arXiv research

Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean Lie algebras.

Develops a bialgebra theory for post-Lie algebras using geometric interpretations and bilinear forms.

In this paper, we present a study on the prolongations of representations of Lie algebras. We show that a tangent bundle of a given Lie algebra attains a Lie algebra structure. Then, we prove that this tangent bundle is algebraically isomorphic to the Lie algebra of a tangent bundle of a Lie group. Using these, we defi…

A Lie-admissible algebra gives by anticommutativity a Lie algebra. In this work we study remarkable classes of Lie-admissible algebras such as Vinberg, PreLie algebras. We compute the corresponding binary quadratic operads and study their Koszul duality. Considering Lie algebras as Lie-admissible algebras we can define…

The paper generalizes para-Kähler Lie algebras to k-para-Kähler Lie algebras and explores their structures.

We introduce post-Lie algebra structures on pairs of Lie algebras $(\Lg,\Ln)$ defined on a fixed vector space $V$. Special cases are LR-structures and pre-Lie algebra structures on Lie algebras. We show that post-Lie algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtai…

Study on pre-Lie structures for semisimple Lie algebras over C.

We construct the Lie algebra of an n-Lie algebra and we also define the notion of cohomology of an n-Lie algebra.

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. …

Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration proce…

Lie algebroids and curved Lie algebras are equivalent categories.

In this paper, we introduce the notion of a (regular) Hom-Lie group. We associate a Hom-Lie algebra to a Hom-Lie group and show that every regular Hom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of …

A pseudo $H$-type Lie algebra naturally gives rise to a conformal pseudo-subriemannian fundamental graded Lie algebras. In this paper we investigate the prolongations of the associated fundamental graded Lie algebra and the associated conformal pseudo-subriemannian fundamental graded Lie algebra. In particular, we show…

If a Lie algebra structure g on a vector space is the sum of a family of mutually compatible Lie algebra structures g_i's, we say that g is simply assembled from the g_i's. Repeating this procedure with a number of Lie algebras, themselves simply assembled from the g_i's, one obtains a Lie algebra assembled in two step…

New Lie algebras from knot homology.

A new category of Lie algebras, called generalized Lie algebras, is presented such that classical Lie algebras and Lie-Rinehart algebras are objects of this new category. A new philosophy over generalized Lie algebroids theory is presented using the notion of generalized Lie algebra and examples of objects of the categ…

Symmetric spaces' connections form Lie admissible triple algebras.

Proofs centerless unimodular contact Lie algebras.

Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and interesting results are obtained. We give prominence to the fact that the Lie groups of automorphisms of cotangent bundles of Lie groups are super symmetric Lie groups. In the cases of orthogonal Lie lgebras, semi-simple Lie…

Characterizes G2-structures on Lie algebras with non-trivial center.

The paper connects Lie bialgebras, Rota-Baxter Lie algebras, and their properties.

Study on generalized derivations in polynomial vector fields Lie algebras.

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

Born Lie algebras classified up to 6D, with integrable metrics studied.

Study cohomology of hemistrict Lie 2-algebras, proving isomorphic results.

Study locally conformally balanced metrics on specific Lie algebras.

Abstract: Connections between Lie algebras and symplectic nilmanifolds explored.

The paper studies the center of the Goldman Lie algebra and its properties.

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

This research introduces Lie brackets on spaces of biderivations in Lie algebras.

2-compatible Lie algebras are quadratic deformations of Lie algebras with specific constraints.

Study on pseudo-Riemannian algebraic Ricci solitons in 4D Lie groups.

We study post-Lie algebra structures on pairs of Lie algebras (g,n), and prove existence results for the case that one of the Lie algebras is semisimple. For semisimple g and solvable n we show that there exist no post-Lie algebra structures on (g,n). For semisimple n and certain solvable g we construct canonical post-…

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…

We study possible cases of complex simple graded Lie algebras of depth 2, which are the Tanaka prolongations of pseudo $H$-type Lie algebras arising through representation of Clifford algebras. We show that the complex simple Lie algebras of type $B_n$ with $|2|$-grading do not contain non-Heisenberg pseudo $H$-type Li…

The paper classifies Lie algebras with special operators.

The paper studies nice bases for Lie algebras and their properties.

Starting with Lie's classification of finite-dimensional transitive Lie algebras of vector fields on $\mathbb C^2$ we construct Lie algebras of vector fields on the bundle $\mathbb C^2 \times \mathbb C$ by lifting the Lie algebras from the base. There are essentially three types of transitive lifts and we compute all o…

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…

We extend the classical characterization of a finite-dimensional Lie algebra g in terms of its Maurer-Cartan algebra-the familiar differential graded algebra of alternating forms on g with values in the ground field, endowed with the standard Lie algebra cohomology operator-to sh Lie-Rinehart algebras. To this end, we …

The study describes the free Lie-Yamaguti algebra.

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…

A flat pseudo-Euclidean Lie algebra is a real Lie algebra with a non degenerate symmetric bilinear form and a left symmetric product whose the commutator is the Lie bracket and such that the left multiplications are skew-symmetric. We show that the center of a flat pseudo-Euclidean nilpotent Lie algebra of signature $(…

Let g be a simplicial Lie algebra with Moore complex Ng of length k. Let G be the simplicial Lie group integrating g, which is simply connected in each simplicial level. We use the 1-jet of the classifying space of G to construct, starting from g, a Lie k-algebra L. The so constructed Lie k-algebra L is actually a diff…

Study SKT and Kähler structures on specific Lie algebras.

The study examines extensions of Lie algebras with specific geometric structures.

If a Lie algebra structures $\gG$ on a vector space is the sum of a family of mutually compatible Lie algebra structures $\gG_i$, we say that $\gG$ is \emph{simply assembled} from $\gG_s$'s. By repeating this procedure several times one gets a family of Lie algebras \emph{assembled} from $\gG_s$'s. The central result o…

The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related: we prove that a linear Poisson structure on the dual of a Lie algebra has a com…