arXiv research

Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.

Study of complex and Hermitian structures on specific Lie groups.

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

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

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct p…

The paper studies a flow on complex Lie groups, showing convergence to solitons.

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

Characterizes hypercomplex Lie groups and their solvmanifolds.

Study on spectral sequence for abelian Lie group actions, with bounds and applications.

The paper classifies invariant structures on complex almost Abelian groups.

Study locally conformally balanced metrics on specific Lie algebras.

The study examines weakly Einstein Lie groups and proves non-existence for certain types.

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…

Lie groups with bi-invariant distance are products of abelian and compact groups.

We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on $\R^8$ which are homogeneous with respect to extensions of Heisenberg type Lie groups. The corresponding hypercomplex structures are of a special kind, called abelian. We prove that on any 2-step nilp…

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

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

We establish a criterion for when an abelian extension of infinite-dimensional Lie algebras integrates to a corresponding Lie group extension $\hat{G}$ of $G$ by $A$, where $G$ is a connected, simply connected Lie group and $A$ is a quotient of its Lie algebra by some discrete subgroup. When $G$ is non-simply connected…

Characterizes symmetric Killing tensors on specific Lie groups.

In the present paper, we describe two geometric notions, holomorphic Norden structures and Kähler-Norden structures on Hom-Lie groups, and prove that on Hom-Lie groups in the left invariant setting, these structures are related to each other. We study Kähler-Norden structures with abelian complex structures and give th…

This note classifies splittable lattices in a specific Lie group.

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…

The paper analyzes Lie symmetries in a specific geometric context.

Discrete connections on abelian Lie groups bundles are studied.

Classifies two-step solvable Lie groups with SKT structures.

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

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

We study the existence of lattices in almost abelian Lie groups that admit left invariant locally conformal Kähler or locally conformal symplectic structures in order to obtain compact solvmanifolds equipped with these geometric structures. In the former case, we show that such lattices exist only in dimension $4$, whi…

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.

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

We introduce and study the notion of the $G$-Tutte polynomial for a list $\mathcal{A}$ of elements in a finitely generated abelian group $Γ$ and an abelian group $G$, which is defined by counting the number of homomorphisms from associated finite abelian groups to $G$. The $G$-Tutte polynomial is a common generalizatio…

Left-invariant metrics force 2-step nilpotent groups, preserving Kähler-like conditions.

The complete affine structures on abelian Lie algebras in small dimensions are well known. In this paper we are interested by the non complete case. In particular we classify all these structures in dimensions 2 and 3.

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

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

We find explicit solutions of the Laplacian coflow of $G_2-$structures on seven-dimensional almost-abelian Lie groups. Moreover, we construct new examples of solitons for the Laplacian coflow which are not eigenforms of the Laplacian and we exhibit a solution, which is not a soliton, having a bounded interval of existe…

We generalize the prequantization central extension of a group of diffeomorphisms preserving a closed 2-form ω(ω-invariant diffeomorphisms) to an abelian extension of a group of diffeomorphisms preserving a closed vector valued 2-form ω, up to a linear isomorphism (ω-equivariant diffeomorphisms). Every abelian extensio…

To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence questio…

The paper generalizes cyclic metrics in homogeneous Finsler geometry.

We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing axioms and smoothness conditions. The smoothness conditions are imposed with respect to a strict Lie 2-group, which plays the …

The note confirms a conjecture for specific Lie groups.

Study Weyl-Einstein structures on conformal solvmanifolds, proving Einstein property and classifying metrics.

The jet bundle $J^kG$ of $k$-jets of curves in a Lie group $G$ has a natural Lie group structure. We present an explicit formula for the group multiplication in the right trivialization and for the group 2-cocycle describing the abelian Lie group extension $\mathfrak{g}\to J^{k}G\to J^{k-1}G$.

Symplectic reduction by abelian subgroups coincides under specific conditions.

Characterizes complex structures on specific Lie groups.

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…

Study characterizes $G_2$-structures on specific Lie groups and identifies harmonic conditions.

Develops Lie algebraic approach for compact complex homogeneous manifolds.