Lie Group Research Articles

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

Non-commutative Barge-Ghys Quasimorphisms

Abstract A (non-commutative) Ulam quasimorphism is a map $q$ from a group $\Gamma $ to a topological group $G$ such that $q(xy)q(y)^{-1}q(x)^{-1}$ belongs to a fixed compact subset of $G$. Generalizing the construction of Barge and Ghys, we build a family of quasimorphisms on a fundamental group of a closed manifold $M$ of negative sectional curvature, taking values in an arbitrary Lie group. This construction, which generalizes the Barge-Ghys quasimorphisms, associates a quasimorphism to any principal $G$-bundle with connection on $M$. Kapovich and Fujiwara have shown that all quasimorphisms taking values in a discrete group can be constructed from group homomorphisms and quasimorphisms taking values in a commutative group. We construct Barge-Ghys type quasimorphisms taking prescribed values on a given subset in $\Gamma $, producing counterexamples to the Kapovich and Fujiwara theorem for quasimorphisms taking values in a Lie group. Our construction also generalizes a result proven by D. Kazhdan in his paper “On $\varepsilon $-representations”. Kazhdan has proved that for any $\varepsilon>0$, there exists an $\varepsilon $-representation of the fundamental group of a Riemann surface of genus 2 which cannot be $1/10$-approximated by a representation. We generalize his result by constructing an $\varepsilon $-representation of the fundamental group of a closed manifold of negative sectional curvature taking values in an arbitrary Lie group.

TWISTED ACTIONS ON COHOMOLOGIES AND BIMODULES

AbstractFor closed subgroups L and R of a compact Lie group G, a left L-space X, and an L-equivariant continuous map $A:X\to G/R$ , we introduce the twisted action of the equivariant cohomology $H_R^{\bullet }(\mathrm {pt},\Bbbk )$ on the equivariant cohomology $H_L^{\bullet }(X,\Bbbk )$ . Considering this action as a right action, $H_L^{\bullet }(X,\Bbbk )$ becomes a bimodule together with the canonical left action of $H_L^{\bullet }(\mathrm {pt},\Bbbk )$ . Using this bimodule structure, we prove an equivariant version of the Künneth isomorphism. We apply this result to the computation of the equivariant cohomologies of Bott–Samelson varieties and to a geometric construction of the bimodule morphisms between them.

A Classification of Compact Cohomogeneity One Locally Conformal Kähler Manifolds

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification GC of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case.

Construction of first-order invariant differential operators

The paper considers the problem of constructing systems of vector fields that are invariant under the action of the local Lie group of transformations. It is shown that there exists a special class of Lie groups for which this problem can be solved elementarily.

Schatten properties of Calderón–Zygmund singular integral commutator on stratified Lie groups

We provide full characterization of the Schatten properties of [Mb,T], the commutator of Calderón–Zygmund singular integral T with symbol b(Mbf(x):=b(x)f(x)) on stratified Lie groups G. We show that, when p is larger than the homogeneous dimension Q of G, the Schatten Lp norm of the commutator is equivalent to the Besov semi-norm BpQ/p of the function b; but when p≤Q, the commutator belongs to Lp if and only if b is a constant. For the endpoint case at the critical index p=Q, we further show that the Schatten LQ,∞ norm of the commutator is equivalent to the Sobolev norm W˙1,Q of b. Our method at the endpoint case differs from existing methods of Fourier transforms or trace formula for Euclidean spaces or Heisenberg groups, respectively, and hence can be applied to various settings beyond.

The five gradients inequality on differentiable manifolds

The goal of this paper is to derive the so-called five gradients inequality for optimal transport theory for general cost functions on two class of differentiable manifolds: locally compact Lie groups and compact Riemannian manifolds.

A Lie symmetry approach to traveling wave solutions, bifurcation, chaos and sensitivity analysis of the geophysical Korteweg–de Vries equation

Strong waves known as tsunamis are caused by earthquakes, landslides, or volcanic eruptions that traverse oceans. This article examines the geophysical Korteweg–de Vries (GPKdV) equation, which controls the propagation of tsunami waves in seas. The study involves exploring symmetry diminution exerted Lie group analysis, examining the properties of the dynamical structure with the help of bifurcation phase pictures, and researching the dynamic demeanor of the perturbed dynamical system utilizing chaos theory. Techniques such as 3D and 2D phase portraits, time series analysis, poincaré maps, looking into the existence of multistability in the autonomous structure under diverse beginning circumstances, lyapunov exponent, and bifurcation diagram are applied to identify chaotic demeanor. Furthermore, the study establishes general forms of solitary wave results, containing periodic, trigonometric, and singular soliton results, by using the unified Riccati equation expansion approach to address the examined problem analytically. These results are visually represented as 2D and 3D graphs with cautiously chosen parameters, along with their accompanying constraint conditions. Moreover, the sensitivity evaluation of the investigated equation is discussed and demonstrated pictorially. The discoveries revealed are intriguing, novel, and potentially helpful in understanding a wide range of physical events in engineering and science.

Dynamics of invariant solutions of the DNA model using Lie symmetry approach

The utilization of the Lie group method serves to encapsulate a diverse array of wave structures. This method, established as a robust and reliable mathematical technique, is instrumental in deriving precise solutions for nonlinear partial differential equations (NPDEs) across a spectrum of domains. Its applications span various scientific disciplines, including mathematical physics, nonlinear dynamics, oceanography, engineering sciences, and several others. This research focuses specifically on the crucial molecule DNA and its interaction with an external microwave field. The Lie group method is employed to establish a five-dimensional symmetry algebra as the foundational element. Subsequently, similarity reductions are led by a system of one-dimensional subalgebras. Several invariant solutions as well as a spectrum of wave solutions is obtained by solving the resulting reduced ordinary differential equations (ODEs). These solutions govern the longitudinal displacement in DNA, shedding light on the characteristics of DNA as a significant real-world challenge. The interactions of DNA with an external microwave field manifest in various forms, including rational, exponential, trigonometric, hyperbolic, polynomial, and other functions. Mathematica simulations of these solutions confirm that longitudinal displacements in DNA can be expressed as periodic waves, optical dark solitons, singular solutions, exponential forms, and rational forms. This study is novel as it marks the first application of the Lie group method to explore the interaction of DNA molecules.

Lie group analysis of bingham plastic flow over a straining surface in the presence of magnetohydrodynamic effects and slip conditions

This paper investigates the two-dimensional flow of a Bingham plastic over a straining surface subjected to an externally applied magnetic field and surface slips. The study aims to understand the behavior of such flows and their response to external factors, which has applications in various industrial processes involving complex fluid dynamics. Through Lie group analysis, a new set of similarity transformations are derived to reduce the number of variables in the governing partial differential equations, facilitating a more tractable analysis. These transformations enable the conversion of the partial differential equations into a self-similar system of ordinary differential equations. A high-order, three-stage Lobatto IIIa formula along with appropriate boundary conditions is applied to solve this system. The solutions obtained for various physical parameters lead to several key deductions. It is found that under constant physical parameter values, the velocity layer thickness of the plastic flow is lower compared to the thermal layer thickness, indicating the dominance of the plastic flow behavior. Additionally, an increase in the magnetic field results in a reduction in the thickness of the plastic boundary layer, highlighting the significant influence of magnetic fields on the flow characteristics. These findings provide valuable insights into the control and optimization of processes involving Bingham plastic flows, particularly in the presence of magnetic fields and surface slips.

Valuations, completions, and hyperbolic actions of metabelian groups

AbstractActions on hyperbolic metric spaces are an important tool for studying groups, and so it is natural, but difficult, to attempt to classify all such actions of a fixed group. In this paper, we build strong connections between hyperbolic geometry and commutative algebra in order to classify the cobounded hyperbolic actions of numerous metabelian groups up to a coarse equivalence. In particular, we turn this classification problem into the problems of classifying ideals in the completions of certain rings and calculating invariant subspaces of matrices. We use this framework to classify the cobounded hyperbolic actions of many abelian‐by‐cyclic groups associated to expanding integer matrices. Each such action is equivalent to an action on a tree or on a Heintze group (a classically studied class of negatively curved Lie groups). Our investigations incorporate number systems, factorization in formal power series rings, completions, and valuations.

Milnor–Wood inequality for klt varieties of general type and uniformization

AbstractWe generalize the definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type due to Koziarz and Maubon to the context of singular klt varieties, where the natural fundamental groups to consider are those of smooth loci. Assuming that the rank of the target Lie group is not greater than two, we show that the Toledo invariant satisfies a Milnor–Wood‐type inequality and we characterize the corresponding maximal representations.

Publisher Correction to: Finite quotients of 3-manifold groups

Publisher Correction to: Finite quotients of 3-manifold groups

Lie group analysis and SQLM of entropy generation of MHD nonlinear radiative micropolar nanofluid flow with thermophoresis and Brownian motion

This study investigates the entropy generation in hydromagnetic mixed convection of a micropolar fluid flowing over a stretchable sheet, taking into account various influential factors such as nonlinear thermal radiation, Ohmic dissipation, uniform heat source/sink effects, chemical reaction, thermophoresis, and Brownian motion. The Spectral Quasilinearization Method (SQLM) is employed, which transforms nonlinear equations into a series of linearized ones, streamlining the solving process. Additionally, Lie group analysis is utilized to identify symmetries and derive similarity transformations crucial for converting partial differential equations into coupled ordinary differential equations, which are then solved numerically. A comprehensive sensitivity analysis is conducted on critical parameters, including the Darcy parameter, Biot number, Brinkman number, nonlinear convection parameter, and chemical reaction parameter. This analysis explores the impacts of these parameters on various flow profiles, such as velocity, temperature, microrotation, concentration, and entropy generation, enriching the depth and practical relevance of the findings. The results reveal that increasing the Brinkman and Reynolds numbers enhances entropy generation profiles while increasing the coupling parameter reduces them. The focus on entropy generation analysis provides valuable insights into energy utilization and system efficiency, particularly in micropolar fluid dynamics. Furthermore, the study underscores the practical implications of its findings by highlighting their relevance in evaluating the effectiveness of industrial processes, especially those related to energy conversion and heat transfer phenomena.

Diffeomorphic structure of evolution equations

In this work the problem of the consistent treatment of multidimensional quadratic Hamiltonians is analyzed and developed from the geometric point of view. To this end, two approaches to the treatment for the problem are studied and developed: the pure matrix representation that involves Madelung type transforms (maps), and the evolution type based in a group manifold endowed with the symplectic groups and their coverings. Some of our goals is to introduce important geometric and group theoretical tools in the proposed approaches, such as Fermi normal coordinates in the first and a generalization of the method of nonlinear realizations in the second one. Several interesting results appear and some examples of application of these concepts in different physical scenarios are developed and presented such as the relationship with the Zeldovich approximation for the dynamic of large-scale cosmological structure, the classic case of Gertsner waves or the evolution problem with an inverted Hamiltonian of Caldirola-Kanai type.

A multiphase eigenvalue problem on a stratified Lie group

AbstractWe consider a multiphase spectral problem on a stratified Lie group. We prove the existence of an eigenfunction of (2, q)-eigenvalue problem on a bounded domain. Furthermore, we also establish a Pohozaev-like identity corresponding to the problem on the Heisenberg group.

Exactly solvable Hamiltonian fragments obtained from a direct sum of Lie algebras.

Exactly solvable Hamiltonians are useful in the study of quantum many-body systems using quantum computers. In the variational quantum eigensolver, a decomposition of the target Hamiltonian into exactly solvable fragments can be used for the evaluation of the energies via repeated quantum measurements. In this work, we apply more general classes of exactly solvable qubit Hamiltonians than previously considered to address the Hamiltonian measurement problem. The most general exactly solvable Hamiltonians we use are defined by the condition that within each simultaneous eigenspace of a set of Pauli symmetries, the Hamiltonian acts effectively as an element of a direct sum of so(N) Lie algebras and can, therefore, be measured using a combination of unitaries in the associated Lie group, Clifford unitaries, and mid-circuit measurements. The application of such Hamiltonians to decomposing molecular electronic Hamiltonians via graph partitioning techniques shows a reduction in the total number of measurements required to estimate the expectation value compared to previously used exactly solvable qubit Hamiltonians.

Normal quantum channels and Markovian correlated two-qubit quantum errors

We study general ‘normally’ (Gaussian) distributed random unitary transformations. These distributions can be defined in terms of a diffusive random walk in the respective group manifold. On the one hand, a Gaussian distribution induces a unital quantum channel, which we will call ‘normal’. On the other hand, the diffusive random walk defines a unital quantum process, generated by a Lindblad master equation. In the single qubit case, we show different distributions may induce the same quantum channel.In the case of two qubits, we study normal quantum channels, induced by Gaussian distributions in SU(2)⊗SU(2) . They provide an appropriate framework for modeling quantum errors with classical correlations. In contrast to correlated Pauli errors, for instance, they conserve their Markovianity, and they lead to very different results in error correcting codes. This is illustrated with an application to entanglement distillation.

Conformal correlators on the Lorentzian torus

The general form of a 2D conformal field theory (CFT) correlator on a Euclidean Riemann surface, Lorentzian plane or Lorentzian cylinder is well known. This paper describes the general form of 2- and 3-point CFT correlators on the Lorentzian torus LT2 which arises as the conformal boundary of the group manifold SL(2,R)≃AdS3/Z. We consider only generic points, thereby omitting an analysis of contact terms, which already exhibits a surprisingly rich structure. The results are relevant to celestial holography, for which the LT2 at the boundary of Klein space is the home of the putative celestial CFT. Published by the American Physical Society 2024

On the spaces of $(d + d^{c})$-harmonic forms and $(d + d^{\Lambda})$-harmonic forms on almost Hermitian manifolds and complex surfaces

In this paper, we study the spaces of (d+d^{c}) -harmonic forms and of (d+d^{\Lambda}) -harmonic forms, a natural generalization of the spaces of Bott–Chern harmonic forms (respectively, symplectic harmonic forms) from complex (respectively, symplectic) manifolds to almost Hermitian manifolds. We apply the same techniques to compact complex surfaces, computing their Bott–Chern and Aeppli numbers and their spaces of (d+d^{\Lambda}) -harmonic forms. We give several applications to compact quotients of Lie groups by a lattice.