Alexey V. Bolsinov, Bozidar Jovanovic
We describe a class of completely integrable $G$-invariant magnetic geodesic flows on (co)adjoint orbits of a compact connected Lie group $G$ with magnetic field given by the Kirillov-Konstant 2-form.
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Isotropic embeddings of coadjoint orbits and magnetic geodesic flows
本文研究紧李群余伴随轨道的各向同性与 Lagrange 嵌入，将 SU(n) 的结果推广到 SO 和 Sp，并证明旗流形磁测地流与特殊经典自旋链系统等价。
We determine all the contractions within the class of finite-dimensional real Lie algebras whose coadjoint orbits have dimensions $\le2$.
Invariant star products are constructed on minimal coadjoint orbits of all the simple Lie algebras. Explicit expressions are given for the generators of the Joseph ideals and the associated infinitesimal characters.
We study the structure of the coadjoint orbits of the 2+1 Poincaré group, using a matricial representation of the group. We also obtain the orbits connected to irreducible representations of the group. Finally we obtain coherent states for the hyperboloidal and conical orbits.
We derive and study stochastic dissipative dynamics on coadjoint orbits by incorporating noise and dissipation into mechanical systems arising from the theory of reduction by symmetry, including a semidirect-product extension. Random attractors are found for this general class of systems when the Lie algebra is semi- simple, provided the top Lyapunov exponent is positive. We study two canonical examples, the free rigid body and the heavy top, whose stochastic integrable reductions are found and numerical simulations of their random attractors are shown.
In this paper we construct quantum analogs of strata of coadjoint orbits and describe their representations. This kind objects play an important role in describing quantum groups as repeated extensions of quantum strata.
We introduce noncommutative phase spaces by minimal couplings (usual one, dual one and their mixing). We then realize some of them as coadjoint orbits of the anisotropic Newton-Hooke groups in two- and three-dimensional spaces. Through these constructions the positions and the momenta of the phase spaces do not commute due to the presence of a magnetic field and a dual magnetic field.
We study the coadjoint orbits of the noncentrally extended (1+1)-affine Galilei group and compute the relevant Wigner functions built on them explicitly. We consider the centrally extended (1+1)-Galilei group and study its coadjoint orbits in the second half of the paper. We also compute the Wigner functions built on the corresponding coadjoint orbits subsequently. Finally, a comparative study of the structure of the coadjoint orbits and corresponding Wigner functions between the extended (1+1)-affine Galilei group and the centrally extended (1+1)-Galilei group is presented along with possible…
Coset symmetries and coadjoint orbits
本讲义比较从对称群陪集空间构造粒子作用量的非线性实现与余伴随轨道两种方法，指出二者在粒子作用量层面等价，并介绍辛几何及庞加莱群余伴随轨道理论。
We show that Schwarzian theories associated to certain hyperbolic and parabolic Virasoro coadjoint orbits admit bosonization, i.e. a global $S^1$-equivariant Darboux chart in which the corresponding path integral becomes Gaussian. In this chart, correlation functions of bilocals, time-ordered and out-of-time ordered, can be computed explicitly. We conjecture that a similar global chart exists for the Teichmüller orbit.
By building on our earlier work, we establish uncertainty principles in terms of Heisenberg inequalities and of the ambiguity functions associated with magnetic structures on certain coadjoint orbits of infinite-dimensional Lie groups. These infinite-dimensional Lie groups are semidirect products of nilpotent Lie groups and invariant function spaces thereon. The recently developed magnetic Weyl calculus is recovered in the special case of function spaces on abelian Lie groups.
For a Lie groupoid $\mathcal{G}$ with Lie algebroid $A$, we realize the symplectic leaves of the Lie-Poisson structure on $A^*$ as orbits of the affine coadjoint action of the Lie groupoid $\mathcal{J}\mathcal{G}\ltimes T^*M$ on $A^*$, which coincide with the groupoid orbits of the symplectic groupoid $T^*\mathcal{G}$ over $A^*$. It is also shown that there is a fiber bundle structure on each symplectic leaf. In the case of gauge groupoids, a symplectic leaf is the universal phase space for a classical particle in a Yang-Mills field.
A Note on Coadjoint Orbits for Multifermion Systems
本文探讨多费米子系统共轭共轭轨道动作的精确描述，提出变量参数化方法以近似该动作，适用于费米面附近展开，回顾了文献中使用的各种动作，并简要讨论了相空间函数与星积及其他截断。
A classification of coadjoint orbits carrying Gibbs ensembles
本文研究了协伴轨道上吉布斯集合的分类，揭示了几何温度与信息几何的关系，并证明了吉布斯测度的存在与协伴轨道的关联。
Orbits of coadjoint representations of classical compact Lie groups have a lot of applications. They appear in representation theory, geometrical quantization, theory of magnetism, quantum optics etc. As geometric objects the orbits were the subject of much study. However, they remain hard for calculation and application. We propose simple solutions for the following problems: an explicit parameterization of the orbit by means of a generalized stereographic projection, obtaining a Kählerian structure on the orbit, introducing basis two-forms for the cohomology group of the orbit.
The extensive analysis of the dynamics of relativistic spinning particles is presented. Using the coadjoint orbits method the Hamiltonian dynamics is explicitly described. The main technical tool is the factorization of general Lorentz transformation into pure boost and rotation. The equivalent constrained dynamics on Poincare group (viewed as configuration space) is derived and complete classification of constraints is performed. It is shown that the first class constraints generate local symmetry corresponding to the stability subgroup of some point on coadjoint orbit. The Dirac brackets for…
In the finite-dimensional setting, every Hermitian-symmetric space of compact type is a coadjoint orbit of a finite-dimensional Lie group. It is natural to ask whether every infinite-dimensional Hermitian-symmetric space of compact type, which is a particular example of an Hilbert manifold, is transitively acted upon by a Hilbert Lie group of isometries. In this paper we give the classification of infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits of L*-groups of compact type using the notion of simple roots of non-compact type. The key step is, given an infinite-dime…
The NHEG algebra is an extension of Virasoro introduced in [arXiv:1503.07861]; it describes the symplectic symmetries of $(n+4)$-dimensional Near Horizon Extremal Geometries with $SL(2,R)\times U(1)^{n+1}$ isometry. In this work we construct the NHEG group and classify the (coadjoint) orbits of its action on phase space. As we show, the group consists of maps from an $n$-torus to the Virasoro group, so its orbits are bundles of standard Virasoro coadjoint orbits over $T^n$. We also describe the unitary representations that are expected to follow from the quantization of these orbits, and displ…
This is the first of a series of two papers devoted to the partition function realization of Wilson surfaces in strict higher gauge theory. A higher version of the Kirillov-Kostant-Souriau theory of coadjoint orbits is presented based on the derived geometric framework, which has shown its usefulness in 4--dimensional higher Chern--Simons theory. An original notion of derived coadjoint orbit is put forward. A theory of derived unitary line bundles and Poisson structures on regular derived orbits is constructed. The proper derived counterpart of the Bohr--Sommerfeld quantization condition is th…
This paper is devoted to the construction of a hyperkaehler structure on the complexification of any Hermitian-symmetric affine coadjoint orbit O of a semi-simple L*-group of compact type, which is compatible with the complex symplectic form of Kirillov-Kostant-Souriau and restricts to the Kaehler structure of O. By the identification of the complexification of O with the cotangent space of O induced by Mostow's Decomposition Theorem, this leads to the existence of a hyperkaehler structure on the cotangent space of O compatible with Liouville's symplectic form and whose restriction to the zero…
A set of all linear transformations with a fixed Jordan structure $J$ is a symplectic manifold isomorphic to the coadjoint orbit $\mathcal O (J)$ of $GL(N,C)$. Any linear transformation may be projected along its eigenspace to (at least one) coordinate subspace of the complement dimension. The Jordan structure $\tilde J$ of the image is defined by the Jordan structure $J$ of the pre-image, consequently the projection $\mathcal O (J)\to \mathcal O (\tilde J)$ is the mapping of the symplectic manifolds. It is proved that the fiber $\mathcal E$ of the projection is a linear symplectic space and t…
The topology of the embedding of the coadjoint orbits of the unitary group U(H) of an in-finite dimensional complex Hilbert space H, as canonically determined subsets of the B-space T_s of symmetric trace class operators, is investigated. The space T_s is identified with the B-space predual of the Lie-algebra L(H)_s of the Lie group U(H). It is proved, that orbits con-sisting of symmetric operators with finite rank are (regularly embedded) closed submanifolds of T_s. An alternative method of proving this fact is given for the `one-dimensional' orbit, i.e. for the projective Hilbert space P(H).…
Mostow's Decomposition Theorem is a refinement of the polar decomposition. It states the following. Let G be a compact connected semi-simple Lie group with Lie algebra g. Given a subspace h of g such that [X, [X, Y]] belongs to h for all X and Y in h, the complexified group G^C with Lie algebra g + ig is homeomorphic to the product G .exp im. exp ih, where m is the orthogonal of h in g with respect to the Killing form. This Theorem is related to geometric properties of the non-positively curved space of positive-definite symmetric matrices and to a characterization of its geodesic subspaces. T…
Do co-adjoint orbits of Lie groups support a Kähler structure? We study this question from a point of view derived from coherent states. We examine three examples of Lie groups: the Weyl-Heisenberg group, $\mathrm{SU(2)}$ and $\mathrm{SU(1,1)}$. In cases, where the orbits admit a Kähler structure, we show that coherent states give us a Kähler embedding of the orbit into projective Hilbert space. In contrast, squeezed states, (which like coherent states, also saturate the uncertainty bound) only give us a symplectic embedding. We also study geometric quantisation of the co-adjoint orbits of the…
Partial convex hulls of coadjoint orbits and degrees of invariants
本文通过引入部分凸包概念定义紧李群余伴随轨道的两个数值不变量，证明其与 Littlewood-Richardson 锥相关，并给出不可约酉表示上不变多项式次数的下界。
This is the second of a series of two papers devoted to the partition function realization of Wilson surfaces in strict higher gauge theory. A higher 2--dimensional counterpart of the topological coadjoint orbit quantum mechanical model computing Wilson lines is presented based on the derived geometric framework, which has shown its usefulness in 4--dimensional higher Chern--Simons theory. Its symmetries are described. Its quantization is analyzed in the functional integral framework. Strong evidence is provided that the model does indeed underlie the partition function realization of Wilson s…
Pseudo-Holomorphic Hamiltonian Systems and Kähler Duality of Complex Coadjoint Orbits
本文研究伪全纯哈密顿系统并引入PHHS以在近复流形上描述经典力学，同时证明复余伴随轨道具有超凯勒与全纯凯勒结构的凯勒对偶性，并猜想其源于双余切丛。
On dynamical invariants of coadjoint orbits of 3D volume-preserving diffeomorphisms
本文研究三维保体积向量场中螺旋度唯一性的失效程度，证明在整同调球面的 $C^1$ 拓扑下存在大量独立于螺旋度的连续全局动力学不变量，从而否定回答了 Arnold 和 Khesin 关于余伴随轨道稠密性的问题。
Dirac operators for matrix algebras converging to coadjoint orbits
该研究针对高能物理中矩阵代数逼近空间结构的需求，提出余伴随轨道及收敛矩阵代数的统一Dirac算子构造，证明对应量子度量空间在强量子Gromov-Hausdorff距离下收敛。
We obtain necessary and sufficient conditions for the integrability in quadratures of geodesic flows on homogeneous spaces $M$ with invariant and central metrics. The proposed integration algorithm consists in using a special canonical transformation in the space $T^*M$ based on constructing the canonical coordinates on the orbits of the coadjoint representation and on the simplectic sheets of the Poisson algebra of invariant functions. This algorithm is applicable to integrating geodesic flows on homogeneous spaces of a wild Lie group.
Oscillator Calculus on Coadjoint Orbits and Index Theorems
本文研究了伴随轨道上的振子微积分与指标定理，证明了自旋链类型的量子力学系统是1D sigma模型的自然截断，并发现计算Witten指标等价于自旋链Hamilton量的对角化。
Following a general method proposed earlier, we construct here Wigner functions defined on coadjoint orbits of a class of semidirect product groups. The groups in question are such that their unitary duals consist purely of representations from the discrete series and each unitary irreducible representation is associated with a coadjoint orbit. The set of all coadjoint orbits (hence UIRs) is finite and their union is dense in the dual of the Lie algebra. The simple structure of the groups and the orbits enables us to compute the various quantities appearing in the definition of the Wigner func…
A strict quantization of a compact symplectic manifold $S$ on a subset $I\subseteq\R$, containing 0 as an accumulation point, is defined as a continuous field of $C^*$-algebras $\{A_{\hbar}\}_{\hbar\in I}$, with $A_0=C_0(S)$, and a set of continuous cross-sections $\{Q(f)\}_{f\in C^{\infty}(S)}$ for which $Q_0(f)=f$. Here $Q_{\hbar}(f^*)=Q_{\hbar}(f)^*$ for all $\hbar\in I$, whereas for $\hbar\to 0$ one requires that $i[Q_{\hbar}(f),Q_{\hbar}(g)]/\hbar\to Q_{\hbar}(\{f,g\})$ in norm. We discuss general conditions which guarantee that a (deformation) quantization in a more physical sense leads …
Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction
本文在实Jordan代数中构建了类似余伴随轨道的叶结构并诱导出伪黎曼度量，揭示了其与经典及量子信息几何中Fisher-Rao和Bures-Helstrom度量的直接联系。
A class of network models with symmetry group $G$ that evolve as a Lie-Poisson system is derived from the framework of geometric mechanics, which generalises the classical Heisenberg model studied in statistical mechanics. We considered two ways of coupling the spins: one via the momentum and the other via the position and studied in details the equilibrium solutions and their corresponding nonlinear stability properties using the energy-Casimir method. We then took the example $G=SO(3)$ and saw that the momentum-coupled system reduces to the classical Heisenberg model with massive spins and t…
$\mathrm{SL}(2,\mathbb{R})$ Gauge Theory, Hyperbolic Geometry and Virasoro Coadjoint Orbits
本文利用柱面上的 $\mathrm{SL}(2,\mathbb{R})$ 规范理论，证明了所有 Virasoro 伴随轨道均可作为双曲度量的模空间，揭示了无常数代表元轨道几何源于大规范变换，并建立了 Virasoro 轨道与双曲度量模空间的完整对应字典。
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