Study connects Lie groups to specific Riemannian manifolds.
problem Understanding Lie groups through Riemannian manifold properties.
method Investigates Lie groups as 3D almost paracontact almost paracomplex Riemannian manifolds.
result Established correspondence between Lie algebra and matrix representation.
Formula for sectional curvatures on matrix groups.
problem Calculating curvatures on matrix groups.
method Simple formula derivation for sectional curvatures.
result Valid formula for general linear and reductive Lie groups.
The paper explores continuous inverse ambiguous functions on various Lie groups.
problem Existence of continuous inverse ambiguous functions on Lie groups.
method Investigation of continuous inverse ambiguous functions on specific Lie groups.
result Existence of continuous inverse ambiguous functions on various Lie groups.
We obtain minimal dimension matrix representations for each indecomposable five-dimensional Lie algebra over R and justify in each case that they are minimal. In each case a matrix Lie group is given whose matrix Lie algebra provides the required representation.
There are studied Lie groups considered as almost hypercomplex Hermitian-Norden manifolds, which are integrable and have the lowest dimension four. It is established a correspondence of the derived Lie algebras of types of invariant hypercomplex structures and the explicit matrix representation of their Lie groups. The…
The object of investigation are Lie groups considered as almost contact B-metric manifolds of the lowest dimension three. It is established a correspondence of all basic-class-manifolds of the Ganchev-Mihova-Gribachev classification of the studied manifolds and the explicit matrix representation of Lie groups. Some kno…
Generative model learns from simpler distributions on Lie groups.
problem Learning from complex Lie group data.
method Substituting exponential curves for line segments on Lie groups.
result Simple, intrinsic, and fast implementation for generative modelling.
Paper connects Stokes phenomena to quantum groups and Poisson-Lie groups.
problem Analyse Stokes phenomena in Poisson-Lie groups and quantum groups.
method Use Ug-valued Stokes phenomena to construct quantum group U_hg and relate it to Poisson-Lie group G*.
result Show that Ug-valued Stokes phenomena can be obtained as a semiclassical limit of the KZ associator.
Attention tokens are group elements, negated algebra norms.
problem Attention mechanism for matrix Lie groups
method Lie-Algebra Attention
result Score matches learned kernel on invariant poses, outperforms vector-token baseline
We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N,g) satisfy the matrix equation Uv = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Ka…
This expository article introduces the topic of roots in a compact Lie group. Compared to the many other treatments of this standard topic, I intended for mine to be relatively elementary, example-driven, and free of unnecessary abstractions. Some familiarity with matrix groups and with maximal tori is assumed. This ar…
Lie bialgebra structures are reviewed and investigated in terms of the double Lie algebra, of Manin- and Gauß-decompositions. The standard R-matrix in a Manin decomposition then gives rise to several Poisson structures on the correponding double group, which is investigated in great detail.
Researchers compute Wodzicki residue for pseudo-differential operators on compact Lie groups.
problem Computing the Wodzicki residue for pseudo-differential operators on compact Lie groups.
method Analytic continuation of traces and matrix-valued symbols.
result Main theorem complementary to [2], removing ellipticity hypothesis.
New preconditioners speed up SGD on Lie groups.
problem Accelerating SGD convergence on Lie groups.
method Online updating of low-rank and matrix-free preconditioners.
result Preconditioners maintain symmetry and invariance.
Let ωg be a Lie algebra valued differential 1-form on a manifold M satisfying the structure equations dωg+21ωg∧ωg=0 where g is solvable. We show that the problem of finding a smooth map ρ:M→G, where G is an n-dimensional so…
The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.
problem Defining projective analogues of Lie bialgebras and Poisson-Lie groups.
method Introducing projective tensor products and adapting classical notions to these structures.
result Every quasi-triangular projective r-matrix gives rise to a projective Banach Lie bialgebra.
The paper classifies vector fields on 5D nilpotent Lie groups.
problem Classifying left-invariant affine and projective vector fields on 5D nilpotent Lie groups.
method Algebraic characterization and case-by-case analysis of vector fields.
result All projective vector fields are affine, extending classical results.
We introduce a systematic method to produce left-invariant, non-Ricci-flat Einstein metrics of indefinite signature on nice nilpotent Lie groups. On a nice nilpotent Lie group, we give a simple algebraic characterization of non-Ricci-flat left-invariant Einstein metrics in both the class of metrics for which the nice b…
Equivalence proven between equivariant K-theory and K-homology for certain matrix group actions.
problem Establishing Poincaré duality for proper cocompact matrix group actions.
method Using equivariant K-theory and K-homology, with geometric models of Baum and Douglas.
result Poincaré duality holds between equivariant K-theory and K-homology for G-spinc manifolds with compact quotient. A 4-dimensional Riemannian manifold equipped with an endomorphism of the tangent bundle, whose fourth power is the identity, is considered. The matrix of this structure in some basis is circulant and the structure acts as an isometry with respect to the metric. Such manifolds are constructed on 4-dimensional real Lie g…
In this article we study the Hofer geometry of a compact Lie group K which acts by Hamiltonian diffeomorphisms on a symplectic manifold M. Generalized Hofer norms on the Lie algebra of K are introduced and analyzed with tools from group invariant convex geometry, functional and matrix analysis. Several global res…
We study two types of preconditioners and preconditioned stochastic gradient descent (SGD) methods in a unified framework. We call the first one the Newton type due to its close relationship to the Newton method, and the second one the Fisher type as its preconditioner is closely related to the inverse of Fisher inform…
Analogous exponential map defined for Hopf algebras.
problem Defining an exponential map for Hopf algebras.
method Analogy with Lie groups, interpretation as states, Hilbert C* bimodules, dual Hopf algebra elements.
result Multiple interpretations of exponential map values in Hopf algebras.
In this paper we show that Galilean group is a matrix Lie group and find its structure. Then provide the invariants of special Galilean geometry of motions, by Olver's method of moving coframes, we also find the corresponding {e}−structure.
The paper extends Chern-Weil theory to simplicial principal bundles.
problem Calculating characteristic classes on simplicial manifolds.
method Using the classifying bundle EGoBG to compute characteristic classes. result First Pontryagin class on Lie matrix groups equals symplectic form up to a constant.
In this paper, we propose an auto-encoder based generative neural network model whose encoder compresses the inputs into vectors in the tangent space of a special Lie group manifold: upper triangular positive definite affine transform matrices (UTDATs). UTDATs are representations of Gaussian distributions and can strai…
Study of logarithms in SVD-closed subgroups of unitary group.
problem Understanding logarithms in SVD-closed subgroups of unitary groups.
method Analysis of generalized principal logarithms and minimizing geodesics.
result Set of generalized principal logarithms is a disjoint union of diffeomorphic subsets.
Study on completeness of metrics on specific Lie groups.
problem Completeness of left-invariant Lorentzian metrics on 3D non-unimodular Lie groups.
method Analyzing metrics with Lie algebra of the form R⋉AR2 for various A. result Determine all geodesically complete and incomplete metrics for different cases of A. The paper simplifies conditions for optimal paths on manifolds avoiding obstacles.
problem Finding optimal paths on manifolds avoiding obstacles.
method Study of sufficient conditions for optimality on Riemannian manifolds and Lie groups.
result New conditions for optimality are provided in terms of matrix invertibility.
A new method for efficiently computing derivatives of skew-symmetric matrix exponentials.
problem Efficient computation of derivatives for skew-symmetric matrices.
method Characterization of invertibility, construction of nearby logarithm, and efficient implementation.
result Explicit formulae for differentiation and its inverse of skew-symmetric matrix exponentials.
Let G be a matrix group. Topological G-manifolds with Palais-proper action have the G-homotopy type of countable G-CW complexes (3.2). This generalizes E Elfving's dissertation theorem for locally linear G-manifolds (1996). Also we improve the Bredon--Floyd theorem from compact groups G (1960).
New model learns graph spectra accurately, outperforming existing methods.
problem Graph diffusion models struggle to distinguish certain graph families and their spectra.
method Leveraged random matrix theory to analytically extract spectral properties, introducing Dyson Diffusion Model.
result Dyson Diffusion Model learns graph spectra accurately and outperforms existing models.
Global analysis of Dixmier traces and Wodzicki residues on compact Lie groups.
problem Computing Dixmier traces and Wodzicki residues on compact Lie groups.
method Global quantisation approach, using global symbols and representation theory.
result Explicit formulae for Dixmier traces and Wodzicki residues on compact Lie groups.
New minimal surfaces found in a specific type of 3D space.
problem Finding minimal surfaces in a particular class of 3D spaces.
method Investigated minimal surfaces invariant under a specific group action in unimodular semidirect products.
result Described new examples of minimal surfaces in a specific 3D space.
In this study, a pairwise comparison matrix is generalized to the case when coefficients create Lie group G, non necessarily abelian. A necessary and sufficient criterion for pairwise comparisons matrices to be consistent is provided. Basic criteria for finding a nearest consistent pairwise comparisons matrix (extend…
This paper develops optimal transport methods on the roto-translation group SE2.
problem Optimal transport on the roto-translation group SE2 for image analysis.
method Develops a computational framework for optimal transportation over Lie groups, focusing on SE2. Uses Sinkhorn-like algorithm with efficient distance approximations.
result Advances in image barycentric interpolation, orientation field interpolation, and Wasserstein flows on SE2.
Discovering transformations for disentangled representations without prior knowledge of the underlying Lie group.
problem Discovering nonlinear transformations for disentangled representations without prior knowledge of the underlying Lie group.
method Approximating target vectors as matrix-vector products of the form \(\boldsymbol{\widetilde{y}}_i = \boldsymbol{\varphi}(t_i) \boldsymbol{x}_i\) where \(\boldsymbol{\varphi}(t_i)\) belongs to a one-parameter subgroup of \(\mathrm{GL}_n (\mathbb{R})\).
result Learning a Lie algebra from unlabeled data pairs without explicit group definition.
The topological string interpretation of homological knot invariants has led to several insights into the structure of the theory in the case of sl(N). We study possible extensions of the matrix factorization approach to knot homology for other Lie groups and representations. In particular, we introduce a new triply gr…
Given a finite dimensional representation of a semisimple Lie algebra there are two ways of constructing link invariants: 1) quantum group invariants using the R-matrix, 2) the Kontsevich universal link invariant followed by the Lie algebra based weight system. Le and Murakami showed that these two link invariants are …
The study investigates linearizability of Poisson structures on groupoids.
problem Linearizing Poisson structures on groupoids around the unit section.
method Extending the Lagrangian neighbourhood theorem to cosymplectic Lie algebroids, integrating triangular Lie bialgebras to symplectic LA-groupoids.
result Poisson structures on groupoids are linearizable under certain conditions.
The paper extends log-Sobolev inequalities to matrix-valued settings using combinatorial methods.
problem Log-Sobolev inequalities for matrix-valued settings.
method Combining noncommutative geometry tools and combinatorial methods.
result Combinatorial methods yield computable lower bounds for matrix-valued log-Sobolev inequalities.
Proposes a neural network for recognizing 3D skeleton-based interactions.
problem Recognizing two-person interactions from 3D skeleton sequences.
method Uses Gaussian distributions and Riemannian geometry of SPD matrices and matrix groups.
result Achieves competitive results on three benchmarks for 3D human activity understanding.
Develops log-Euclidean Lie groups for SPD and correlation matrices.
problem Unifies various log-Euclidean constructions for SPD and correlation matrices.
method Theory and explicit isometries linking different log-Euclidean metrics.
result Explicit log-Euclidean metrics on SPD and correlation matrices.
The paper extends ternary algebra concepts using cube roots of unity.
problem Extending algebraic structures from binary to ternary multiplication.
method Introducing ternary associator, commutator, and Lie algebra at cube roots of unity.
result Derived an identity for ternary commutator based on GA(1,5). Representations of coherent state Lie algebras on coherent state manifolds as first order differential operators are presented. The explicit expressions of the differential action of the generators of semisimple Lie groups determine for linear Hamiltonians in the generators of the groups first order differential equati…
Estimates Laplace eigenvalues and diameter for Lie group metrics.
problem Estimating Laplace eigenvalues and diameter for left-invariant metrics on compact Lie groups.
method Relates left-invariant metrics to positive definite matrices and uses eigenvalue properties.
result Partial answers to Eldredge's conjecture on Laplace eigenvalues and diameter.
Extends ESGVI for UWB localization with skewed noise, improving state estimation accuracy.
problem Improving state estimation accuracy in UWB localization with skewed noise.
method Generalizes ESGVI to matrix Lie groups and introduces non-Gaussian factors.
result Improved accuracy in UWB localization with NLOS and multipath effects.
Researchers use shape analysis to recover protein structures from Cryo-EM data.
problem Recovering the three-dimensional backbone structure of single polypeptide proteins from noisy tomographic projections.
method Shape analysis and matrix Lie group actions to deform point clouds to match 2D tomography data.
result Optimal deformations are computed to recover the three-dimensional backbone structure of proteins.