arXiv subjects

Isabel Hernández

Publications and source records attributed to Isabel Hernández.

7 recordsLinked to original sources

arXiv2026

Engel's Theorem for Alternative Superalgebras

In this paper, a nilpotency criterion is given for finite dimensional alternative superalgebras in the spirit of Engel's Theorem for Jordan superalgebras over infinite fields provided by Shestakov and Okunev. For alternative superalgebras, no restrictions on the cardinality of the ground field are required. Furthermore, we establish some connections between the concepts of graded-nil and nilpotent alternative superalgebras, and we also exhibit an example of an Engelian commutative power-associative superalgebra of dimension $4$ which is not nilpotent.

math.RA↗

arXiv2026

Varieties of nilpotent Jordan superalgebras of dimension five

The paper is devoted to the description of the varieties of complex 5-dimensional nilpotent Jordan superalgebras. We find all representatives for the isomorphism classes, using the Jordan normal form, results of simultaneous matrix triangularization, the Jordan-Kronecker theorem for a pair of skew-symmetric bilinear forms and similar arguments developed for $Δ$-modules by Burde and Grunewald. We also provide a complete geometric classification, determining the irreducible components of the corresponding varieties and describing all possible degenerations and non-degenerations between these superalgebras, in particular, applying some $\mathbb{Z}_2$-graded subspaces as invariants.

math.RA↗

arXiv2025

Classification of almost abelian Lie groups admitting left-invariant complex or symplectic structures

We classify the almost abelian Lie algebras $\mathfrak g_A=\mathbb R e_0 \ltimes_A \mathbb R^{2n-1}$ admitting complex or symplectic structures. The matrix $A\in M(2n-1,\mathbb R )$ encodes the adjoint action of $e_0$ on the abelian ideal $\mathbb R^{2n-1}$, and the existence of complex or symplectic structures on $\mathfrak g_A$ imposes restrictions on the Jordan normal form of $A$. The classification essentially reduces to the case when $A$ is nilpotent, so we start by considering this case. It turns out that if $A$ is nilpotent and $\mathfrak g_A$ admits a complex structure, then $\mathfrak g_A$ necessarily admits a symplectic structure. This is not true in general when $A$ is non-nilpotent. Finally, several consequences of the classification theorems are obtained.

math.DG↗

arXiv2025

The Variety of Jordan Superalgebras of dimension four and even part of dimension two

We describe the variety of Jordan superalgebras of dimension $4$ whose even part is a Jordan algebra of dimension $2$ over an algebraically closed field $\mathbb{F}$ of characteristic $0$. We prove that the variety has $25$ irreducible components, $24$ of them correspond to the Zariski closure of the $GL_2(\mathbb{F})\times GL_2(\mathbb{F})$-orbits of rigid superalgebras and the other one is the Zariski closure of an union of orbits of an infinite family of superalgebras, none of them being rigid. Furthermore, it is known that the question of the existence of a rigid Jordan algebra whose second cohomology group does not vanish is still an open problem. We solve this problem in the context of superalgebras, showing a four-dimensional rigid Jordan superalgebra whose second cohomology group does not vanish.

math.RA↗

arXiv2025

Irreducible Components of the Varieties of Jordan Superalgebras of Types $(1,3)$ and $(3,1)$

We describe the variety of Jordan superalgebras of dimension $4$ whose even part is a Jordan algebra of dimension $1$ or $3$. We prove that the variety is the union of Zariski closures of the orbits of $11$ and $21$ rigid superalgebras, respectively. In both cases, the irreducible components of the varieties are described. Furthermore, we exhibit a four-dimensional solvable rigid Jordan superalgebra, showing that an analogue to the Vergne conjecture for Jordan superalgebras does not hold.

math.RA↗

arXiv2018

On Degenerations of Lie Superalgebras

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 irreducible components, three of which are the Zariski closures of rigid Lie superalgebras. As byproduct, we obtain an example of a nilpotent rigid Lie superalgebra, in contrast to the classical case where no example is known.

math.RA↗

arXiv2017

Degenerations of Jordan Superalgebras

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

math.RA↗