In this paper, we study the geometric property (T) for discretized warped cones of actions on a compact Lie group M by its finitely generated subgroups. We show that if a subgroup G is dense in M , then the associated discretized warped cone \bigsqcup_{n} M\times \{t(n)\} does not have geometric property (T) for any sequence of positive numbers \{t(n)\}_{n\in \mathbb{N}} converging to \infty . This result applies to certain ergodic actions of groups with property (T), for example, the action of \mathrm{SO}\bigl(d,\mathbb{Z}\bigl[\frac{1}{5}\bigr]\bigr) on \mathrm{SO}(d) with d\geq 5 . As an application, we obtain new examples of expanders without geometric property (T), including certain superexpanders.

ABSTRACT We investigate various properties of the superposition operator on harmonic Fock spaces. First, we show the operator admits a nontrivial order‐bounded structure only when it acts on the growth‐type harmonic Fock space . Next, we establish several local invertibility conditions that ensure the operator's global invertibility on the spaces. Building on these results, we analyze the group of homeomorphisms and the Lie group generated by the operators and characterize the closed subspaces that remain invariant under the action of the group.

For a $G$ -equivariant fibration $p \colon E\to B$ , we introduce and study the invariant analogue of Cohen, Farber, and Weinberger’s parametrized topological complexity, called the invariant parametrized topological complexity. This notion generalizes the invariant topological complexity introduced by Lubawski and Marzantowicz. When $G$ is a compact Lie group acting freely on $E$ , we show that the invariant parametrized topological complexity of the $G$ -fibration $p \colon E\to B$ coincides with the parametrized topological complexity of the induced fibration $\overline{p} \colon \overline{E} \to \overline{B}$ between the orbit spaces. Furthermore, we compute the invariant parametrized topological complexity of equivariant Fadell–Neuwirth fibrations, which measures the complexity of motion planning in the presence of obstacles with unknown positions, where the order of their placement is irrelevant. In addition, we study the equivariant sectional category and the equivariant parametrized topological complexity, which serve as essential tools for obtaining several results in this paper.

Abstract We present necessary and sufficient conditions to have global hypoellipticity for a class of complex-valued coefficient first-order evolution equations defined on $$\mathbb {T}^1 \times G$$ T 1 × G , where G is a compact Lie group. First, we show that the global hypoellipticity of the constant coefficient operator related to this operator is a necessary condition, but not a sufficient condition. Under certain hypothesis, we show that the global hypoellipticity of this class of operator is completely characterized by Nirenberg–Treves’ condition $$(\mathcal {P})$$ ( P ) .

ABSTRACT In this paper, we introduce a linearly polarized light wave in an optical fiber and rotation of the polarization plane through the Frenet‐type frame with Myller configuration. Since the geometric evaluation and interpretations of a polarized light wave are associated with geometric phase, a new type of geometric phase model has been constructed with Myller configuration. Also, the rotation of the polarization plane is determined by the Fermi–Walker parallel transportation law. Then, this is examined with the Rytov parallel transportation along with the direction of the state of the polarization plane in an optical fiber with Myller configuration. Moreover, the electromagnetic curves obtained by the electric field along the polarization plane of a light wave traveling in an optical fiber are given. Furthermore, we examine the Lorentz force equations and scrutinize the electromagnetic trajectories constructed by the electric field of the light wave traveling in the optical fiber with Myller configuration. Then, we give a brief introduction to the geometric phase model for a curve with a Frenet‐type frame with Myller configuration in a 3D Lie group. Additionally, we give some physical interpretations based on the direction of the electromagnetic curves thanks to the unique and natural structure of Myller configuration in both Euclidean 3‐space and 3D Lie group.

Similarity analysis for cylindrical shock wave in a low conducting rotating non-ideal gas in the presence of axial and azimuthal magnetic inductions using the Lie group theoretic method

Abstract In the context of information geometry, the concept known as left-invariant statistical structure on Lie groups is defined by Furuhata–Inoguchi–Kobayashi (Inf Geom 4(1):177–188, 2021). In this paper, we introduce the notion of the moduli space of left-invariant statistical structures on a Lie group. We study the moduli spaces for three particular Lie groups, each of which has a moduli space of left-invariant Riemannian metrics that is a singleton. As applications, we classify left-invariant conjugate symmetric statistical structures and left-invariant dually flat structures (which are equivalent to left-invariant Hessian structures) on these three Lie groups. A characterization of the Amari–Chentsov $$\alpha $$ α -connections on the Takano Gaussian space is also given.

Abstract This study investigates the dynamical properties of a generalized extended (3+1)-dimensional nonlinear evolution equation. A variety of ansatz strategies will be employed to accomplish this. The initial solution will be the soliton solution or shock wave soliton solution. Furthermore, the singular soliton solution of the extended (3+1)-dimensional nonlinear evolution problem will be presented. Based on the invariance surface condition, we will derive more analytical solutions. Subsequently, we employ the multiplier method to construct conservation laws. Graphical representations of the obtained solutions will be shown based on suitable selections of the arbitrary parameters included in the solutions to enhance comprehension of the underlying physics.

This research focuses on the Heisenberg Lie group. The aim is to determine the coadjoint orbits and their parametrizations. The method used in this research involves constructing the parametrization of coadjoint orbit for Heisenberg Lie group corresponding to the Heisenberg Lie algebra of dimension 2n+1. Furthermore, the obtained results are specialized to the cases of n=1, 2, and 3 which correspond to the Heisenberg Lie algebras of dimensions 3, 5, and 7. The main results are the explicit formulas of coadjoint orbits for the Heisenberg Lie group H_1, H_2, and H_3 which are expressed by the equations (〖Ad〗^* H_1 ) l_(α,β,γ)={l_(α^',β^',γ^' ):α^',β^',γ^'∈R}, (〖Ad〗^* H_2 ) l_(α,β,γ)={l_(α^',β^',γ^' ):α^',β^'∈R^2,γ^'∈R}, and (〖Ad〗^* H_3 ) l_(α,β,γ)={l_(α^',β^',γ^' ):α^',β^'∈R^3,γ^'∈R}. In addition, their associated parametrizations are given by the explicit formulas ψ(γZ^*,u)=∑_(i=1)^n▒(u_i X_i^*+u_(n+i) Y_i^* ) +γZ^* for n=1, 2, and 3. As a further study, various types of Lie groups can be explored to determine coadjoint orbits and their parametrization. Two Lie groups that are interesting to investigate further regarding their coadjoint orbits and parametrization are the diamond and Jacobi groups.

Ricci flows play an important role in studies of geometry and topology of manifolds and were first studied for the Levi-Civita connection by R. Hamilton and other mathematicians. A natural generalization of the Levi-Civita connection is the class of metric connections with vector torsion, or the class of semisymmetric connections, first discovered by E. Cartan. The Ricci tensor of such connections is, generally speaking, not symmetric. Therefore, when studying Ricci flows for semisymmetric connections, it is necessary to consider semisymmetric equiaffine connections, or such semisymmetric connections for which the Ricci tensor is symmetric. In the case of Lie groups, this is equivalent to the fulfillment of a certain system of algebraic equations. In this paper, we study the Ricci flow on three-dimensional unimodular Lie groups with a semisymmetric equiaffine connection. The flow equation in the coordinate system of J. Milnor is reduced to a mixed system consisting of algebraic and differential equations. By solving a subsystem of algebraic equations and substituting the obtained solutions into a subsystem of differential equations, we find the Ricci flow on a three-dimensional unimodular Lie group with the Milnor metric with respect to a se-misymmetric equiaffine connection.

This paper presents the main results that allow obtaining Hermitian and Kahler homogeneous spaces using subtwistor structures. A subtwistor structure is related to degenerate skew-symmetric 2-form and Riemannian metrics on a manifold. Such a structure is a generalized classic construction of a twistor structure, a symplectic structure, and a Kahler structure for manifolds of arbitrary dimensions with a degenerate skew-symmetric 2-form. It is proved that a subtwistor structure with a vanishing torsion tensor on a Lie group produces the invariant Hermitian structure or Kahler structure on homogeneous space, which is generated by this subtwistor structure. There is a description of an important construction that allows obtaining an invariant Hermitian structure on a homogeneous space of a Lie group of arbitrary dimensions from a left-invariant skew-symmetric degenerate 2-form being a radical that is ideal in a Lie algebra. The mentioned homogeneous space is obtained as a quotient of this Lie group over the radical subgroup.

Deep learning systems deployed in regulated settings require explanations that are accurate and stable under nuisance transformations, yet classical post hoc transition matrices rely on fidelity-only fitting that fails to guarantee consistent explanations under spatial rotations or other group actions. In this work, we propose Equivariant Transition Matrices, a post hoc approach that augments transition matrices with Lie-group-aware structural constraints to bridge this research gap. Our method estimates infinitesimal generators in the formal and mental feature spaces, enforces an approximate intertwining relation at the Lie algebra level, and solves the resulting convex Least-Squares problem via singular value decomposition for small networks or implicit operators for large systems. We introduce diagnostics for symmetry validation and an unsupervised strategy for regularization weight selection. On a controlled synthetic benchmark, our approach reduces the symmetry defect from 13,100 to 0.0425 while increasing the mean squared error marginally from 0.00367 to 0.00524. On the MNIST dataset, the symmetry defect decreases by 72.6 percent (141.19 to 38.65) with changes in structural similarity and peak signal-to-noise ratio below 0.03 percent and 0.06 percent, respectively. These results demonstrate that explanation-level equivariance can be reliably imposed post-training, providing geometrically consistent interpretations for fixed deep models.

Abstract The Euler--Poincaré equations, firstly introduced by Henri Poincaré in 1901, arise from the application of Lagrangian mechanics to systems on Lie groups that exhibit symmetries, particularly in the contexts of classical mechanics and fluid dynamics. These equations have been extended to various frameworks, such as semidirect products, advected parameters, and field theory, with broad applications across physics and engineering. In this paper, we develop a discrete variational framework by establishing the discrete Euler--Poincaré reduction for discrete Lagrangian systems on Lie groups with advected parameters and additional dynamics. The reduction is achieved through the use of group difference maps, specifically those defined using the Cayley transform or the matrix exponential. Furthermore, we extend the Kelvin--Noether theorem and its discrete analogue to characterize the Kelvin--Noether quantities for the resulting continuous and discrete Euler--Poincaré equations. As a specific application, we derive both the continuous and discrete Euler--Poincaré equations for the dynamics of underwater vehicles. Numerical simulations validate the proposed discrete scheme, demonstrating excellent long-term preservation of both total energy and Kelvin--Noether quantities. These results highlight the potential of the framework for high-fidelity simulation, control, and navigation of underwater vehicles and other complex mechanical systems.

Invariant Gibbs measures for $(1+1)$-dimensional wave maps into Lie groups

Spectral multipliers on two-step stratified Lie groups with degenerate group structure

Joint distribution alignment on Lie group manifolds for domain adaptation

This paper classifies all left-invariant Hermitian and hyper-Hermitian structures on Lie groups whose commutator groups are one-dimensional. Furthermore, for each natural number n = 2k and n = 4k (k > 1 for the case of hyper-Hermitian structures), we construct an n-dimensional Lie group with a one-dimensional commutator group that possesses left-invariant Hermitian and hyper-Hermitian structures, respectively.

We describe four-dimensional Lorentzian algebraic Ricci solitons. In sharp contrast with the Riemannian situation, any four-dimensional Lie group admits a left-invariant Lorentz metric which is a Ricci soliton.

If is a connected semisimple Lie group with finite center and is a maximal compact subgroup of G, then the Lie algebra of admits a Cartan decomposition = .This allows us to define the Cartan motion group = .In this paper, we study the regularity of -finite matrix coefficients of unitary representations of .We prove that the optimal exponent () for which all such coefficients are ()-Hlder continuous coincides with the optimal regularity of all -finite coefficients of the group itself.Our approach relies on stationary phase techniques that were previously employed by the author to study regularity in the setting of (, ).Furthermore, we provide a general framework to reduce the question of regularity from -finite coefficients to -bi-invariant coefficients.

Abstract We investigate the integrability of polynomial vector fields through the lens of duality in parameter spaces. We examine formal power series solutions annihilated by differential operators and explore the properties of the integrability variety in relation to the invariants of the associated Lie group. Our study extends to differential operators on affine algebraic varieties, highlighting the intrinsic connection between these operators and local analytic first integrals. To illustrate the duality, the case of quadratic vector fields is considered in detail.