arXiv subjects

Edison Alberto Fernández-Culma

Publications and source records attributed to Edison Alberto Fernández-Culma.

13 recordsLinked to original sources

arXiv2026

Associative Structures in Pseudo-Riemannian Lie Algebras

This paper investigates the algebraic and geometric consequences of the associativity of the symmetric part $U$ of the Levi-Civita connection on a pseudo-Riemannian Lie algebra $(\mathfrak{g}, \langle \cdot, \cdot \rangle)$. We demonstrate that in the Riemannian (positive-definite) setting, the associativity of $U$ is an extremely restrictive condition that forces the tensor to vanish identically, thereby recovering the class of bi-invariant metrics. In contrast, in the pseudo-Riemannian setting, we focus on the subclass where $(\mathfrak{g}, U)$ is associative and unimodular. As a primary result, we establish that every connected Lie group endowed with a left-invariant pseudo-Riemannian metric whose $U$-tensor is associative and unimodular is geodesically complete. Finally, we explore the families of 2-step nilpotent and almost-abelian Lie algebras. For the latter, we obtain some rigid structural classifications, showing that the paradigmatic models are the $3$-dimensional Heisenberg algebra with certain (anti)-Lorentzian metrics or a semi-direct extension involving a nondegenerate infinitesimal $β$-transformation on the canonical neutral space $W \oplus W^*$.

math.DG↗

arXiv2025

On the variety of Lie algebras endowed with complex structures: degenerations and deformations

We study the space of Lie algebras equipped with left-invariant complex structures, $\mathcal{L}_{ J_{\tiny{\mbox{cn}}} }(\mathbb{R}^{2n}) $, with particular attention to their degenerations and deformations. To this end, we identify certain invariants that remain well-behaved under degenerations while preserving the complex structure. These concepts are then applied to the four-dimensional case. Additionally, we explore applications to the study of left-invariant Hermitian structures on Lie groups, and we discuss some aspects of the deformation theory within $ \mathcal{L}_{ J_{\tiny{\mbox{cn}}} }(\mathbb{R}^{2n}) $.

math.RT↗

arXiv2024

On $(α,1,0)$-derivations of anti-commutative algebras

We solve an open problem concerning the well-known $(α,β,γ)$-derivations, proving that the spaces of $(α,1,0)$-derivations of any Lie algebra are isomorphic ($α\neq 0,1$). Also, we prove sharp bounds for the invariants functions defined by such spaces.

math.RA↗

arXiv2024

Extended derivations of algebras

We study the concept of extended derivations of algebras which expands diverse definitions of generalized derivations given in the literature. We concentrate on the family of the anti-commutative algebras and classify such spaces of derivations by considering a suitable notion of equivalence when we restrict attention to such algebras. Afterwards, we investigate a link between the well-known $(α,β,γ)$-derivations and some new extended derivations obtained in the above-mentioned classification. Finally, we present applications of extended derivations to study degenerations of algebras.

math.RA↗

arXiv2023

Classification of orbit closures in the variety of 4-dimensional symplectic Lie algebras

The aim of this paper is to study the natural action of the real symplectic group, $\operatorname{Sp}(4, \mathbb{R})$, on the algebraic set of $4$-dimensional Lie algebras admitting symplectic structures and to give a complete classification of orbit closures. We present some applications of such classification to the study of the Ricci curvature of left-invariant almost Kähler structures on four dimensional Lie groups.

math.RT↗

arXiv2019

On the classification of $3$-dimensional complex hom-Lie algebras

We classify hom-Lie structures with nilpotent twisting map on $3$-dimensional complex Lie algebras, up to isomorphism, and classify all degenerations in such family. The ideas and techniques presented here can be easily extrapolated to study similar problems in other algebraic structures and provide different perspectives from where one can tackle classical open problems of interest in rigid Lie algebras.

math.RA↗

arXiv2017

Anti-Kählerian geometry on Lie groups

Let $G$ be a Lie group of even dimension and let $(g,J)$ be a left invariant anti-Kähler structure on $G$. In this article we study anti-Kähler structures considering the distinguished cases where the complex structure $J$ is abelian or bi-invariant. We find that if $G$ admits a left invariant anti-Kähler structure $(g,J)$ where $J$ is abelian then the Lie algebra of $G$ is unimodular and $(G,g)$ is a flat pseudo-Riemannian manifold. For the second case, we see that for any left invariant metric $g$ for which $J$ is an anti-isometry we obtain that the triple $(G, g, J)$ is an anti-Kähler manifold. Besides, given a left invariant anti-Hermitian structure on $G$ we associate a covariant $3$-tensor $θ$ on its Lie algebra and prove that such structure is anti-Kähler if and only if $θ$ is a skew-symmetric and pure tensor. From this tensor we classify the real 4-dimensional Lie algebras for which the corresponding Lie group has a left invariant anti-Kähler structure and study the moduli spaces of such structures (up to group isomorphisms that preserve the anti-Kähler structures).

math.DG↗

arXiv2015

Interpolation of geometric structures compatible with a pseudo Riemannian metric

Let (M, g) be a pseudo Riemannian manifold. We consider four geometric structures on M compatible with g: two almost complex and two almost product structures satisfying additionally certain integrability conditions. For instance, if r is a product structure and symmetric with respect to g, then r induces a pseudo Riemannian product structure on M. Sometimes the integrability condition is expressed by the closedness of an associated two-form: if j is almost complex on M and ω(x, y) = g(jx, y) is symplectic, then M is almost pseudo Kähler. Now, product, complex and symplectic structures on M are trivial examples of generalized (para)complex structures in the sense of Hitchin. We use the latter in order to define the notion of interpolation of geometric structures compatible with g. We also compute the typical fibers of the twistor bundles of the new structures and give examples for M a Lie group with a left invariant metric.

math.DG↗

arXiv2013

Classification of Nilsoliton metrics in dimension seven

The aim of this paper is to classify Ricci soliton metrics on $7$-dimensional nilpotent Lie groups. It can be considered as a continuation of our paper in [Transformation Groups, Volume 17, Number 3 (2012), 639--656]. To this end, we use the classification of $7$-dimensional real nilpotent Lie algebras given by Ming-Peng Gong in his Ph.D thesis and some techniques from the results of Michael Jablonski in [Rocky Mtn. Journal of Math. 42 (2012), 1521--1549] and [Münster J. Math. 3 (2010), 67--88], and of Yuri Nikolayevsky in [Trans. Amer. Math. Soc. 363 (2011), 3935--3958]. Of the $9$ one-parameter families and $140$ isolated $7$-dimensional indecomposable real nilpotent Lie algebras, we have $99$ nilsoliton metrics given in an explicit form and $7$ one-parameter families admitting nilsoliton metrics. Our classification is the result of a case-by-case analysis, so many illustrative examples are carefully developed to explain how to obtain the main result.

math.DG↗

arXiv2013

Soliton almost Kähler structures on 6-dimensional nilmanifolds for the symplectic curvature flow

The aim of this paper is to study self-similar solutions to the symplectic cuvature flow on 6-dimensional nilmanifolds. For this purpose, we focus our attention in the family of symplectic Two- and Three-step nilpotent Lie algebras admitting a "minimal compatible metric" and we give a complete classification of these algebras together with their respective metric. Such classification is given by using our generalization of Nikolayevsky's nice basis criterium (arXiv:1301.4949), which will be repeated here in the context of canonical compatible metrics for geometric structures on nilmanifolds, for the convenience of the reader. By computing the Chern-Ricci operator $P$ in each case, we show that the above distinguished metrics define a soliton almost Kähler structure. Many illustrative examples are carefully developed.

math.DG↗

arXiv2013

Classification of 7-dimensional Einstein nilradicals II

This paper contains all computations supporting the classification of 7-dimensional Einstein nilradicals given in the article "Classification of 7-dimensional Einstein nilradicals" (arXiv). Each algebra is analyzed in detail here.

math.DG↗

arXiv2013

Classification of 7-dimensional Einstein nilradicals

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 moment map. In dimension 7, there are 148 complex nilpotent Lie algebras and 6 curves of pairwise non-isomorphic nilpotent Lie algebras, and we give in this paper a complete classification of the aforementioned distinguished orbits.

math.DG↗

arXiv2013

On distinguished orbits of reductive representations

Let $G$ be a real reductive Lie group and $τ:G \longrightarrow GL(V)$ be a real reductive representation of $G$ with (restricted) moment map $m_{\ggo}: V-{0} \longrightarrow \ggo$. In this work, we introduce the notion of "nice space" of a real reductive representation to study the problem of how to determine if a $G$-orbit is "distinguished" (i.e. it contains a critical point of the norm squared of $m_{\ggo}$). We give an elementary proof of the well-known convexity theorem of Atiyah-Guillemin-Sternberg in our particular case and we use it to give an easy-to-check sufficient condition for a $G$-orbit of a element in a nice space to be distinguished. In the case where $G$ is algebraic and $τ$ is a rational representation, the above condition is also necessary (making heavy use of recent results of M. Jablonski), obtaining a generalization of Nikolayevsky's nice basis criterium. We also provide useful characterizations of nice spaces in terms of the weights of $τ$. Finally, some applications to ternary forms and minimal metrics on nilmanifolds are presented.

math.RT↗