arXiv research

Lie algebroids and curved Lie algebras are equivalent categories.

Study on complex curves in hypercomplex nilmanifolds with quaternionic-solvable Lie algebras.

A Lie algebra structure on variation vector fields along an immersed curve in a $2$-dimensional real space form is investigated. This Lie algebra particularized to plane curves is the cornerstone in order to define a Hamiltonian structure for plane curve motions. The Hamiltonian form and the integrability of the planar…

The paper extends a geometric model using singular curves.

The paper extends Heintze-Kobayashi-Wolf theory to negatively curved homogeneous Finsler manifolds.

The only known examples of noncompact Einstein homogeneous spaces are standard solvmanifolds (special solvable Lie groups endowed with a left invariant metric), and according to a long standing conjecture, they might be all. The classification of Einstein solvmanifolds is equivalent to the one of Einstein nilradicals, …

A Lie bracket defined on the linear span of the free homotopy classes of undirected closed curves was discovered in stages passing through Thurston's earthquake deformations, Wolpert's corresponding calculations with Hamiltonian vector fields and Goldman's algebraic treatment of the latter leading to a Lie bracket on t…

Paper constructs connections on curves with specific Galois groups.

Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an `evolution operator' exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisi…

Geometric proof of curve characterization using loop-bundles.

We consider the Lie algebra consisting of all derivations on the free associative algebra, generated by the first homology group of a closed oriented surface, which kill the symplectic class. We find the first non-trivial abelianization of this Lie algebra and discuss its relation to unstable cohomology classes of the …

Geometrically deforms $L_\infty$ algebras to Lie algebroids, revealing new invariants.

Lectures explore how differential methods improve understanding of algebraic group orbit spaces.

Classifies geodesic vectors in low-dimensional Lie algebras.

The paper studies the center of the Goldman Lie algebra and its properties.

Abstract: Generalizes prequantization map for 2-plectic manifolds.

The paper extends a theorem to Lie-Rinehart algebras and provides new decompositions of universal enveloping algebras.

We classify extremal curves in free nilpotent Lie groups. The classification is obtained via an explicit integration of the adjoint equation in Pontryagin Maximum Principle. It turns out that abnormal extremals are precisely the horizontal curves contained in algebraic varieties of a specific type. We also extend the r…

Introduces new construction for Courant algebroids and curved structures.

The paper studies automorphisms of generalized Kähler manifolds and their Lie algebras.

Covariance is shown as a commutator in random variable calculus.

We apply the Nash-Moser theorem for exact sequences of R. Hamilton to the context of deformations of Lie algebras and we discuss some aspects of the scope of this theorem in connection with the polynomial ideal associated to the variety of nilpotent Lie algebras. This allows us to introduce the space $H_{k-nil}^2(\math…

Contact Lie systems analyze integral curves of Hamiltonian vector fields.

A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie algebra. This work pioneers the analysis of Lie systems admitting a Vessiot--Guldberg …

Classifies hexagonal circular 3-webs with cubic polar curves.

We introduce a Lie algebra associated with a non-orientable surface, which is an analogue for the Goldman Lie algebra of an oriented surface. As an application, we deduce an explicit formula of the Dehn twist along an annulus simple closed curve on the surface as in Kawazumi-Kuno and Masseyeau-Turaev.

The problem of classifying Einstein solvmanifolds, or equivalently, Ricci soliton nilmanifolds, is known to be equivalent to a question on the variety of n-dimensional complex nilpotent Lie algebra laws. Namely, one has to determine which GL(n)-orbits in this variety have a critical point of the squared norm of the mom…

We construct a new grading on the Goldman Lie algebra of a closed oriented surface by the winding number. This grading induces a grading on the HOMFLY-PT skein algebra and related algebras. Our work supports the conjectures of B. Cooper and P. Samuelson

A new mathematical approach to general covariance using stacks and Lie algebras.

Study Lie algebroid connections on principal bundles over complex projective varieties.

Decomposes Goldman-Turaev Lie bialgebra via cutting a surface.

A Lie system is a system of first-order ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a so-called Vessiot-Guldberg Lie algebra. We suggest the definition of a particular class of Lie systems, the $k…

Study the Lax equation in infinite-dimensional Lie algebras and Lie groups.

In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) Condition), which means that in each tangent plane, we can find a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact …

New Lie group approach for envelope surface computation.

To each non-isotropic almost-complex immersion of a 2-torus into $S ^ 6$ we associate an algebraic curve, called the spectral curve, and a linear flow in the intersection of two Prym varieties on this spectral curve. We show that generically the spectral curve is smooth and compute the dimension of the moduli space o…

We prove that one can obtain natural bundles of Lie algebras on rank two s-Kähler manifolds, whose fibres are isomorphic to so(s+1,s+1), su(s+1,s+1) and sl(2s + 2,\R). In the most rigid case (which includes complex tori and abelian varieties) these bundles have natural flat connections, whose flat global sections act n…

Reduces multisymplectic Lie systems through symmetry analysis.

Reconstructing signature features from randomized vector fields in differential equations.

The study connects conic connections and torsion-free principal connections on G-structures.

We propose a generalization of tropical curves by dropping the rationality and integrality requirements while preserving the balancing condition. An interpretation of such curves as critical points of a certain quadratic functional allows us to settle the existence and uniqueness problem. The machinery of dual polygons…

The paper constructs a new Ricci-flat metric on almost abelian Lie groups.

Given compact Lie groups H\subset G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K,H) is a …

The Bergman kernel and period map for curves are studied.

We represent algebraic curves via commuting matrix polynomials. This allows us to show that the Hilbert scheme of cohomologically stable twisted rational curves of degree $d$ in ${\Bbb P}^3\backslash {\Bbb P}^1$ is isomorphic to a complexified hyperkähler quotient of an open subset of a vector space by a non-reductive …

We construct an explicit bundle with flat connection on the configuration space of n points of a complex curve. This enables one to recover the `formality' isomorphism between the Lie algebra of the prounipotent completion of the pure braid group of n points on a surface and an explicitly presented Lie algebra t_{g,n} …

A stratified Lie system is a nonautonomous system of first-order ordinary differential equations on a manifold $M$ described by a $t$-dependent vector field $X=\sum_{α=1}^rg_αX_α$, where $X_1,\ldots,X_r$ are vector fields on $M$ spanning an $r$-dimensional Lie algebra that are tangent to the strata of a stratification …

Geometry of holomorphic curves from point of view of open Toda systems is discussed. Parametrization of curves related this way to non-exceptional simple Lie algebras is given. This gives rise to explicit formulas for minimal surfaces in real, complex and quaternionic projective spaces or complex quadrics. The paper ge…