arXiv research

Paper proves zero stability for one-row colored sl₃-Jones polynomials.

Introduces new stability concept for Fano fibrations.

The paper studies $\mathfrak{g}$-stability of surfaces with boundary and derives area estimates.

Uniform criteria for stability of fixed points in various geometric structures.

Stability of cut locus under metric perturbations in compact Riemannian manifolds.

Contact Lie algebras have specific properties related to stabilizers and invariant polynomials.

The paper develops stability criteria for real reductive Lie groups acting on manifolds.

Let $X$ be a manifold with a bi-Poisson structure $\{η^t\}$ generated by a pair of $G$-invariant symplectic structures $ω_1$ and $ω_2$, where the Lie group $G$ acts properly on $X$. Let $H$ be some isotropy subgroup for this action representing the principle orbit type and $X^r_\mathfrak{h}$ be the submanifold of $X$ c…

The paper studies deformations of Lie ideals in Lie algebras.

Let $\mathfrak{g}$ be a Leibniz algebra and $E$ a vector space containing $\mathfrak{g}$ as a subspace. All Leibniz algebra structures on $E$ containing $\mathfrak{g}$ as a subalgebra are explicitly described and classified by two non-abelian cohomological type objects: ${\mathcal H}{\mathcal L}^{2}_{\mathfrak{g}} \, (…

Let $\mathfrak{g}$ be a Lie algebra, $E$ a vector space containing $\mathfrak{g}$ as a subspace. The paper is devoted to the \emph{extending structures problem} which asks for the classification of all Lie algebra structures on $E$ such that $\mathfrak{g}$ is a Lie subalgebra of $E$. A general product, called the unifi…

Extends results on marginally outer trapped surfaces to general null expansion.

Homology of torus knots stabilizes to loop space homology.

New stability theorems for H-type Carnot groups established.

Stability of extremal Reissner-Nordström black holes proven in spherical symmetry.

Classifies geodesic vectors in low-dimensional Lie algebras.

In this paper, we consider a connected Riemannian manifold $M$ where a connected Lie group $G$ acts effectively and isometrically. Assume $X\in\mathfrak{g}=\mathrm{Lie}(G)$ defines a bounded Killing vector field, we find some crucial algebraic properties of the decomposition $X=X_r+X_s$ according to a Levi decompositio…

The paper studies dynamical properties in semigroups modulo ideals.

Let $(G,h)$ be a nilpotent Lie group endowed with a left invariant Riemannian metric, $\mathfrak{g}$ its Euclidean Lie algebra and $Z(\mathfrak{g})$ the center of $\mathfrak{g}$. By using an orthonormal basis adapted to the splitting $\mathfrak{g}=(Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}])\oplus O^+\oplus (Z(\mat…

Let $\mathfrak{L}$ be a Leibniz algebra, $E$ a vector space and $π: E \to \mathfrak{L}$ an epimorphism of vector spaces with $\mathfrak{g} = {\rm Ker} (π)$. The global extension problem asks for the classification of all Leibniz algebra structures that can be defined on $E$ such that $π: E \to \mathfrak{L}$ is a morph…

If $\mathfrak{n}$ is a $\mathbb{Z}^d_+$-graded nilpotent finite dimensional Lie algebra over a field of characteristic zero, it is well known that $\dim H^{\ast }(\mathfrak{n})\geq L(p)$ where $p$ is the polynomial associated to the grading and $L(p)$ is the sum of the absolute values of the coefficients of $p$. From …

A subalgebra of a Lie algebra $\mathfrak{h}\subset\mathfrak{g}$ determines $\mathfrak{h}$-representation $ρ$ on $\mathfrak{m}=\mathfrak{g}/\mathfrak{h}$. In this note we discuss how to reconstruct $\mathfrak{g}$ from $(\mathfrak{h},\mathfrak{m},ρ)$. In other words, we find all the ingredients for building non-reductive…

A well known result of Drinfeld classifies Poisson Lie groups $(H,Π)$ in terms of Lie algebraic data in the form of Manin triples $(\mathfrak{d},\mathfrak{g},\mathfrak{h})$; he also classified compatible Poisson structures on $H$-homogeneous spaces $H/K$ in terms of Lagrangian subalgebras $\mathfrak{l}\subset\mathfrak{…

Let ${\mathfrak M}$ be a closed, orientable, hyperbolic 3-orbifold whose singular set is a link, and such that $π_1({\mathfrak M})$ contains no hyperbolic triangle group. We show that if the underlying manifold $|{\mathfrak M}|$ is irreducible, and $|{\mathfrak M}|$ is irreducible for every two-sheeted (orbifold) cover…

We recover the classification of the maximally supersymmetric bosonic backgrounds of eleven-dimensional supergravity by Lie algebraic means. We classify all filtered deformations of the $\mathbb Z$-graded subalgebras $\mathfrak{h}=\mathfrak{h}_{-2}\oplus\mathfrak{h}_{-1}\oplus\mathfrak{h}_{0}$ of the Poincaré superalge…

The paper proves metrizability and dynamics of Weil bundles.

New proof shows Lie algebras are rigid under certain conditions.

Solves classification of compact Clifford-Klein forms for specific Lie groups.

Maps Kähler cones to moduli spaces of stable manifolds.

Study Lie bialgebra structures on flat metric Lie algebras, leading to explicit Poisson-Lie groups.

The paper explores maximal nilpotent complex structures on Lie algebras.

Flat hypercomplex nilmanifolds have a specific solvability property.

The paper finds formulas for flat models of certain Lie algebras.

Quantum groups give lower genus bounds for links.

Extends Kostant's results to symmetric pairs in Clifford algebras.

The paper explores properties of Lie algebra g2 and related geometric structures.

Let ${\mathfrak M}$ be a closed, orientable, hyperbolic 3-orbifold such that $π_1({\mathfrak M})$ contains no hyperbolic triangle group. We show that strict upper bounds of 0.07625, 0.1525 and 0.22875 for ${\rm vol}\ {\mathfrak M}$ imply respective upper bounds of 23, 43 and 79 for $\dim H_1({\mathfrak M};{\mathbb F}_2…

The paper investigates gradings of complex simple Lie algebras, focusing on $|3|$-gradings and their algebraic structures.

Existence and uniqueness of the solution to the discrete Lp Minkowski problem for $\mathfrak{p}$-capacity are proved when $p \geq 1$ and $1<\mathfrak{p}<n$. For general Lp Minkowski problem for $\mathfrak{p}$-capacity, existence and uniqueness of the solution are given when $p \geq 1$ and $1<\mathfrak{p}\le 2$. These r…

For a real, non-singular, 2-step nilpotent Lie algebra $\mathfrak{n}$, the group \Aut(\mathfrak{n})/\Aut_0(\mathfrak{n})$, where$\Aut_0(\mathfrak{n})$ is the group of automorphisms which act trivially on the center, is the direct product of a compact group with the 1-dimensional group of dilations. Maximality of some …

Gradient maps of real reductive group actions on manifolds studied.

Quantum traces map skein algebras to Fock-Goncharov spaces.

Extends knot invariant computation to symmetrically colored sl_N.

We define a homology $\mathcal{H}_N$ for closed braids by applying Khovanov and Rozansky's matrix factorization construction with potential $ax^{N+1}$. Up to a grading shift, $\mathcal{H}_0$ is the HOMFLYPT homology defined in arXiv:math/0505056. We demonstrate that, for $N \geq 1$, $\mathcal{H}_N$ is a $\mathbb{Z}_2\o…

Study unbounded $\mathfrak{sl}_3$-laminations around punctures.

Until a couple of years ago, the only known examples of Lie groups admitting left-invariant metrics with negative Ricci curvature were either solvable or semisimple. We use a general construction from a previous article of the second named author to produce a great amount of examples with compact Levy factor. Given a c…

New model for rational tropical points using $\mathfrak{sp}_4$-webs and measures.

Study extremals on Lie groups with asymmetric polyhedral Finsler structures using Pontryagin's Maximal Principle.