arXiv research

Classifies Spin(7) structures using spinorial equations.

Study on a specific type of Lie algebras with Kähler and contact properties.

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

Study locally conformally balanced metrics on specific Lie algebras.

We define the C^*-action on moduli spaces of reductive representations of fundamental groups of quasi-compact Kaehler manifolds by solving Hermitian-Yang-Mills equation. As applications in algebraic geometry we show a non-abelian Hodge (p,q)-type theorem for families of quasi-projective manifolds. We also prove that an…

We study the natural map eta between a group of binary planar trees whose leaves are labeled by elements of a free abelian group H and a certain group D(H) derived from the free Lie algebra over H. Both of these groups arise in several different topological contexts. The map eta is known to be an isomorphism over Q, bu…

Study on CKY forms on almost abelian Lie groups, proving parallelism and characterizing non-parallel cases.

The paper generalizes the number of complex structures on metric Lie algebras.

Study abelian factors in Lie algebras from graph edge labels.

Paper confirms conjecture for specific Lie algebras.

Classifies complex symplectic structures on Lie algebras with large abelian ideals.

We obtain a characterization of the real Lie algebras admitting abelian complex structures in terms of certain affine Lie algebras $\frak a \frak f \frak f (A)$, where $A$ is a commutative algebra. These affine Lie algebras are natural generalizations of $\frak a \frak f \frak f (\Bbb C)$ and the corresponding Lie grou…

Study of abelian structures on odd-dimensional Lie algebras and their geometric properties.

Study on Hermitian metrics on Lie algebras with specific ideals.

A Lie version of Turaev's $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a \textit{$\frak{g}$-quasi-Frobenius Lie algebra} for $\frak{g}$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius…

Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.

Study pseudo-Kähler structures on almost abelian solvmanifolds.

New Lie 2-algebra structure for multiplicative forms on quasi-Poisson groupoids.

This paper classifies LCSKT almost abelian Lie algebras in 6 dimensions.

Study of presymplectic forms on almost abelian Lie algebras, determining moduli spaces and their finiteness.

Study coKähler structures on Lie algebras using Fino-Vezzoni correspondence.

Unique complex structures on specific Lie algebras.

Characterizes almost abelian Lie algebras with integrable complex structure

New methods for solving hydrodynamic-type equations using quasi-rectifiable Lie algebras.

We construct an abelian quotient of the symplectic derivation Lie algebra $\mathfrak{h}_{g,1}$ of the free Lie algebra generated by the fundamental representation of $\mathrm{Sp}(2g,\mathbb{Q})$. More specifically, we show that the weight $12$ part of the abelianization of $\mathfrak{h}_{g,1}$ is $1$-dimensional for $g…

Resolves conjectures on non-abelian Hodge loci for quasi-projective varieties.

Classifies Lie symmetry algebras for 2D quasilinear equations, linking symmetry to linearity.

The paper classifies hypercomplex Lie algebras and solvmanifolds using quaternionic Jordan form.

The study explores Lie superalgebras constructed from Lie algebras using Schouten-like brackets.

The aim of this note is to introduce the notion of a $\operatorname{D}$-Lie algebra and to prove some elementary properties of $\operatorname{D}$-Lie algebras, the category of $\operatorname{D}$-Lie algebras, the category of modules on a $\operatorname{D}$-Lie algebra and extensions of $\operatorname{D}$-Lie algebras. …

For finite dimensional real Lie algebras, we investigate the existence of an inner product having a basis comprised of geodesic elements. We give several existence and non-existence results in certain cases: unimodular solvable Lie algebras having an abelian nilradical, algebras having an abelian derived algebra, algeb…

We introduce hom-Lie-Rinehart algebras as an algebraic analogue of hom-Lie algebroids, and systematically describe a cohomology complex by considering coefficient modules. We define the notion of extensions for hom-Lie-Rinehart algebras. In the sequel, we deduce a characterisation of low dimensional cohomology spaces i…

We classify the 6-dimensional Lie algebras that can be endowed with an abelian complex structure and parameterize, on each of these algebras, the space of such structures up to holomorphic isomorphism.

The paper explores geometric structures on Hom-Lie groups and algebras.

We present the classification of real nilpotent quasi-filiform Lie algebras endowed with a complex structure. A nilpotent Lie algebra g is called quasi-filiform is the nilindex is equal to dim(n)-2. We recall that the filiform case (nilindex =dim(g)-1) has already been studied.

It is shown that every abelian regular Lie group is a quotient of its Lie algebra via the exponential mapping.

Let K be a compact Lie group. We compute the abelianization of the Lie algebra of equivariant vector fields on a smooth K-manifold X. We also compute the abelianization of the Lie algebra of strata preserving smooth vector fields on the quotient X/K.

The paper analyzes Lie symmetries in a specific geometric context.

It was shown by Samelson and Wang that each compact Lie group K of even dimension admits left-invariant complex structures. When K has odd dimension it admits a left-invariant CR-structure of maximal dimension. This has been proved recently by Charbonnel and Khalgui who have also given a complete algebraic description …

Cocalibrated G_2-structures and cocalibrated G_2^*-structures are the natural initial values for Hitchin's evolution equations whose solutions define (pseudo)-Riemannian manifolds with holonomy group contained in Spin(7) or Spin_0(3,4), respectively. In this article, we classify which seven-dimensional real Lie algebra…

We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra and symplectic derivations of the free associative algebra. In each case, we consider both the whole derivation Lie algebra and …

Classifies two-step solvable Lie groups with SKT structures.

Develops Lie algebraic approach for compact complex homogeneous manifolds.

Characterizes almost Abelian Lie algebras with special $H$-structures.

We associate Hamiltonian homological evolutionary vector fields --which are the non-Abelian variational Lie algebroids' differentials-- with Lie algebra-valued zero-curvature representations for partial differential equations.

New algebraic structure derived from Kähler manifolds.

Algebraic dimension is zero for generic complex structures on hypercomplex nilmanifolds.

We present an explicit realization of abelian extensions of infinite dimensional Lie groups using abelian extensions of path groups, by generalizing Mickelsson's approach to loop groups and the approach of Losev-Moore-Nekrasov-Shatashvili to current groups. We apply our method to coupled cocycles on current Lie algebra…