Published: 01 January 1996 Received: 25 January 1996

CLASSICAL YANG-MILLS THEORY WITH FERMIONS I: GENERAL PROPERTIES OF A SYSTEM WITH CONSTRAINTS

A review is made of the basic properties of the Hamiltonian description of classical Yang-Mills theory with fermions described by Grassmann-valued Dirac spinors, in the case of a trivial principal bundle with a compact, semisimple, connected, simply connected Lie structure group over Minkowski space-time. The Poincaré group is assumed to be globally implementable and only the field configurations producing finite Poincaré generators are considered. A detailed study of the Hamiltonian group of gauge transformations is made, trying to elucidate the meaning of the global gauge transformations (connected with the non-Abelian charges and with the center of the gauge group), of the winding number (connected with the large gauge transformations and with the topological charge) and of the small gauge transformations generated by the first class constraints. This leads to the identification of boundary conditions on the gauge potentials and their conjugate momenta suitable for the Hamiltonian description and allowing covariance of the non-Abelian charges. Finally, a review is made of the problem of the Gribov ambiguity, whose basis is connected with the existence of stability subgroups of gauge transformations for certain gauge potentials (gauge symmetries) and/or certain field strengths (gauge copies) in generic Sobolev spaces.

Some features of blown-up non-linear -models

We describe, in terms of the gauged non-linear -models, some results and implications of solving the following problem: given a smooth symplectic manifold as a target space with a quasi-free Hamiltonian group action, perform the symplectic blowing up of the point singularity and identify the blow-up modes in the corresponding (gauged) -model. Both classical and quantum aspects of the construction are explained, along with illustrative examples from the toric projective space and the Kähler manifold. We also discuss related problems such as the origin of mirror symmetry and quantum cohomologies.

Units of integral group rings of hamiltonian groups*

We give a finite set of generators for a subgroup of finite index of the unit group of an integral group ring of an Hamiltonian group the order of which is only divisible by 2 and primes that are congruent to 3 modulo 8. An application is given to the unit group of the integral group ring of nilpotent groups.

Nilpotent Orbits, Normality, and Hamiltonian Group Actions

Let $M$ be a $G$-covering of a nilpotent orbit in $\g$ where $G$ is a complex semisimple Lie group and $\g=\text{Lie}(G)$. We prove that under Poisson bracket the space $R[2]$ of homogeneous functions on $M$ of degree 2 is the unique maximal semisimple Lie subalgebra of $R=R(M)$ containing $\g$. The action of $\g'\simeq R[2]$ exponentiates to an action of the corresponding Lie group $G'$ on a $G'$-cover $M'$ of a nilpotent orbit in $\g'$ such that $M$ is open dense in $M'$. We determine all such pairs $(\g\subset\g')$.

Nilpotent orbits, normality and Hamiltonian group actions

The theory of coadjoint orbits of Lie groups is central to a number of areas in mathematics. A list of such areas would include (1) group representation theory, (2) symmetry-related Hamiltonian mechanics and attendant physical theories, (3) symplectic geometry, (4) moment maps, and (5) geometric quantization. From many points of view the most interesting cases arise when the group G in question is semisimple. For semisimple G the most familiar of the orbits are orbits of semisimple elements. In that case the associated representation theory is pretty much understood (the Borel-Weil-Bott Theorem and noncompact analogs, e.g., Zuckerman functors). The isotropy subgroups are reductive and the orbits are in one form or another related to flag manifolds and their natural generalizations. A totally different experience is encountered with nilpotent orbits of semisimple groups. Here the associated representation theory (unipotent representations) is poorly understood and there is a loss of reductivity of isotropy subgroups. To make matters worse (or really more interesting) orbits are no longer closed and there can be a failure of normality for orbit closures. In addition simple connectivity is generally gone but more seriously there may exist no invariant polarizations. The interest in nilpotent orbits of semisimple Lie groups has increased sharply over the last two decades. This perhaps may be attributed to the recurring experience that sophisticated aspects of semisimple group theory often lead one to these orbits (e.g., the Springer correspondence with representations of the Weyl group). This paper presents new results concerning the symplectic and algebraic geometry of the nilpotent orbits 0 and the symmetry groups of that geometry. The starting point is the recognition (made also by others) that the ring R of regular functions on any G-homogeneous cover M of 0 is not only a Poisson algebra (the case for any coadjoint orbit) but that R is also naturally graded. The key theme is that the same nilpotent orbit may be by more than one simple group. The key result is the determination of all pairs of simple Lie groups having a shared nilpotent orbit. Furthermore there is then a unique maximal such group and this group is encoded in the symplectic and algebraic

Nilpotent orbits, normality,\\ and Hamiltonian group actions

The theory of coadjoint orbits of Lie groups is central to a number of areas in mathematics. A list of such areas would include (1) group representation theory, (2) symmetry-related Hamiltonian mechanics and attendant physical theories, (3) symplectic geometry, (4) moment maps, and (5) geometric quantization. From many points of view the most interesting cases arise when the group G in question is semisimple. For semisimple G the most familiar of the orbits are orbits of semisimple elements. In that case the associated representation theory is pretty much understood (the Borel-Weil-Bott Theorem and noncompact analogs, e.g., Zuckerman functors). The isotropy subgroups are reductive and the orbits are in one form or another related to flag manifolds and their natural generalizations. A totally different experience is encountered with nilpotent orbits of semisimple groups. Here the associated representation theory (unipotent representations) is poorly understood and there is a loss of reductivity of isotropy subgroups. To make matters worse (or really more interesting) orbits are no longer closed and there can be a failure of normality for orbit closures. In addition simple connectivity is generally gone but more seriously there may exist no invariant polarizations. The interest in nilpotent orbits of semisimple Lie groups has increased sharply over the last two decades. This perhaps may be attributed to the recurring experience that sophisticated aspects of semisimple group theory often lead one to these orbits (e.g., the Springer correspondence with representations of the Weyl group). This paper presents new results concerning the symplectic and algebraic geometry of the nilpotent orbits 0 and the symmetry groups of that geometry. The starting point is the recognition (made also by others) that the ring R of regular functions on any G-homogeneous cover M of 0 is not only a Poisson algebra (the case for any coadjoint orbit) but that R is also naturally graded. The key theme is that the same nilpotent orbit may be by more than one simple group. The key result is the determination of all pairs of simple Lie groups having a shared nilpotent orbit. Furthermore there is then a unique maximal such group and this group is encoded in the symplectic and algebraic

Hamiltonian group representations and phase actions

An action of a group G on a set S is a (surjective) composition law ƒ: G × S → S, associative with respect to the group multiplication m: G × G → G. We show that an action of a Lie group G on a symplectic manifold P together with an equivariant momentum mapping can be characterized as a symplectic reduction ϱ: T* G × P → P, associative with respect to Pm, where P is the phase functor [1]. Applications to a study of symplectic analogues of pseudogroups [4] (called also quantum groups [5]) and their representations will be presented in subsequent publications.

The genus of low rank hamiltonian groups

Any finite hamiltonian group H can be written as a direct product of three groups: the quaternion group Q, an abelian group Ar of odd order and rank r⩾0, and an abelian 2-group Zs2, s⩾0. The genus of H is computed for r=2, s=1, and, when 3 divides the order of H, for r=2, s=2, 3 and r=3, s=1, 2. New bounds are given for the other low rank cases of r=2, s⩽3 and r=3, s⩽2. Tables are given indicating the status of the genus problem for all cases of r and s and listing all unknown cases of order 1000 or less.

Nonorientable Embeddings of Groups

Embeddings of Cayley graphs into nonorientable surfaces are studied. Some lower and upper bounds for the nonorientable genus of abelian and hamiltonian groups are obtained. In the case when the lower and upper bounds coincide the nonorientable genus is computed. For example, let A = Z m 1 × Z m 2 × ⋯ × Z m r , where m i + 1 ∣ m i (1 ⩽ i ⩽ r - 1); if 4 divides m 1 , r ⩾ 3, and m r ⩾ 5, m r odd, then A has nonorientable genus 2 + ( r - 2)∣ A ∣/2. In some cases the method is applied to orientable embeddings. For example, if the abelian factor A of a hamiltonian group H has rank r ⩾ 6, then both the genus and the nonorientable genus of H are determined unless either ∣ A ∣ is odd or m r = 3.

Reduced phase spaces of models of collective motion

The phase spaces T*GL+(3, R) and T*SL(3, R), which underlie the dynamics of certain models of collective motion, are reduced with respect to the symmetry group of spatial rotations. It is shown that the resulting reduced phase spaces are generated by the orbits of Hamiltonian group actions of the groups GCM(3)=Ms(3) os GL+(3, R) or CM(3)=Ms(3) os SL(3, R) on the cotangent bundles.

Quantization onV-manifolds

V-manifolds are spaces which generalize the notion of differential manifold with a certain type of singularities. They arise naturally as orbit spaces of Hamiltonian group actions on symplectic manifolds, when the action is not free. The quantization of the nonisotropic harmonic oscillator, whose orbit space is aV-manifold, is discussed by using Maslov’s quantization condition.

On the variation in the cohomology of the symplectic form of the reduced phase space

is called the momentum mapping of the Hamiltonian T-action. Given (1.1), the condition (1.2) just means that T acts along the fibers of J. For the basic definitions and properties of non-commutative Hamiltonian group actions, see [AM]. The results of this paper can easily be extended to Hamiltonian actions of arbitrary compact connected Lie groups, by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping. For some more details, see the remarks at the end of Sect. 2. We will assume throughout this paper that the momentum map is proper, that is J I(U) is compact for each compact subset U of t*. Now let r be a regular value of J, that is T,,J: TmM-~t* is surjective for all meYe=J 1(4 ). Then Ye is a smooth submanifold of M, compact because

Hamiltonian groups are color‐graph‐hamiltonian

AbstractA group Γ is said to be color ‐graph ‐hamiltonian if Γ has a minimal generating set Δ such that the Cayley color graph DΔ(Γ) is hamiltonian. It is shown that every hamiltonian group is color ‐graph ‐hamiltonian.

Hamiltonian group actions and integrable systems

The purpose of this paper is to show how some ideas from the theory of Hamiltonian systems with symmetry can be applied to dynamical systems like the nonperiodic Toda Lattice of Moser to yield striking information about their trajectories. Our presentation of these ideas is as self-contained as possible. We try to keep abstract machinery in its proper place, and emphasize the concrete computations which are implicit in some parts of modern symplectic geometry. Most of the results we shall derive are known in one form or another. The only novelties, perhaps, are some proofs and examples.