arXiv research

Study locally conformally balanced metrics on specific Lie algebras.

Study transcendence of abelian differential periods from bi-algebraic perspective.

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

Quantum trace maps abelian character varieties to SL2 character varieties.

Paper confirms conjecture for specific Lie algebras.

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

Study abelian factors in Lie algebras from graph edge labels.

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

Study pseudo-Kähler structures on almost abelian solvmanifolds.

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

Study on Hermitian metrics on Lie algebras with specific 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 presymplectic forms on almost abelian Lie algebras, determining moduli spaces and their finiteness.

Unique complex structures on specific Lie algebras.

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

Characterizes almost abelian Lie algebras with integrable complex structure

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…

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.

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…

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

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

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

Computes the component group of arbitrary real algebraic groups.

This thesis explores algebraic cycles and moduli spaces over real numbers.

Introduces non-abelian Hodge correspondence linking algebraic structures to geometry.

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…

Extends Chern character to non-abelian cohomology, linking to physics.

For every fibration $f : X \to B$ with $X$ a compact Kähler manifold, $B$ a smooth projective curve, and a general fiber of $f$ an abelian variety, we prove that $f$ has an algebraic approximation.

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…

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

Classifies two-step solvable Lie groups with SKT structures.

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

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

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

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. …

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 …

In this article, we determine the seven-dimensional almost Abelian Lie algebras which admit calibrated or parallel G_2-/G_2^*-structures. Along the way, we show that certain well-established curvature restrictions for calibrated and parallel G_2-structures are not valid in the G_2^* case. In more detail, we provide the…

The Streets-Tian conjecture is confirmed for Lie algebras with specific abelian ideals.

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.

Develops Lie algebraic approach for compact complex homogeneous manifolds.

We study the abelianization of Kontsevich's Lie algebra associated with the Lie operad and some related problems. Calculating the abelianization is a long-standing unsolved problem, which is important in at least two different contexts: constructing cohomology classes in $H^k(\mathrm{Out}(F_r);\mathbb Q)$ and related g…

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…

We study Lie algebras endowed with an abelian complex structure which admit a symplectic form compatible with the complex structure. We prove that each of those Lie algebras is completely determined by a pair (U,H) where U is a complex commutative associative algebra and H is a sesquilinear hermitian form on U which ve…

Local classification of 4D Ricci solitons with specific algebra properties.

We review recent works concerning deformation quantization of abelian supergroups. Indeed, we expose the construction of an induced representation of the Heisenberg supergroup and an associated pseudodifferential calculus by using Kirillov's orbits method. Then, a star-product is built on the abelian supergroup R^{m|n}…

The paper analyzes Lie symmetries in a specific geometric context.