arXiv research

Classifies complex structures on specific nilpotent Lie algebras.

New findings on complex structures in nilpotent Lie algebras and pseudo-Kähler geometry.

We study complex product structures on nilpotent Lie algebras, establishing some of their main properties, and then we restrict ourselves to 6 dimensions, obtaining the classification of 6-dimensional nilpotent Lie algebras admitting such structures. We prove that any complex structure which forms part of a complex pro…

The paper characterizes and examines nilpotent complex structures on stratified Lie algebras.

This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Kählerian complex structures on six-dimensional nilpotent algebras such that the differential Gerstenhaber a…

Study para-complex structures on specific Lie groups, finding explicit forms and properties.

Proves a complex structure conjecture for a specific type of Lie groups.

Characterizes complex structures on specific Lie groups.

Study on existence of $p$-Kähler structures on nilmanifolds with nilpotent complex structures.

The paper explores maximal nilpotent complex structures on Lie algebras.

We classify real 6-dimensional nilpotent Lie algebras for which the corresponding Lie group has a left-invariant complex structure, and estimate the dimensions of moduli spaces of such structures.

CR embeddings in complex spaces for specific Lie groups.

Study on pseudo-Hermitian quadratic nilpotent Lie algebras with methods and classifications.

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.

Unique complex structures on specific Lie algebras.

Study of complex structures on specific solvmanifolds, proving existence and non-existence results.

Study semi-Kähler structures on specific Lie groups without symplectic structures.

Abstract classifies Lie algebras with complex or symplectic structures.

The paper studies nilpotent structures in oriented neutral vector bundles and neutral hyperKähler structures.

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

We investigate Lie algebras endowed with a complex symplectic structure and develop a method, called \emph{complex symplectic oxidation}, to construct certain complex symplectic Lie algebras of dimension $4n+4$ from those of dimension $4n$. We specialize this construction to the nilpotent case and apply complex symplec…

We obtain several restrictions on the terms of the ascending central series of a nilpotent Lie algebra $\mathfrak g$ under the presence of a complex structure $J$. In particular, we find a bound for the dimension of the center of $\mathfrak g$ when it does not contain any non-trivial $J$-invariant ideal. Thanks to thes…

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

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

Holomorphic actions on complex spaces for nilpotent groups.

This study explores complex structures on Lie algebras from graph perspectives.

We prove that no nilpotent Lie algebra admits an invariant generalized Kaehler structure. This is done by showing that a certain differential graded algebra associated to a generalized complex manifold is formal in the generalized Kaehler case, while it is never formal for a generalized complex structure on a nilpotent…

It is known that there are 34 classes of isomorphic connected simply connected six-dimensional nilpotent Lie groups. Of these, only 26 classes suppose left-invariant symplectic structures \cite{Goze-Khakim-Med}. In \cite{CFU2} it is shown that 14 classes of symplectic six-dimensional nilpotent Lie groups suppose compat…

Higher nilpotent analogues of the $A-\infty$-structure are explicitly defined on arbitrary simplicial complexes, generalizing explicit construction of /hep-th/0704.2609. These structures are associated with the higher nilpotent differential $d_n$, satisfying $d_n^n =0$, which is naturally defined on triangulated manifo…

We find a one-parameter family of non-isomorphic nilpotent Lie algebras $\mathfrak{g}_a$, with $a \in [0,\infty)$, of real dimension eight with (strongly non-nilpotent) complex structures. By restricting $a$ to take rational values, we arrive at the existence of infinitely many real homotopy types of $8$-dimensional ni…

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

Let (N,J) be a real 2n-dimensional nilpotent Lie group endowed with an invariant complex structure. A left-invariant Riemannian metric on N compatible with J is said to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics on (N,J) with the same scalar curvature. In…

For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kaehler space. This notion es…

We classify non-nilpotent complex structures on 6-nilmanifolds and their associated invariant balanced metrics. As an application we find a large family of solutions of the heterotic supersymmetry equations with non-zero flux, non-flat instanton and constant dilaton satisfying the anomaly cancellation condition with re…

The aim of this work is to prove the nonexistence of complex structures over nilpotent Lie algebras of maximal class (also called filiform).

There are five six-dimensional nilpotent Lie groups G, which do not admit neither symplectic, nor complex structures and, therefore, can be neither almost pseudo-Kahler, nor almost Hermitian. In this work, these Lie groups are being studied. The aim of the paper is to define new left-invariant geometric structures on t…

We give a necessary and sufficient condition on the 1-jet of a field of nilpotent endomorphisms to be integrable. Together with the well known corresponding condition for an almost complex structure, the nullity of its Nijenhuis tensor, this gives an integrability condition for any field of endomorphisms.

The paper explores SKT, balanced, and generalized Kähler structures on specific Lie groups.

We present structural properties of Lie algebras admitting symmetric, invariant and nondegenerate bilinear forms. We show that these properties are not satisfied by nilradicals of parabolic subalgebras of real split forms of complex simple Lie algebras, neither by 2-step nilpotent Lie algebras associated with graphs, w…

New definition of Rumin complex for nilpotent Lie groups.

In this work we introduce an obstruction for the existence of symplectic structures on nilpotent Lie algebras. Indeed, a necessary condition is presented in terms of the cohomology of the Lie algebra. Using this obstruction we obtain both positive and negative results on the existence of symplectic structures on a larg…

The paper studies gradings on nilpotent Lie algebras linked to smooth algebraic varieties.

Completes classification of G2-structures on specific nilpotent Lie groups.

This paper completes the classification of certain nilpotent Lie groups with specific geometric structures.

Let $(N, J)$ be a simply connected $2n$-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on $N$ compatible with $J$ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar…

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

We study evolution of (strong Kähler with torsion) SKT structures via the pluriclosed flow on complex nilmanifolds, i.e. on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for stu…

Study of curvature flow on specific Lie groups, leading to soliton solutions.