María Alejandra Alvarez

• PhD
• Professor (Full) at University of Antofagasta

We exhibit in this article a contraction of the direct product Lie algebra $g\oplus g$ of a finite-dimensional complex Lie algebra $g$ onto the semi-direct product Lie algebra $g\rtimes g$, where the first factor $g$ is viewed as a trivial Lie algebra and as the adjoint $g$-module. This contraction gives rise to a non-zero cohomology class in the s...

The aim of this work is to study the relation between S-expansions and other transformations of Lie algebras. In particular we prove that contractions, deformations and central extensions of Lie algebras are preserved by S-expansions. We also provide several examples in low-dimension and give conditions so transformations of reduced subalgebras of...

This paper is devoted to the complete algebraic and geometric classification of complex $5$-dimensional Zinbiel algebras. In particular, we proved that the variety of complex $5$-dimensional Zinbiel algebras has dimension $24$, it is defined by $16$ irreducible components and it has $11$ rigid algebras.

This paper is devoted to the complete algebraic and geometric classification of complex $4$-dimensional nilpotent weakly associative, complex $4$-dimensional symmetric Leibniz algebras, and complex $5$-dimensional nilpotent symmetric Leibniz algebras. In particular, we proved that the variety of complex $4$-dimensional symmetric Leibniz algebras ha...

This paper is devoted to the complete algebraic and geometric classification of complex 5-dimensional Zinbiel algebras. In particular, we proved that the variety of complex 5-dimensional Zinbiel algebras has dimension 24, it is defined by 16 irreducible components and it has 11 rigid algebras.

In this work we consider 2-step nilradicals of parabolic subalgebras of the simple Lie algebra A n and describe a new family of faithful nil-representations of the nilradicals n a,c , a, c ∈ N. We obtain a sharp upper bound for the minimal dimension µ(n a,c) and for several pairs (a, c) we obtain µ(n a,c).

In this note we compute all deformations of the 4-dimensional classical Galilei algebra &. In particular, we find examples of quadratic, cubic and quartic Lie algebra deformations.

In this work, we consider degenerations between 8-dimensional 2-step nilpotent Lie algebras over ℂ and obtain the geometric classification of the variety N82.

This paper is devoted to the complete algebraic and geometric classification of complex 4-dimensional nilpotent weakly associative, complex 4-dimensional symmetric Leibniz algebras, and complex 5-dimensional nilpotent symmetric Leibniz algebras. In particular, we proved that the variety of complex 4-dimensional symmetric Leibniz algebras has no Ver...

In this work we obtain all rigid complex 3-dimensional multiplicative Hom-Lie algebras. This is done by studying all deformations of multiplicative Hom-Lie algebras which are also Lie algebras. As a byproduct, we obtain the well-known classification of 3-dimensional multiplicative (non-Lie) Hom-Lie algebras. https://authors.elsevier.com/a/1dizu4~FP...

We describe all central extensions of all 3-dimensional nontrivial complex Zinbiel algebras. As a corollary, we have a full classification of 4-dimensional nontrivial complex Zinbiel algebras and a full classification of 5-dimensional nontrivial complex Zinbiel algebras with 2-dimensional annihilator, which gives the principal step in the algebraic...

We describe all central extensions of all $3$-dimensional non-zero complex Zinbiel algebras. As a corollary, we have a full classification of $4$-dimensional non-trivial complex Zinbiel algebras and a full classification of $5$-dimensional non-trivial complex Zinbiel algebras with $2$-dimensional annihilator, which gives the principal step in the a...

The purpose of this work is to completely characterize contact Lie algebras, i.e., linear and quadratic deformations of the Heisenberg Lie algebra, by means of double extensions.

The aim of this work is to provide a criterion for the rigidity of 2-step nilpotent Lie superalgebras in the variety N ( m | n ) 2 of 2-step nilpotent Lie superalgebras of dimension ( m | n ) . We give several examples of rigid 2-step nilpotent Lie superalgebras of any dimension.

In this note we compute all deformations of the 3-dimensional Heisenberg Lie algebra ℌ 3 . This shows that ℌ 3 deforms to almost all Lie algebras of dimension 3.

The aim of this work is to explicitly compute the semi-invariants of low-dimensional Lie algebras by reducing the amount of work, i.e., we can prove that almost every irreducible Lie algebra g , of dimension less than or equal to 5, satisfies the following: It is either a contact Lie algebra or there exists a torus T ⊂ Der ( g ) such that T ⋉ g is...

In this note we compute the Betti numbers for the Heisenberg Lie algebras. This is a well-known result due to Santharoubane. We re-derive this by considering Heisenberg Lie algebras as nilradicals of a certain subalgebra of a special linear Lie algebra and computing the dimensions of certain irreducible modules by representing them using Young diag...

A mathematical first-order difference equation was designed to predict the dynamics of the phage-bacterium adsorption process in aquatic environments, under laboratory conditions. Our model requires knowledge of bacteria and bacteriophage concentrations and the measurements of bacterial size and velocity to predict both the number of bacteriophages...

There exist pentadiagonal matrices which are diagonally similar to symmetric matrices. In this work we describe explicitly the diagonal matrix that gives this transformation for certain pentadiagonal matrices. We also consider particular classes of pentadiagonal matrices and obtain recursive formulas for the characteristic polynomial and explicit f...

In this paper we compute the Betti numbers for complex nilpotent Lie superalgebras of dimension ≤5.

The aim of this work is to explicitly compute the semi-invariants of low-dimensional Lie algebras by reducing the amount of work, i.e., we can prove that every irreducible Lie algebra g, of dimension less or equal to 5, satisfies that: it is a contact Lie algebra or there exists a torus T ⊂ Der(g) such that T g is a contact Lie algebra. Therefore,...

In this work we associate a Lie algebra to every finite simple graph and obtain a necessary and sufficient condition for the existence of degenerations between these Lie algebras. As a corollary, we obtain that all these algebras belong to the irreducible component within the variety of 2-step nilpotent Lie algebras, given by the orbit closure of t...

In this paper we study the varieties of nilpotent Lie superalgebras of dimension ≤ 5. We provide the algebraic classification of these superalgebras and obtain the irreducible components in every variety. As a byproduct we construct rigid nilpotent Lie superalgebras of arbitrary dimension.

In this work, we consider degenerations between 8-dimensional 2-step nilpotent Lie algebras over $\mathbb{C}$ and obtain the geometric classification of the variety $\mathcal{N}^2_8$.

In this paper we study the varieties of nilpotent Lie superalgebras of dimension $\leq 5$. We provide the algebraic classification of these superalgebras and obtain the irreducible components in every variety. As a by product we construct rigid nilpotent Lie superalgebras of arbitrary dimension.

The aim of this work is to provide explicit calculations that describe any 7-dimensional contact nilpotent Lie algebra as a double extension of a 5-dimensional contact nilpotent Lie algebra. In particular, we describe an arbitrary (2n+ 1)-dimensional contact filiform Lie algebra as a double extension of a (2n − 1)-dimensional contact nilpotent Lie...

In this work, we consider the Heisenberg Lie algebra with all its Hom-Lie structures. We completely characterize the cohomology and deformations of any order of all Heisenberg Hom-Lie algebras of dimension 3.

In this note we consider 2-step nilpotent Lie algebras and give a criterion for the rigidity of this class in the variety N 2 n of 2-step nilpotent Lie algebras of dimension n. We apply this criterion to prove that every rigid Lie algebra in N 2 n is indecomposable, except for h3 ⊕ C and h3 ⊕ h3.

We give necessary conditions for the existence of degenerations between two complex Lie superalgebras of dimension $(m,n)$. As an application, we study the variety $\mathcal{LS}^{(2,2)}$ of complex Lie superalgebras of dimension $(2,2)$. First we give the algebraic classification and then obtain that $\mathcal{LS}^{(2,2)}$ is the union of seven irr...

We give necessary conditions for the existence of degenerations between two complex Lie superalgebras of dimension $(m,n)$. As an application, we study the variety $\mathcal{LS}^{(2,2)}$ of complex Lie superalgebras of dimension $(2,2)$. First we give the algebraic classification and then obtain that $\mathcal{LS}^{(2,2)}$ is the union of seven irr...

In this note, we consider degenerations between complex 2-step nilpotent Lie algebras of dimension 7 within the variety N 7 2 . This allows us to obtain the rigid algebras in N 7 2 , whose closures give the irreducible components of the variety.

The aim of this work is to characterize the families of Frobenius (respectively, contact) solvable Lie algebras that satisfies the following condition: $g = h \ltimes V$ , where $h \subset gl(V )$, $\vert dim V - dim g \vert \leq 1$ and $NilRad(g) = V$ , $V$ being a finite dimensional vector space. In particular, it is proved that every complex...

We describe degenerations of three-dimensional Jordan superalgebras over $\mathbb{C}$. In particular, we describe all irreducible components in the corresponding varieties.

We describe degenerations of three-dimensional Jordan superalgebras over $\mathbb{C}$. In particular, we describe all irreducible components in the corresponding varieties.

In this work we show that for n>=1, every finite (2n + 3)-dimensional contact nilpotent Lie algebra g can be obtained as a double extension of a contact nilpotent Lie algebra h of codimension 2. As a consequence, for n >=1, every (2n + 3)-dimensional contact nilpotent Lie algebra g can be obtained from the 3-dimensional Heisenberg Lie algebra h_3...

By considering the Heisenberg Lie algebra h2n-1 as the nilradical of a parabolic subalgebra p of An , we give a full description of its adjoint homology as a module over a Levi factor of p.

We consider a family of parabolic subalgebras pp of a simple Lie algebra of type AnAn and give a full description of the adjoint homology of the nilradicals of pp as a module over its Levi factor. All the nilradicals considered are 2-step nilpotent.