Generalized Symplectic Action and Symplectomorphism Groups of Coadjoint Orbits

Let M be a quantizable symplectic manifold. If ψ_t is a loop in the group {Ham}(M) of Hamiltonian symplectomorphisms of M and A is a 2k-cycle in M, we define a symplectic action κ_A(ψ)∊ U(1) around ψ_t(A), which is invariant under deformations of ψ, and such that κ_A(ψ) depends only on the homology class of A. Using properties of κ_A( ) we determine a lower bound for ♯π₁(Ham(O)), where O is a quantizable coadjoint orbit of a compact Lie group. In particular we prove that ♯π₁(Ham(CPⁿ)) ≥ n+1. Mathematics Subject Classifications (2000): 53D05, 57S05, 57R17, 57T20.     

Coadjoint Orbits of Three-Step Free Nilpotent Lie Groups and Time-Optimal Control Problem

Doklady Mathematics - We describe coadjoint orbits for three-step free nilpotent Lie groups. It turns out that two-dimensional orbits have the same structure as coadjoint orbits of the Heisenberg...     

Coadjoint Orbits and Time-Optimal Problems for Step- $$2$$ Free Nilpotent Lie Groups

Mathematical Notes -     

Noise and Dissipation on Coadjoint 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 in details 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.     

Adjoint and Coadjoint Orbits of the Poincaré Group

In this paper we give an effective method for finding a unique representative of each orbit of the adjoint and coadjoint action of the real affine orthogonal group on its Lie algebra. In both cases there are orbits which have a modulus that is different from the usual invariants for orthogonal groups. We find an unexplained bijection between adjoint and coadjoint orbits. As a special case, we classify the adjoint and coadjoint orbits of the Poincaré group.The author (R.C) was partially supported by European Community funding for the Research and Training Network MASIE (HPRN-CT-2000-00113).     

A Generalized Beurling Theorem for Some Lie Groups

Mathematical Notes - For any real numbers p,q ≥ 1, we present in this paper a (p, q)-generalized version of Beurling’s uncertainty principle for ?n, which largely extends the...     

Orbits of Real Semisimple Lie Groups on Real Loci of Complex Symmetric Spaces

Let G be a complex semisimple algebraic group and X be a complex symmetric homogeneous G-variety. Assume that both G, X as well as the G-action on X are defined over real numbers.Then G(R) acts on X(R) with finitely many orbits. We describe these orbits in combinatorial terms using Galois cohomology, thus providing a patch to a result of Borel and Ji.     

Generalized Polar Decompositions on Lie Groups with Involutive Automorphisms

The polar decomposition, a well-known algorithm for decomposing real matrices as the product of a positive semidefinite matrix and an orthogonal matrix, is intimately related to involutive automorphisms of Lie groups and the subspace decomposition they induce. Such generalized polar decompositions, depending on the choice of the involutive automorphism σ , always exist near the identity although frequently they can be extended to larger portions of the underlying group. In this paper, first of all we provide an alternative proof to the local existence and uniqueness result of the generalized polar decomposition. What is new in our approach is that we derive differential equations obeyed by the two factors and solve them analytically, thereby providing explicit Lie-algebra recurrence relations for the coefficients of the series expansion. Second, we discuss additional properties of the two factors. In particular, when σ is a Cartan involution, we prove that the subgroup factor obeys similar optimality properties to the orthogonal polar factor in the classical matrix setting both locally and globally, under suitable assumptions on the Lie group G . September 12, 2000. Final version received: April 16, 2001.     

Geometry of Coadjoint Orbits and Multiplicity-one Branching Laws for Symmetric Pairs

Consider the restriction of an irreducible unitary representation π of a Lie group G to its subgroup H. Kirillov’s revolutionary idea on the orbit method suggests that the multiplicity of an irreducible H-module ν occurring in the restriction π|_H could be read from the coadjoint action of H on \(\mathcal {O}^{G} \cap \text {pr}^{-1}({\mathcal {O}}^{H})\), provided π and ν are ‘geometric quantizations’ of a G-coadjoint orbit \(\mathcal {O}^{G}\) and an H-coadjoint orbit \(\mathcal {O}^{H}\), respectively, where \(\text {pr} \colon \sqrt {-1}\mathfrak {g}^{\ast } \to \sqrt {-1}\mathfrak {h}^{\ast }\) is the projection dual to the inclusion \(\mathfrak {h} \subset \mathfrak {g}\) of Lie algebras. Such results were previously established by Kirillov, Corwin and Greenleaf for nilpotent Lie groups. In this article, we highlight specific elliptic orbits \(\mathcal {O}^{G}\) of a semisimple Lie group G corresponding to highest weight modules of scalar type. We prove that the Corwin–Greenleaf number \(\sharp (\mathcal {O}^{G} \cap \text {pr}^{-1}({\mathcal {O}}^{H}))/H\) is either zero or one for any H-coadjoint orbit \(\mathcal {O}^{H}\), whenever (G,H) is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits \(\mathcal {O}^{H}\) with nonzero Corwin–Greenleaf number. Our results coincide with the prediction of the orbit philosophy, and can be seen as ‘classical limits’ of the multiplicity-free branching laws of holomorphic discrete series representations (Kobayashi [Progr. Math. 2007]).     

Explicit Classifications of Orbits in Jordan Algebra and Freudenthal Vector Space Over the Exceptional Lie Groups

ABSTRACT

We give the explicit classifications of orbits in the Jordan algebra 𝔍 over the group E _6(?26) and the Freudenthal, R -vector space 𝔓 over the group E _7(?25).

Communicated by E. Zelmanov     

可解群的大轨道

G可解群,G忠实的作用在有限群H上,且(|G|,|H|)=1,那么存在x,y∈H满足C_G(x)∩C_G(y)=1,即G有大轨道.     

Orbits of Inner Mapping Groups

 Several new tools are developed dealing with multiplication groups of loops. One of them is based upon the theorem that the inner mapping group of a loop never acts faithfully on any of its orbits, unless it is trivial. The applications include an analysis of loops whose inner mapping group is nonabelian of order pq. Some important earlier results are proved anew in a shorter way.     

On Orbits of Automorphism Groups

If G is a finite group and if A is a group of automorphisms of G whose fixed point subgroup is C _G (A) then every subgroup F of C _G (A) acts on the set of orbits of A in G. The peculiarities of this action are used here to derive several results on the number of orbits of A in an economical manner.Original Russian Text Copyright © 2005 Deaconescu M. and Walls G. L.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 533–537, May–June, 2005.     

Orbits of Inner Mapping Groups

 Several new tools are developed dealing with multiplication groups of loops. One of them is based upon the theorem that the inner mapping group of a loop never acts faithfully on any of its orbits, unless it is trivial. The applications include an analysis of loops whose inner mapping group is nonabelian of order pq. Some important earlier results are proved anew in a shorter way. (Received 5 March 2001; in revised form 11 June 2001)     

Unitary Representations and Coadjoint Orbits for a Group of Germs of Real Analytic Diffeomorphisms

The method of coadjoint orbits is developed for the group of real analytic germs of diffeomorphisms φ with φ(0) = 0 and φ′(0) = 1. The form of all infinite dimensional coadjoint orbits is described. Classes U of unitary representations are constructed. In the case n = 2 these representations are related to coadjoint orbits.     

Generalized Fuzzy Lie Ideals of Lie Algebras

In this paper we introduce new generalized fuzzy Lie ideals of Lie algebras and study some of their important properties.We characterize these generalized Lie ideals of Lie algebras by their level subsets.Some characterization of the generalized fuzzy Lie ideals of Lie algebras are also established.     

Generalized Splines for Radon Transform on Compact Lie Groups with Applications to Crystallography

The Radon transform $\mathcal{R}f$ of functions f on SO(3) has recently been applied extensively in texture analysis, i.e. the analysis of preferred crystallographic orientation. In practice one has to determine the orientation probability density function f∈L ₂(SO(3)) from $\mathcal{R}f\in L_{2}(S^{2}\times S^{2})$ which is known only on a discrete set of points. Since one has only partial information about $\mathcal{R}f$ the inversion of the Radon transform becomes an ill-posed inverse problem. Motivated by this problem we define a new notion of the Radon transform $\mathcal{R}f$ of functions f on general compact Lie groups and introduce two approximate inversion algorithms which utilize our previously developed generalized variational splines on manifolds. Our new algorithms fit very well to the application of Radon transform on SO(3) to texture analysis.