Connected Lie Group Research Articles

Boundedness of Differential of Symplectic Vortices in Open Cylinder Model

Let G be a compact connected Lie group, (X,ω,μ) a Hamiltonian G-manifold with moment map μ, and Z a codimension-2 Hamiltonian G-submanifold of X. We study the boundedness of the differential of symplectic vortices (A,u) near Z, where A is a connection 1-form of a principal G-bundle P over a punctured Riemann surface Σ˚, and u is a G-equivariant map from P to an open cylinder model near Z. We show that if the total energy of a family of symplectic vortices on Σ˚≅[0,+∞)×S1 is finite, then the A-twisted differential dAu(r,θ) is uniformly bounded for all (r,θ)∈[0,+∞)×S1.

Quantization in fibering polarizations, Mabuchi rays and geometric Peter–Weyl theorem

In this paper we use techniques of geometric quantization to give a geometric interpretation of the Peter–Weyl theorem. We present a novel approach to half-form corrected geometric quantization in a specific type of non-Kähler polarizations and study one important class of examples, namely cotangent bundles of compact connected Lie groups K. Our main results state that this canonically defined polarization occurs in the geodesic boundary of the space of K×K-invariant Kähler polarizations equipped with Mabuchi's metric, and that its half-form corrected quantization is isomorphic to the Kähler case. An important role is played by invariance of the limit polarization under a torus action.Unitary parallel transport on the bundle of quantum states along a specific Mabuchi geodesic, given by the coherent state transform of Hall, relates the non-commutative Fourier transform for K with the Borel–Weil description of irreducible representations of K.

Lorentzian G-manifolds of Constant Positive Curvature

We study the orbits arising from isometric actions of connected Lie groups on Lorentzian manifolds with constant positive curvature.

Index theorem for homogeneous spaces of Lie groups

We study index theory for homogeneous spaces associated to an almost connected Lie group in terms of the topological and analytic aspects. For the topological aspect, we obtain a topological formula as a result of the Riemann–Roch formula for proper cocompact actions of the Lie group, inspired by the work of Paradan and Vergne. For the analytic aspect, we apply heat kernel methods to obtain a local index formula representing the higher indices of equivariant elliptic operators with respect to a proper, cocompact action of the Lie group.

Wave front sets of nilpotent Lie group representations

Let G be a nilpotent, connected, simply connected Lie group with Lie algebra g, and π a unitary representation of G. In this article we prove that the wave front set of π coincides with the asymptotic cone of the orbital support of π, i.e. WF(π)=AC(⋃σ∈supp(π)Oσ), where Oσ⊂ig⁎ is the coadjoint Kirillov orbit associated to the irreducible unitary representation σ∈Gˆ.

Errors-in-variables regression for mixed Euclidean and non-Euclidean predictors

In this paper, we explore a novel regression problem encompassing both Euclidean and non-Euclidean predictors, all of which are subject to measurement errors. Specifically, we focus on a non-Euclidean predictor taking values in a compact and connected Lie group. We propose a nonparametric estimator and establish its asymptotic properties, including rates of convergence and an asymptotic distribution. We validate the practical efficacy of our estimator through simulation studies and real data analysis.

Automatic Continuity of every Locally Bunded Homomorphism of a Perfect Connected Lie Group to a Connected Lie Group

Automatic Continuity of every Locally Bunded Homomorphism of a Perfect Connected Lie Group to a Connected Lie Group

Hardy–Sobolev–Rellich, Hardy–Littlewood–Sobolev and Caffarelli–Kohn–Nirenberg Inequalities on General Lie Groups

In this paper, we establish a number of geometrical inequalities such as Hardy, Sobolev, Rellich, Hardy–Littlewood–Sobolev, Caffarelli–Kohn–Nirenberg, Gagliardo-Nirenberg inequalities and their critical versions for an ample class of sub-elliptic differential operators on general connected Lie groups, which include both unimodular and non-unimodular cases in compact and noncompact settings. We also obtain the corresponding uncertainty type principles.

Smooth actions of connected compact Lie groups with a free point are determined by two vector fields

Consider a smooth action G×M→M of a compact connected Lie group G on a connected manifold M. Assume the existence of a point of M whose isotropy group has a single element (a free point). Then we prove that there exist two complete vector field X,X1 such that their group of automorphisms equals G regarded as a group of diffeomorphisms of M (the existence of a free point implies that the action of G is effective). Moreover, some examples of effective actions with no free point where this result fails are exhibited.

Minimal extensions in smooth dynamics

A classical result of Fathi and Herman from 1977 states that a smooth compact connected manifold without boundary admitting a locally free action of a 1-torus, respectively, an almost free action of a 2-torus, admits a minimal diffeomorphism, respectively, a minimal flow. In the first part of our paper we study the existence of locally free and almost free actions of tori on homogeneous spaces of compact connected Lie groups, thus providing new examples of spaces admitting minimal diffeomorphisms or flows. In the second part we combine the ideas of Fathi and Herman with our recent ideas to study the existence of minimal skew products over certain minimal flows with general connected Lie groups as acting groups. Our results apply to so called flows with free cycles. In the last part of our work we study the existence of free cycles in homogeneous flows.

A note on p-Kähler structures on compact quotients of Lie groups

A p-Kähler structure on a complex manifold of complex dimension n is given by a d-closed transverse real (p, p)-form. In the paper, we study the existence of p-Kähler structures on compact quotients of simply connected Lie groups by discrete subgroups endowed with an invariant complex structure. In particular, we discuss the existence of p-Kähler structures on nilmanifolds, with a focus on the case p=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p =2$$\\end{document} and complex dimension n=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 4$$\\end{document}. Moreover, we prove that a (n-2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-2)$$\\end{document}-Kähler almost abelian solvmanifold of complex dimension n≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 3$$\\end{document} has to be Kähler.

On the existence and properties of left invariant k-symplectic structures on Lie groups with bi-invariant peudo-Riemannian metric

k-symplectic manifolds are a convenient framework to study classical field theories and they are a generalization of polarized symplectic manifolds. This paper focus on the existence and the properties of left invariant k-symplectic structures on Lie groups having a bi-invariant pseudo-Riemannian metric. We show that compact semi-simple Lie groups and a large class of Lie groups having a bi-invariant pseudo-Riemannian metric does not carry any left invariant k-symplectic structure. This class contains the oscillator Lie groups which are the only solvable non abelian Lie groups having a bi-invariant Lorentzian metric. However, we built a natural left invariant n-symplectic structure on SL ( n , R ) . Moreover, up to dimension 6, only three connected and simply connected Lie groups have a bi-invariant indecomposable pseudo-Riemannian metric and a left invariant k-symplectic structure, namely, the universal covering of SL ( 2 , R ) with a 2-symplectic structure, the universal covering of the Lorentz group SO ( 3 , 1 ) with a 2-symplectic structure, and a 2-step nilpotent 6-dimensional connected and simply connected Lie group with both a 1-symplectic structure and a 2-symplectic structure.

Weights for compact connected Lie groups

Let ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} be a prime. If G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{G}$$\\end{document} is a compact connected Lie group, or a connected reductive algebraic group in characteristic different from ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}, and ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} is a good prime for G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{G}$$\\end{document}, we show that the number of weights of the ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}-fusion system of G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{G}$$\\end{document} is equal to the number of irreducible characters of its Weyl group. The proof relies on the classification of ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}-stubborn subgroups in compact Lie groups.

Symmetric pairs and branching laws

Let G be a compact connected Lie group and let H be a subgroup fixed by an involution. A classical result assures that the Hℂ-action on the flag variety F of G admits a finite number of orbits. In this article we propose a formula for the branching coefficients of the symmetric pair (G,H) that is parametrized by Hℂ∖F.

Extension of Characters from the Radical of a Connected Lie Group to a One-Dimensional Pure Pseudorepresentation of the Group Revisited

Extension of Characters from the Radical of a Connected Lie Group to a One-Dimensional Pure Pseudorepresentation of the Group Revisited

Error bounds for Lie group representations in quantum mechanics

We provide state-dependent error bounds for strongly continuous unitary representations of connected Lie groups. That is, we bound the difference of two unitaries applied to a state in terms of the energy with respect to a reference Hamiltonian associated with the representation and a left-invariant metric distance on the group. Our method works for any connected Lie group, and the metric is independent of the chosen representation. The approach also applies to projective representations and allows us to provide bounds on the energy-constrained diamond norm distance of any suitably continuous channel representation of the group.

Conformal vector fields on Lie groups: The trans-Lorentzian signature

A pseudo-Riemannian Lie group is a connected Lie group endowed with a left-invariant pseudo-Riemannian metric of signature (p,q). In this paper, we study pseudo-Riemannian Lie groups (G,〈⋅,⋅〉) with non-Killing left-invariant conformal vector fields. Firstly, we prove that if h is a Cartan subalgebra for a semisimple Levi factor of the Lie algebra g, then dim⁡h≤max⁡{0,min⁡{p,q}−2}. Secondly, we classify trans-Lorentzian Lie groups (i.e., min⁡{p,q}=2) with non-Killing left-invariant conformal vector fields, and prove that [g,g] is at most three-step nilpotent. Thirdly, based on the classification of the trans-Lorentzian Lie groups, we show that the corresponding Ricci operators are nilpotent, and consequently the scalar curvatures vanish. As a byproduct, we prove that four-dimensional trans-Lorentzian Lie groups with non-Killing left-invariant conformal vector fields are necessarily conformally flat, and construct a family of five-dimensional ones which are not conformally flat.

PERFECTING GROUP SCHEMES

Abstract We initiate a systematic study of the perfection of affine group schemes of finite type over fields of positive characteristic. The main result intrinsically characterises and classifies the perfections of reductive groups and obtains a bijection with the set of classifying spaces of compact connected Lie groups topologically localised away from the characteristic. We also study the representations of perfectly reductive groups. We establish a highest weight classification of simple modules, the decomposition into blocks, and relate extension groups to those of the underlying abstract group.

Левоинвариантная параконтактная метрическая структура на группе Sol

Among Thurston's famous list of eight three-dimensional geometries is the geometry of the manifold Sol. The variety Sol is a connected simply connected Lie group of real matrices of a special form. The manifold Sol has a left-invariant pseudo-Riemannian metric for which the group of left shifts is the maximal simply transitive isometry group. In this paper, we prove that on the manifold Sol there exists a left-invariant differential 1-form, which, together with the left-invariant pseudo-Riemannian metric, defines a paracontact metric structure on Sol. A three-parameter family of left-invariant paracontact metric connections is found, that is, linear con­nec­tions invariant under left shifts, in which the structure tensors of the par­acontact structure are covariantly constant. Among these connections, a flat connection is distinguished. It has been established that some geo­desics of a flat connection are geodesics of a truncated connection, which is an orthogonal projection of the original connection onto a 2n-dimen­sional con­tact distribution. This means that this connection is con­sistent with the con­tact distribution. Thus, the manifold Sol has a pseudo-sub-Riemannian structure determined by a completely non-holonomic contact distribution and the restriction of the original pseudo-Riemannian metric to it.

Полная группа изометрий специальной трёхмерной группы Ли

We consider a special three-dimensional Lie algebra 3 of Bianchi type V. A matrix representation of this Lie algebra and the corresponding connected simply connected Lie group 𝑆3 is found and natural coordinates are introduced, which are determined by the matrix representation. Formulas for exponential mapping with respect to natural coordinates are found. Complete groups of autoisometries of the special Lie algebra with respect to the Euclidean or Lorentz scalar product are written out. The leftinvariant Riemannian metric of the Lie group 𝑆3 and the complete group of isometries of the resulting homogeneous manifold are found. This manifold turned out to be a space of constant Ricci curvature.