We study Lie algebras endowed with a complex structure admitting, at the same time, a pseudo-Hermitian quadratic structure and a (possibly indefinite) Kähler structure. These algebras have an invertible derivation that is skew-symmetric with respect to the quadratic structure and commutes with the complex structure. We propose several methods to construct such Lie algebras and use a generalized method of double extension to provide an inductive description of all of them. This double extension method allows us to prove that Lorentz-Hermitian quadratic Lie algebras cannot admit a Kähler structure for the same complex structure unless they are abelian. Additionally, we show that there is only one non-abelian Lie algebra of that kind for dimension less than or equal to 8 and prove that 2-step nilpotent ones with quadratic metrics of index 4 are necessarily either abelian or a trivial central extension of the aforementioned 8-dimensional Lie algebra.

Almost abelian Lie groups are those admitting a normal subgroup which is commutative and of codimension one. We determine all Lie algebras of almost abelian Lie groups admitting a left-invariant Lorentzian metric of vanishing curvature. For each of those algebras, we completely classify the set of inequivalent flat Lorentzian products.

We show that there are no symmetric non-zero biderivations on perfect Lie algebras of finite dimension over a field of characteristic zero. We show that this is equivalent to show that every symmetric biderivation on a finite-dimensional perfect Lie algebra over such a field with values in a finite-dimensional module vanishes identically. This answers an open question posed by Brešar and Zhao (J Lie Theory 28, 885–900, 2018).

We give a complete classification of flat Lorentzian nilpotent Lie algebras, this is to say of pseudo-Euclidean Lie algebras associated to nilpotent Lie groups endowed with a left-invariant Lorentzian metric of vanishing curvature. We prove that every such a Lie algebra is a direct sum of an indecomposable flat Lorentzian Lie algebra and an abelian Euclidean summand and show that, if h(2k+1) denotes the 2k+1-dimensional Heisenberg Lie algebra, then the only non-abelian Lie algebras admitting flat Lorentzian metrics which are indecomposable are h(3) and the semidirect products n(1)(k) = R(sic)(F1) h(2k+1) and n(2)(k)= R(sic)(F2) (h(2k+1) circle plus R) F-1,F-2. In all those cases we also find the equivalence classes of flat Lorentzian products.

We give a complete classification of flat Lorentzian nilpotent Lie algebras, this is to say of pseudo-Euclidean Lie algebras associated to nilpotent Lie groups endowed with a left-invariant Lorentzian metric of vanishing curvature. We prove that every such a Lie algebra is a direct sum of an indecomposable flat Lorentzian Lie algebra and an abelian Euclidean summand and show that, if h 2 k + 1 ${\mathfrak {h}}_{2k+1}$ denotes the 2 k + 1 $2k+1$ -dimensional Heisenberg Lie algebra, then the only non-abelian Lie algebras admitting flat Lorentzian metrics which are indecomposable are h 3 ${\mathfrak {h}}_3$ and the semidirect products N 1 ( k ) = R ⋉ F 1 h 2 k + 1 ${\mathfrak {N}}_1(k)={\mathbb {R}}\ltimes _{ F_1}{\mathfrak {h}}_{2k+1}$ and N 2 ( k ) = R ⋉ F 2 ( h 2 k + 1 ⊕ R ) ${\mathfrak {N}}_2(k)={\mathbb {R}}\ltimes _{ F_2}({\mathfrak {h}}_{2k+1}\oplus {\mathbb {R}})$ , defined by some particular derivations F 1 , F 2 $F_1,F_2$ . In all those cases we also find the equivalence classes of flat Lorentzian products.

We study the existence of 2-plectic structures on Lie algebras which admit an ad-invariant non-degenerate symmetric bilinear form, frequently called quadratic Lie algebras. It is well-known that every centerless quadratic Lie algebra admits a 2-plectic form but not many quadratic examples with nontrivial center are known. In this paper we give several constructions to obtain large families of 2-plectic quadratic Lie algebras with nontrivial center, many of them among the class of nilpotent Lie algebras. We give some sufficient conditions to assure that certain extensions of 2-plectic quadratic Lie algebras result to be 2-plectic as well. We prove that every quadratic and symplectic Lie algebra with dimension greater than 4 also admits a 2-plectic form. Further, conditions to assure that one may find a 2-plectic which is exact on certain quadratic Lie algebras are also obtained.

We study nilpotent Lie algebras endowed with a complex structure and a quadratic structure which is pseudo-Hermitian for the given complex structure. We propose several methods to construct such Lie algebras and describe a method of double extension by planes to get an inductive description of all of them. As an application, we give a complete classification of nilpotent quadratic Lie algebras where the metric is Lorentz-Hermitian and we fully classify all nilpotent pseudo-Hermitian quadratic Lie algebras up to dimension 8 and their inequivalent pseudo-Hermitian metrics.

This article proposes a method for computing the Moore-Penrose inverse of a complex matrix using polynomials in matrices. Such a method is valid for all matrices and does not involve spectral calculation, which could be infeasible when the size of the matrix is large. We first study under which conditions the Moore-Penrose inverse of a square matrix A is a polynomial in A. As an application, we also see that the Moore-Penrose inverse of an arbitrary matrix A is an element of M-m,M-n(C) may be factored as the product of its conjugate transpose A* and a polynomial in either AA* or A*A. The article is self-contained so that it can be understood by all readers with a basic knowledge of linear algebra. We have illustrated most of the relevant results with examples.

We consider Lie groups G endowed with a pair of anticommuting left-invariant abelian complex structures (J1, J2) and a left-invariant (possibly indefinite) metric g such that (G, J1, J2, g) results to be a hyperkähler manifold. We give a classification of their Lie algebras up to dimension 12 and study some of their geometric properties. In particular, we show that all such groups are locally symmetric and complete and that the metric is flat if and only if the group is 2-step nilpotent.

Hyper-para-Kähler structures on Lie algebras where the complex structure is abelian are studied. We show that there is a one-to-one correspondence between such hyper-para-Kähler Lie algebras and complex commutative (hence, associative) symplectic left-symmetric algebras admitting a semilinear map \(K_s\) verifying certain algebraic properties. Such equivalence allows us to give a complete classification, up to holomorphic isomorphism, of pairs \(({\mathfrak g},J)\) of 8-dimensional Lie algebras endowed with abelian complex structures which admit hyper-para-Kähler structures.

We study a classical model for a population that reproduces and disperses in a landscape of heterogeneous patches. Under symmetrical dispersal, we provide a sufficient condition to ensure the existence of a globally attracting fixed point. This condition is used in order to prove that certain patches with complex dynamics can be stabilized by the combination with stable patches. Specifically, given a patch with complex dynamics, we estimate the necessary number of patches with simple dynamics so that the whole metapopulation has a globally attracting equilibrium.

We study the existence of invariants for the family of systems in an open domain \(\mathcal {D}\) of \(\mathbb {R}^n\) or \(\mathbb {C}^n\) whose components are linear fractionals sharing denominator. Such systems can be written with the aid of homogeneous coordinates as the composition of a linear map in \(\mathbb {K}^{n+1}\) with a certain projection and their behaviour is strongly determined by the spectral properties of the corresponding linear map.The paper is committed to prove that if \(n\ge 2\) then every system of this kind admits an invariant, both in the real and in the complex case. In fact, for a sufficiently large n several functionally independent invariants can be obtained and, in many cases, the invariant can be chosen as the quotient of two quadratic polynomials.

We study the existence of invariant quadrics for a class of systems of difference equations in ${\mathbb R}^n$ defined by linear fractionals sharing denominator. Such systems can be described in terms of some square matrix $A$ and we prove that there is a correspondence between non-degenerate invariant quadrics and solutions to a certain matrix equation involving $A$. We show that if $A$ is semisimple and the corresponding system admits non-degenerate quadrics, then every orbit of the dynamical system is contained either in an invariant affine variety or in an invariant quadric.

The forbidden sets of systems of first order rational difference equations in the plane in which the denominators are common for all the components of the system is studied. Such forbidden sets are composed of lines which, depending of some spectral properties of an associated matrix, can either be a finite number or lines or an infinity of lines converging to either an invariant line or to a finite number of lines itersecting in a fixed point or else it can be dense in a large subset of R-2.

We study Lie algebras endowed with an Abelian complex structure which admit a symplectic form compatible with the complex structure. We prove that each of those Lie algebras is completely determined by a pair (U, H) where U is a complex commutative associative algebra and H is a sesquilinear Hermitian form on U which verifies certain compatibility conditions with respect to the associative product on U. The Riemannian and Ricci curvatures of the associated pseudo-Kahler metric are studied and a characterization of those Lie algebras which are Einstein but not Ricci flat is given. It is seen that all pseudo-Kahler Lie algebras can be inductively described by a certain method of double extensions applied to the associated complex associative commutative algebras.

We consider a rational system of first-order difference equations in the plane with four parameters such that all fractions have a common denominator. We study, for the different values of the parameters, the global and local properties of the system. In particular, we discuss the boundedness and the asymptotic behavior of the solutions, the existence of periodic solutions, and the stability of equilibria.

We describe the asymptotic behaviour and the stability properties of the solutions to a second order rational difference equation.

Para-Kähler Lie algebras which decompose as the sum of two abelian Lagrangian subalgebras are studied. We propose several constructions and provide an inductive description of such Lie algebras. The curvatures of the para-Kähler metric are computed and sufficient conditions to ensure flatness or Ricci-flatness are given. The Lie algebras for which the para-Kähler metric is Einstein and non-Ricci-flat are completely characterized.

We describe the asymptotic behaviour and the stability properties of the solutions to the nonlinear second order dierence equation xn+1 = xn 1 a + bxnxn 1 ; n 0; for all values of the real parameters a;b, and any initial condition (x 1;x0)2 R 2 .

A subset X of an algebra A is said to generate A, if A is the smallest linear space closed under the product containing X. In this talk we describe generating sets for some classes of finite dimensional nonassociative algebras defined over a field of characteristic 0. The first result about this subject is due to Kuranishi [4], who proved that any finite dimensional semisimple Lie algebra over a field of characteristic 0 can be generated by two elements. We have generalized this result for any Lie algebra [2] and adapted the reasoning of Kuranishi to the case of semisimple Jordan algebras [1] where Cartan decomposition was replaced by Peirce decomposition. We prove that any classical simple Lie superalgebra is generated by one element [3] and we finalize this talk studying this problem for simple Malcev superalgebras.