We study algebraic properties of matrices whose rows are mutual neighbours, and are also neigbours of 0 ("neighbour" in the sense of a certain nilpotency condition). The intended application is in synthetic differential geometry. For a square matrix of this kind, the product of the diagonal entries equals the determina…
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48 results for “nilpotent matrices”
Study linear perturbations of Spin(7) metrics, finding only rank one nilpotent matrices.
problem Linear perturbations of Spin(7) metrics.
method Applying the method of linear perturbations to Spin(7)-structures.
result Only rank one nilpotent matrices determine nontrivial perturbations.
Study nilpotent Higgs bundles and Calabi-Yau moduli metrics.
problem Understand nilpotent Higgs bundles and their metrics.
method Algebraic inequality for nilpotent matrices, geometric applications.
result Sharp upper bound of holomorphic sectional curvatures on Calabi-Yau moduli.
We consider Ricci flow on two classes of nilpotent Lie groups that generalize the three-dimensional Heisenberg group: the higher-dimensional classical Heisenberg groups, and the groups of real unitriangular matrices. Each group is known to admit a Ricci soliton, but we construct them \textit{explicitly} on each group. …
Abstract classifies Lie algebras with complex or symplectic structures.
problem Classifying Lie algebras with specific structures.
method Analyzing Jordan normal form and restrictions on matrix .
result Classification reduces to nilpotent case, with specific structure implications.
Coupled nonlinear integrable systems are generated from usual zero curvature equation. The relevant Maurer-Cartan forms are constructed by combining suitably chosen matrices (nilpotent, Hadamard, idempotent and k-idempotent) and Lie algebraic elements via Kronecker product. In each case a closure type property among th…
The aim of our paper is to construct pseudo -type algebras from the covering free nilpotent two-step Lie algebra as the quotient algebra by an ideal. We propose an explicit algorithm of construction of such an ideal by making use of a non-degenerate scalar product. Moreover, as a bypass result, we recover the existe…
New metrics defined for full-rank correlation matrices, ensuring unique operations.
problem No suitable problem statement as the abstract does not describe a problem to be solved.
method New Riemannian metrics defined on full-rank correlation matrices, providing unique operations.
result Unique Riemannian logarithm and Fréchet mean defined for full-rank correlation matrices.
Let T be the nilpotent group of 4 x 4 real upper triangular matrices. In this note we show that the Euler equations of certain left-invariant riemannian metrics on T have a horseshoe. We also show, with the aid of a numerical computation of a Melnikov-type integral, that the Euler equations of the sub-riemannian Carnot…
New algorithm for Coxeter connections with maximally ramified singularities.
problem Constructing connections on the projective line with a maximally ramified irregular singularity.
method Numerical algorithm for matrix completions to solve the Upper Nilpotent Completion Problem.
result Explicit constructions of Coxeter connections with specified singularities.
The group of real 4 by 4 upper triangular matrices with 1s on the diagonal has a left-invariant subRiemannian (or Carnot-Caratheodory) structure whose underlying distribution corresponds to the superdiagonal. We prove that the associated subRiemannian geodesic flow is not completely integrable. This provides the first …
Nilpotent quandlesmath.GT
Nilpotent quandles have simple characterizations and are closely related to nilpotency.
problem Characterizing and understanding nilpotent quandles.
method Characterization of generating sets, Hopf property, construction of free nilpotent quandles, simple presentations of associated groups.
result Nilpotent quandles have the Hopf property and are equivalent to reduced peripheral systems.
Nilpotent Lie algebras obtained from ordered sets and quivers are algebraic Ricci solitons.
problem Constructing nilpotent Lie algebras as algebraic Ricci solitons
method Using transitively and antisymmetrically ordered sets (TAOSs) and incidence algebras
result Nilpotent Lie algebras with arbitrarily high degrees of nilpotency are algebraic Ricci solitons
New method constructs nilpotent Lie algebras from quivers.
problem Constructing nilpotent Lie algebras from quivers.
method Using paths within quivers to construct nilpotent Lie algebras.
result Constructs a broad family of Ricci soliton nilmanifolds.
Study knot invariants using automorphism groups of free nilpotent groups.
problem Developing knot invariants using automorphism groups.
method Nilpotently -localization of knot groups and automorphism groups of free nilpotent groups.
result Maps from outer automorphism groups yield knot invariants.
Nilpotent Sasakian manifolds are Heisenberg nilmanifolds.
problem Characterizing Sasakian manifolds with nilpotent fundamental groups.
method Proved diffeomorphism to Heisenberg nilmanifolds.
result Compact aspherical Sasakian manifolds with nilpotent fundamental groups are Heisenberg nilmanifolds.
We exhibit pseudo Riemannian manifolds which are Szabó nilpotent of arbitrary order, or which are Osserman nilpotent of arbitrary order, or which are Ivanov-Petrova nilpotent of order 3.
Classifies complex structures on specific nilpotent Lie algebras.
problem Classifying complex structures on nilpotent Lie algebras with minimal center.
method Classification through algebraic and geometric methods.
result Space of complex structures on specific Lie algebras up to isomorphism.
We compute the characteristic varieties and the Alexander polynomial of a finitely generated nilpotent group. We show that the first characteristic variety may be used to detect nilpotence. We use the Alexander polynomial to deduce that the only torsion-free, finitely generated nilpotent groups with positive deficiency…
This paper classifies Ricci soliton subgroups in a specific type of nilpotent group.
problem Classifying Ricci soliton subgroups in a specific type of nilpotent group.
method Using the properties of nilpotent Iwasawa groups and Lie subgroups.
result Classification of codimension one Lie subgroups of nilpotent Iwasawa groups that are Ricci solitons.
Study limits of adjoint orbits for Lie groups, describing nilpotent orbits.
problem Understanding limits of adjoint orbits for Lie groups.
method Systematic and topological study of limits of continuous families of adjoint orbits for non-compact simple Lie groups.
result Explicit description of nilpotent orbits in terms of Richardson orbits for hyperbolic semisimple elements.
Completes classification of G2-structures on specific nilpotent Lie groups.
problem Classifying seven-dimensional nilpotent Lie groups with purely coclosed G2-structures.
method Analyzing nilpotent Lie groups of various steps and dimensions.
result Classification of indecomposable 5- and 6-step nilpotent Lie groups.
New findings on complex structures in nilpotent Lie algebras and pseudo-Kähler geometry.
problem Classifying complex structures on nilpotent Lie algebras.
method Analysis of Lie algebras in dimensions eight and higher, identifying structures with pseudo-Kähler metrics.
result Identification of pseudo-Kähler nilmanifolds with invariant complex structures, providing counterexamples and topological restrictions.
The paper studies gradings on nilpotent Lie algebras linked to smooth algebraic varieties.
problem Understanding gradings on nilpotent Lie algebras associated with algebraic varieties.
method Analyzing lattice structures in nilpotent Lie groups and their fundamental groups.
result Conditions for a lattice to be the fundamental group of a smooth complex algebraic variety.
This paper completes the classification of certain nilpotent Lie groups with specific geometric structures.
problem Classifying nilpotent Lie groups with purely coclosed G2-structures.
method Analyzing seven-dimensional nilpotent Lie groups of various steps.
result Classification of indecomposable 5- and 6-step nilpotent Lie groups with these structures.
Criterion for nilpotent Lie groups to have nilsolitons.
problem Existence of nilsolitons in nilpotent Lie groups.
method Algebraic criterion for nilpotent Lie algebras, proving necessary and sufficient condition for nilsolitons.
result Criterion provides a necessary and sufficient condition for nilpotent Lie groups to admit nilsolitons.
We study complex product structures on nilpotent Lie algebras, establishing some of their main properties, and then we restrict ourselves to 6 dimensions, obtaining the classification of 6-dimensional nilpotent Lie algebras admitting such structures. We prove that any complex structure which forms part of a complex pro…
Study proves equality of LS-category and cohomological dimension for specific group homomorphisms.
problem Proving equality of LS-category and cohomological dimension for specific group homomorphisms.
method Analyzing epimorphisms and homomorphisms between specific types of almost nilpotent and virtually nilpotent groups.
result Equality of LS-category and cohomological dimension for specified group homomorphisms.
This paper classifies strongly nilpotent special multi-flags and their Goursat counterparts.
problem Local classification of strongly nilpotent special multi-flags and Goursat distributions.
method Study of special multi-flags in homogeneous case, focusing on their weights.
result Strongly nilpotent germs of multiflags from different singularity classes are pairwise inequivalent.
Study on pseudo-Hermitian quadratic nilpotent Lie algebras with methods and classifications.
problem Characterizing and classifying pseudo-Hermitian quadratic nilpotent Lie algebras.
method Construction methods and double extension by planes.
result Complete classification of nilpotent quadratic Lie algebras and pseudo-Hermitian metrics up to dimension 8.
This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Kählerian complex structures on six-dimensional nilpotent algebras such that the differential Gerstenhaber a…
Extends Floquet-Bloch theory to nilpotent groups for geometric applications.
problem Asymptotic problems on nilpotent covers of negatively curved manifolds.
method Generalized Floquet-Bloch theory using Malcev completions.
result Branching formula relating finite and infinite-dimensional representations.
Researchers determine Dehn functions of specific nilpotent groups.
problem Understanding the Dehn functions of central products of nilpotent groups.
method Analyzing families of filiform and Lie groups to determine Dehn functions.
result Confirms conjecture and provides evidence for lower Dehn functions in central products.
Gromov proposed an averaged version of the Dehn function and claimed that in many cases it should be subasymptotic to the Dehn function. Using results on random walks in nilpotent groups, we confirm this claim for most nilpotent groups. In particular, if a nilpotent group satisfies the isoperimetric inequality $δ(l)<Cl…
We study the question of the existence of left-invariant Sasaki contact structures on the seven-dimensional nilpotent Lie groups. It is shown that the only Lie group allowing Sasaki structure with a positive definite metric tensor is the Heisenberg group. We find a complete list of the 22 classes of seven-dimensional n…
Classifies nilpotent Lie groups with specific -structures.
problem Identifying 7D nilpotent Lie groups with purely coclosed -structures.
method Examined all 7D nilpotent Lie algebras, classified them, and verified the existence or non-existence of -structures.
result Provided a complete classification of 7D nilpotent Lie groups with purely coclosed -structures.
Study torsion-free nilpotent fundamental groups of smooth varieties up to rank 7.
problem Characterize fundamental groups of smooth quasi-projective varieties.
method Analyzes fundamental groups of smooth quasi-projective varieties using topological and Lie group theory.
result Determine fundamental groups for smooth quasi-projective varieties up to rank 7.
A near-identity nilpotent pseudogroup of order m >= 1 is a family f_1, ..., f_n: (-1,1) -> R of C^2 functions for which: |f_i - id|_{C^1} < epsilon for some small positive real number epsilon < 1/10^{m+1} and commutators of the functions f_i of order at least m equal the identity. We present a classification of near-id…
A nonpolycyclic nilpotent-by-cyclic group Gamma can be expressed as the HNN extension of a finitely-generated nilpotent group N. The first main result is that quasi-isometric nilpotent-by-cyclic groups are HNN extensions of quasi-isometric nilpotent groups. The nonsurjective injection defining such an extension induces…
The study characterizes G₂-structures on 2-step nilpotent Lie groups.
problem Characterizing G₂-structures on 2-step nilpotent Lie groups.
method Analyzing left-invariant purely coclosed G₂-structures on 7-dimensional 2-step nilpotent Lie groups.
result Criteria for Riemannian metrics induced by these structures and determination of isomorphism classes.
In this paper we attempt to give a systematic account on privileged coordinates and the nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold with a distinguished filtration of subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. This paper lies …
New definition of Rumin complex for nilpotent Lie groups.
problem No new problem introduced.
method Alternative definition of Rumin complex on nilpotent Lie groups.
result Direct application of cohomology results to all nilpotent Lie groups.
The paper studies nilpotent structures in oriented neutral vector bundles and neutral hyperKähler structures.
problem Nilpotent structures in oriented neutral vector bundles and their relation to neutral hyperKähler structures.
method Defined -nilpotent structures for Lie subgroups of related to neutral hyperKähler structures.
result Existence of complex and paracomplex structures forming neutral hyperKähler structures if and only if there exists an -nilpotent structure.
The study defines a canonical nilpotent structure for certain collapsed manifolds.
problem Understanding the structure of collapsed Riemannian manifolds.
method Analyzes the nilpotent structure of manifolds with bounded Ricci curvature and Reifenberg local covering geometry.
result A canonical nilpotent structure can be defined and uniquely determined over regular limit spaces.
The study classifies nilpotent Lie foliations with cohomological obstructions.
problem Understanding rigidity of nilpotent Lie foliations under solvable deformations.
method Development of a cohomological framework and algebraic criterion for rigidity.
result Established a necessary and sufficient algebraic criterion for rigidity in generalized Heisenberg groups.
The paper characterizes and examines nilpotent complex structures on stratified Lie algebras.
problem Characterizing and understanding nilpotent complex structures on stratified Lie algebras.
method Introduced a new descending series to prove a new characterization of nilpotent complex structures and examined whether these structures preserve the strata.
result Found that there exists a -invariant stratification on a step 2 nilpotent Lie algebra with a complex structure.
The paper establishes analogs of Stallings' theorem for group homomorphisms and their nilpotent quotients.
problem Understanding the structure of fundamental groups of geometric objects.
method Develops analogs of Stallings' theorem for group homomorphisms and their nilpotent quotients.
result Derives applications including non-isomorphic number fields and hyperbolic manifolds with isomorphic universal nilpotent quotients.
Holomorphic actions on complex spaces for nilpotent groups.
problem Understanding polynomial actions on complex spaces for nilpotent groups.
method Explicit construction of biholomorphisms by polynomial maps.
result Simply connected nilpotent Lie groups are biholomorphic to .