arXiv research

Classifies Killing forms of arbitrary degree on specific nilpotent Lie groups.

The paper studies Riemannian metrics on tangent Lie groups using two left-invariant metrics.

Survey on invariant conformal Killing forms on Lie groups.

New method classifies symplectic structures on Lie groups.

Study finds only complex 2-step nilpotent Lie groups have Killing-Yano forms.

Computational techniques calculate dimensions of complex structures.

We study left invariant contact forms and left invariant symplectic forms on Lie groups. We give the classification of all symplectic structures on nilpotent Lie algebras up the dimension 6.

A $n$-dimensional Lie group $G$ equipped with a left invariant symplectic form $\om^+$ is called a symplectic Lie group. It is well-known that $\om^+$ induces a left invariant affine structure on $G$. Relatively to this affine structure we show that the left invariant Poisson tensor $π^+$ corresponding to $\om^+$ is po…

We use the theory of dual of Fréchet-Schwartz (DFS) spaces to establish a sufficient condition for top-degree solvability for the differential complex associated to a hypocomplex locally integrable structure. As an application, we show that the top-degree cohomology of left-invariant hypocomplex structures on a compact…

Study finds optimal loops in hyperbolic space with Finsler structure.

We are interested in the class, in the Elie Cartan sense, of left invariant forms on a Lie group. We construct the class of Lie algebras provided with a contact form and classify the frobeniusian Lie algebras up to a contraction. We also study forms which are invariant by a subgroup. We show that the simple group SL(2n…

Study Killing forms on 2-step nilpotent Lie groups, finding their structure and dimensions.

Study Dolbeault harmonic forms on Lie group quotients with specific structures.

We consider nilmanifolds with left-invariant complex structure and prove that small deformations of such structures are again left invariant if the Dolbeault-cohomology of the nilmanifold can be calculated using left-invariant forms. By a result of Console and Fino this is generically the case. Our main tool is an anal…

We study the geometry of a family of Lie groups, which contained the classical affine Lie groups, endowed with an exact left invariant symplectic form. We show that this family is closed by symplectic reduction and symplectic double extension in the sense of Dardié and Medina. We prouve also that these groups are endow…

Study conformal Killing forms on specific nilpotent Lie groups.

Study odd generalized Einstein metrics on 3D Lie groups.

The paper finds a contact form on SL(2p) for p > 1.

In this paper, we investigate left-invariant geodesic orbit metrics on connected simple Lie groups, where the metrics are formed by the structures of generalized flag manifolds. We prove that all these left-invariant geodesic orbit metrics on simple Lie groups are naturally reductive.

A special symplectic Lie group is a triple $(G,ω,\nabla)$ such that $G$ is a finite-dimensional real Lie group and $ω$ is a left invariant symplectic form on $G$ which is parallel with respect to a left invariant affine structure $\nabla$. In this paper starting from a special symplectic Lie group we show how to ``defo…

The paper classifies invariant structures on complex almost Abelian groups.

Study invariant CKY 2-forms on 5D Lie groups, classifying and determining their properties.

This work reviews left-invariant optimal control problems on Lie groups.

We extend the Nambu bracket to 1-forms. Following the Poisson-Lie case, we define Nambu-Lie groups as Lie groups endowed with a multiplicative Nambu structure. A Lie group G with a Nambu structure P is a Nambu-Lie group iff P=0 at the unit and the Nambu bracket of left (right) invariant forms is left (right) invariant.…

The paper constructs a family of SKT metrics on the exceptional Lie group G2.

The paper describes metrics on left Leibniz algebras, linking them to quadratic Lie algebras.

We provide a classification of $ts$-invariant sub-Lorentzian structures on $3$ dimensional contact Lie groups. Our approach is based on invariants arising form the construction of a normal Cartan connection.

In this work we deal with left invariant complex and symplectic structures on simply connected four dimensional solvable real Lie groups. We search the general form of such structures, when they exist and we make use of this information to determine all left invariant Kaehler structures. Finally, as an appendix we comp…

We study Lie algebras admitting para-Kähler and hyper-para-Kähler structures. We give new characterizations of these Lie algebras and we develop many methods to build large classes of examples. Bai considered para-Kähler Lie algebras as left symmetric bialgebras. We reconsider this point of view and improve it in order…

Study of zero-divisors in sedenions via determinant factorization.

Two specific Einstein metrics found on a product of SL(2,R) groups.

The study lists low-dimensional stratified groups and their properties.

The paper studies conjugate points on Lie groups with specific metrics.

Study on vector fields on Lie groups reveals surprising algebraic coincidences.

For a connected Lie group G, we show that a complex structure on the total space TG of the tangent bundle of G that is left invariant and has the property that each left translation G-orbit is a totally real submanifold is induced from a smooth immersion of TG into the complexification of G. For G compact and connected…

Study of Hermitian and Gauduchon connections on Lie groups with almost Hermitian structures.

The real Jacobi group $G^J_1(\mathbb{R})$, defined as the semi-direct product of the group ${\rm SL}(2,\mathbb{R})$ with the Heisenberg group $H_1$, is embedded in a $4\times 4$ matrix realisation of the group ${\rm Sp}(2,\mathbb{R})$. The left-invariant one-forms on $G^J_1(\mathbb{R})$ and their dual orthogonal left-i…

Study of complex and Hermitian structures on specific Lie groups.

The main aim of this paper is the description of a large class of lattices in some nilpotent Lie groups, sometimes filiformes, carrying a flat left invariant linear connection anf often a left invariant symplectic form. As a consequence we obtain an infinity of, non homeomorphic, compact affine or symplectic manifolds.…

The paper studies invariants of almost complex and almost Kähler manifolds, proving properties and relationships between them.

Classifies nilpotent Lie groups with specific $G_2$-structures.

Proves cohomology of elliptic structures on Lie groups can be algebraic.

We study 4-dimensional simply connected Lie groups $G$ with left-invariant Riemannian metric $g$ admitting non-trivial conformal Killing 2-forms. We show that either the real line defined by such a form is invariant under the group action, or the metric is half conformally flat. In the first case, the problem reduces t…

Constructs special Kähler structures on Lie groups.

The Kodaira-Thurston manifold is a quotient of a nilpotent Lie group by a cocompact lattice. We compute the family Gromov-Witten invariants which count pseudoholomorphic tori in the Kodaira-Thurston manifold. For a fixed symplectic form the Gromov-Witten invariant is trivial so we consider the twistor family of left-in…

Flat Hermitian Lie algebras are always Kähler.

Study proves all left-invariant contact structures on 3D Lie groups are tight.

This paper studies moduli spaces of statistical structures on Lie groups.