Kirillov Kostant Souriau Research Articles

Geometric uncertainty relations on Wigner–Yanase skew information

We formulate uncertainty relations based on Wigner–Yanase skew information. By using the Kirillov–Kostant–Souriau Kähler structure on the quantum phase space, we present a new geometric uncertainty relation associated to the skew information, which is shown to be tighter than the existing ones. Furthermore, we provide a skew information-based product uncertainty relation, in which the lower bound can also be used to capture the non-commutativity of the observables. We also attempt to generalize the geometric uncertainty inequalities to the case of arbitrary three observables, where the Kähler structure plays a vital role in the proof.

Some Aspects of Positive Kernel Method of Quantization

We discuss various aspects of the positive kernel method of quantization of the one-parameter groups tau _t in text{ Aut }(P,vartheta ) of automorphisms of a G-principal bundle P(G,pi ,M) with a fixed connection form vartheta on its total space P. We show that the generator {hat{F}} of the unitary flow U_t = e^{it {hat{F}}} being the quantization of tau _t is realized by a generalized Kirillov–Kostant–Souriau operator whose domain consists of sections of some vector bundle over M, which are defined by a suitable positive kernel. This method of quantization applied to the case when G=hbox {GL}(N,{mathbb {C}}) and M is a non-compact Riemann surface leads to quantization of the arbitrary holomorphic flow tau _t^{mathrm{hol}} in text{ Aut }(P,vartheta ). For the above case, we present the integral decompositions of the positive kernels on Ptimes P invariant with respect to the flows tau _t^{mathrm{hol}} in terms of the spectral measure of {hat{F}}. These decompositions generalize the ones given by Bochner’s Theorem for the positive kernels on {mathbb {C}} times {mathbb {C}} invariant with respect to the one-parameter groups of translations of complex plane.

A critical point analysis of Landau–Ginzburg potentials with bulk in Gelfand–Cetlin systems

Using the bulk deformation of Floer cohomology by Schubert classes and non-Archimedean analysis of Fukaya–Oh–Ohta–Ono’s bulk-deformed potential function, we prove that every complete flag manifold Fl(n) (n≥3) with a monotone Kirillov–Kostant–Souriau (KKS) symplectic form carries a continuum of nondisplaceable Lagrangian tori which degenerates to a nontorus fiber in the Hausdorff limit. In particular, the Lagrangian S3-fiber in Fl(3) is nondisplaceable, answering a question raised by Nohara and Ueda who computed its Floer cohomology to be vanishing.

Goldman–Turaev formality implies Kashiwara–Vergne

Let \Sigma be a compact connected oriented 2-dimensional manifold with non-empty boundary. In our previous work, we have shown that every solution of the higher genus Kashiwara–Vergne equations for an automorphism F \in \mathrm {aut}(L) of a free Lie algebra induces an isomorphism between the Goldman–Turaev Lie bialgebra \mathfrak{g}(\Sigma) and its associated graded gr \mathfrak{g}(\Sigma) . In this paper, we prove the converse: if F induces an isomorphism \mathfrak{g}(\Sigma) \cong \mathrm{gr} \mathfrak{g}(\Sigma) , then it satisfies the Kashiwara–Vergne equations up to conjugation. As an application of our results, we compute the degree one non-commutative Poisson cohomology of the Kirillov–Kostant–Souriau double bracket. The main technical tool used in the paper is a novel characterization of conjugacy classes in the free Lie algebra in terms of cyclic words.

Poisson brackets in Kontsevich’s “Lie World”

In this note we develop the theory of double brackets in the sense of van den Bergh (2008) in Kontsevich’s non-commutative “Lie World”. These double brackets can be thought of as Poisson structures defined by formal expressions only involving the structure maps of a quadratic Lie algebra. The basic example is the Kirillov–Kostant–Souriau (KKS) Poisson bracket.We introduce a notion of non-degenerate double brackets. Surprisingly, in this framework the KKS bracket turns out to be non-degenerate. The main result of the paper is the uniqueness theorem for double brackets with a given moment map. As applications, we establish a monoidal equivalence between Hamiltonian quasi-Poisson spaces and Hamiltonian spaces and give a new proof of the theorem by L. Jeffrey in Jeffrey (1994) on symplectic structure on the moduli space of flat g-connections on a surface of genus 0.

A bi-Hamiltonian system on the Grassmannian

Considering the recent result that the Poisson–Nijenhuis geometry corresponds to the quantization of the symplectic groupoid integrating a Poisson manifold, we discuss the Poisson–Nijenhuis structure on the Grassmannian defined by the compatible Kirillov–Kostant–Souriau and Bruhat–Poisson structures. The eigenvalues of the Nijenhuis tensor are Gelfand–Tsetlin variables, which, as was proved, are also in involution with respect to the Bruhat–Poisson structure. Moreover, we show that the Stiefel bundle on the Grassmannian admits a bi-Hamiltonian structure.

SUPERORBITS

We study actions of Lie supergroups, in particular, the hitherto elusive notion of orbits through odd (or more general) points. Following categorical principles, we derive a conceptual framework for their treatment and therein prove general existence theorems for the isotropy (or stabiliser) supergroups and orbits through general points. In this setting, we show that the coadjoint orbits always admit a (relative) supersymplectic structure of Kirillov–Kostant–Souriau type. Applying a family version of Kirillov’s orbit method, we decompose the regular representation of an odd Abelian supergroup into an odd direct integral of characters and construct universal families of representations, parametrised by a supermanifold, for two different super variants of the Heisenberg group.

Reduction of polysymplectic manifolds

The aim of this paper is to generalize the classical Marsden–Weinstein reduction procedure for symplectic manifolds to polysymplectic manifolds in order to obtain quotient manifolds which inherit the polysymplectic structure. This generalization allows us to reduce polysymplectic Hamiltonian systems with symmetries, such as those appearing in certain kinds of classical field theories. As an application of this technique, an analogue to the Kirillov–Kostant–Souriau theorem for polysymplectic manifolds is obtained and some other mathematical examples are also analyzed. Our procedure corrects some mistakes and inaccuracies in previous papers (Günther 1987 J. Differ. Geom. 25 23–53; Munteanu et al 2004 J. Math. Phys. 45 1730–51) on this subject.

A HAMILTONIAN ACTION OF THE SCHRÖDINGER–VIRASORO ALGEBRA ON A SPACE OF PERIODIC TIME-DEPENDENT SCHRÖDINGER OPERATORS IN (1 + 1)-DIMENSIONS

Let be the space of Schrödinger operators in (1 + 1)-dimensions with periodic time-dependent potential. The action on of a large infinite-dimensional reparametrization group SV with Lie algebra [8, 10], called the Schrödinger–Virasoro group and containing the Virasoro group, is proved to be Hamiltonian for a certain Poisson structure on . More precisely, the infinitesimal action of appears to be part of a coadjoint action of a Lie algebra of pseudo-differential symbols, , of which is a quotient, while the Poisson structure is inherited from the corresponding Kirillov–Kostant–Souriau form.

Cohomogeneity-Three HyperKähler Metrics on Nilpotent Orbits

Let \({\cal O}\) be a nilpotent orbit in ℊℂ where G is a compact, simple group and ℊ = Lie(G). It is known that \({\cal O}\) carries a unique G-invariant hyperKahler metric admitting a hyperKahler potential compatible with the Kirillov–Kostant–Souriau symplectic form. In this work, the hyperKahler potential is explicitly calculated when \({\cal O}\) is of cohomogeneity three under the action of G. It is found that such a structure lies on a one-parameter family of hyperKahler metrics with G-invariant Kahler potentials if and only if ℊ is Sp3, su6, So7, So12 or e7 and otherwise is the unique G-invariant hyperKahler metric with G-invariant Kahler potential.

Quantum Coadjoint Orbits of GL(n) and Generalized Verma Modules

In a previous Letter, we constructed an explicit GL(n)-equivariant quantization of the Kirillov—Kostant—Souriau bracket on a semisimple coadjoint orbit. We realize here that quantization as a subalgebra of endomorphisms of a generalized Verma module. As a corollary, we obtain an explicit description of the annihilators of generalized Verma modules over $$U({\text{gl(}}n{\text{))}}$$ . As an application, we construct real forms of the quantum orbits and classify their finite-dimensional representations.

Explicit Equivariant Quantization on Coadjoint Orbits of GL( n , C)

We present an explicit Uh(gl(n, C))-equivariant quantization on coadjoint orbits of GL(n, C). It forms a two-parameter family quantizing the Poisson pair of the reflection equation and Kirillov–Kostant–Souriau brackets.

Formula omitted]) invariant quantization of coadjoint orbits and vector bundles over them

Let M be a coadjoint semisimple orbit of a simple Lie group G. Let U h( g) be a quantum group corresponding to G. We construct a universal family of U h( g) invariant quantizations of the sheaf of functions on M and describe all such quantizations. We also describe all two parameter U h( g) invariant quantizations on M, which can be considered as U h( g) invariant quantizations of the Kirillov–Kostant–Souriau (KKS) Poisson bracket on M. We also consider how those quantizations relate to the natural polarizations of M with respect to the KKS bracket. Using polarizations, we quantize the sheaves of sections of vector bundles on M as one- and two-sided U h( g) invariant modules over a quantized function sheaf.

A UNIFORM MODEL OF THE MASSIVE SPINNING PARTICLE IN ANY DIMENSION

The general model of an arbitrary spin massive particle in any dimensional space–time is derived on the basis of Kirillov–Kostant–Souriau approach. It is shown that the model allows consistent coupling to an arbitrary background of electromagnetic and gravitational fields.

Dynamical and symmetry groups of the SU(2) Kepler problem

It is well known that the MIC–Kepler problem, an extension of the three-dimensional Kepler problems, admits the same dynamical and symmetry groups as the Kepler problem. This paper aims to study dynamical and symmetry groups of the SU(2) Kepler problem, where the SU(2) Kepler problem is defined to be the dynamical system reduced from the eight-dimensional conformal Kepler problem through an SU(2) symmetry and turns out to be an extension of the five-dimensional Kepler problem. It is shown that the SU(2) Kepler problem admits a dynamical group SO *(8) and that the phase space of the SU(2) Kepler problem is symplectomorphic with a co-adjoint orbit of SO *(8), on which the Kirillov–Kostant–Souriau form is defined. It is further shown that the subgroups, SU(4), SU *(4), and Sp(2)× S R 5, of SO *(8) provide the symmetry groups, SU(4)/ Z 2≅ SO(6), SU *(4)/ Z 2≅ SO 0(1,5), and ( Sp(2)× S R 5)/ Z 2≅ SO(5)× S R 5, of the SU(2) Kepler problem with negative, positive, and zero energies, respectively, where × S denotes a semi-direct product. Furthermore, constants of motion for the SU(2) Kepler problem are found together with their Poisson brackets. The symmetry Lie algebra formed by constants of motion is shown to be isomorphic with so(6)≅ su(4), so(1,5)≅ su *(4), or so(5)⊕ S R 5≅ sp(2)⊕ S R 5, depending on whether the energy is negative, positive, or zero, where ⊕ S denotes a semi-direct sum. These Lie algebras are subalgebras of so *(8)≅ so(2,6).

A geometric approach to quantum vortices

In this paper a geometrical description is given of the theory of quantum vortices first developed by Rasetti and Regge [Physica A 80, 217 (1975)] relying on the symplectic techniques of Marsden and Weinstein [J. Phys. D 7, 305 (1983)], and Kirillov–Kostant–Souriau geometric quantization. The RR-current algebra is interpreted as the natural Hamiltonian algebra associated to a certain coadjoint orbit of the group G=SDiff(R3), the KKS prequantization condition of which is related to the Feynman–Onsager relation. This orbit is also shown to possess a G-invariant Kaehler structure, whence, in principle, it is possible to quantize it in a natural way.