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Miguel Gómez Lozano

Publications and source records attributed to Miguel Gómez Lozano.

7 recordsLinked to original sources
arXiv2026

Leibniz algebras and their connection to Jordan pair disystems

In this paper we present a generalization of Jordan pairs, called Jordan pair disystem, which appear naturally as the wings of a Leibniz algebra with a finite $\mathbb{Z}$-grading; conversely, we give the Tits-Kantor-Koecher construction for Jordan pair disystems, obtaining a Leibniz algebra with a short $\mathbb{Z}$-grading. This construction extends the construction given by Gubarev and Kolesnikov for Jordan dialgebras (arXiv:0907.1740). We introduce homotopes for Jordan pair disystems at elements and we show that we obtain again a Jordan dialgebra. Moreover, we introduce abelian inner ideals and their kernels for Leibniz algebras, and prove that the subquotient of a Leibniz algebra with respect to an abelian inner ideal is a Jordan pair disystem. This notion of subquotient extends the construction of Jordan dialgebras at $Q$-Jordan elements given by Felipe and Velásquez.

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arXiv2026

Matrices over Finite Fields of Characteristic 2 as Sums of Diagonalizable and Square-Zero Matrices

We investigate the problem asking when any square matrix whose entries lie in a finite field of characteristic 2 is decomposable into the sum of a diagonalizable matrix and a nilpotent matrix with index of nilpotency at most 2 and, as a result, we completely resolve this question in the affirmative for any finite field of characteristic 2 having strictly more than three elements. Our main theorem of that type, combined with results from our recent publication in Linear Algebra & Appl. (2026) (see [7]), totally settle this problem for all finite fields different from $\mathbb{F}_2$ and $\mathbb{F}_3$. However, in this paper we also prove that each matrix over $\mathbb{F}_2$ is expressible as the sum of a potent matrix with index of potency not exceeding 4 and a nilpotent matrix with index of nilpotency not exceeding 2, thus substantiating recent examples due to Šter in Linear Algebra & Appl. (2018) and Shitov in Indag. Math. (2019) (see, respectively, [9] and [8]).

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arXiv2025

Matrices over finite fields of odd characteristic as sums of diagonalizable and square-zero matrices

Let $\mathbb{F}$ be a finite field of odd characteristic. When $|\mathbb{F}|\ge 5$, we prove that every matrix $A$ admits a decomposition into $D+M$ where $D$ is diagonalizable and $M^2=0$. For $\mathbb{F}=\mathbb{F}_3$, we show that such decomposition is possible for non-derogatory matrices of order at least 5, and more generally, for matrices whose first invariant factor is not a non-zero trace irreducible polynomial of degree 3; we also establish that matrices consisting of direct sums of companion matrices, all of them associated to the same irreducible polynomial of non-zero trace and degree 3 over $\mathbb{F}_3$, never admit such decomposition. These results completely settle the question posed by Breaz in Lin. Algebra & Appl. (2018) asking if it is true that for big enough positive integers $n\ge 3$ all matrices $A$ over a field of odd cardinality $q$ admit decompositions of the form $E+M$ with $E^q=D$ and $M^2=0$: the answer is {\it yes} for $q\ge 5$, but there are counterexamples for $q=3$ and each order $n=3k$, $k\ge 1$.

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arXiv2024

Decompositions of Periodic Matrices into a Sum of Special Matrices

We study the problem of when a periodic square matrix of order $n$ over an arbitrary field $\mathbb{F}$ is decomposable into the sum of a square-zero matrix and a torsion matrix, and show that this decomposition can always be obtained for matrices of rank at least $\frac n2$ when $\mathbb{F}$ is either a field of prime characteristic, or the field of rational numbers, or an algebraically closed field of zero characteristic. We also provide a counterexample to such a decomposition when $\mathbb{F}$ equals the field of the real numbers. Moreover, we prove that each periodic square matrix over any field is a sum of an idempotent matrix and a torsion matrix.

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arXiv2024

On prescribed characteristic polynomials

Let $\mathbb{F}$ be a field. We show that given any $n$th degree monic polynomial $q(x)\in \mathbb{F}[x]$ and any matrix $A\in\mathbb{M}_n(\mathbb{F})$ whose trace coincides with the trace of $q(x)$ and consisting in its main diagonal of $k$ 0-blocks of order one, with $k<n-k$, and an invertible non-derogatory block of order $n-k$, we can construct a square-zero matrix $N$ such that the characteristic polynomial of $A+N$ is exactly $q(x)$. We also show that the restriction $k<n-k$ is necessary in the sense that, when the equality $k=n-k$ holds, not every characteristic polynomial having the same trace as $A$ can be obtained by adding a square-zero matrix. Finally, we apply our main result to decompose matrices into the sum of a square-zero matrix and some other matrix which is either diagonalizable, invertible, potent or torsion.

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arXiv2023

Decompositions of Matrices Into a Sum of Torsion Matrices and Matrices of Fixed Nilpotence

For $n\ge 2$ and fixed $k\ge 1$, we study when a square matrix $A$ over an arbitrary field $\mathbb{F}$ can be decomposed as $T+N$ where $T$ is a torsion matrix and $N$ is a nilpotent matrix with $N^k=0$. For fields of prime characteristic, we show that this decomposition holds as soon as the characteristic polynomial of $A\in \mathbb{M}_{n}(\mathbb{F})$ is algebraic over its base field and the rank of $A$ is at least $\frac nk$, and we present several examples that show that the decomposition does not hold in general. Furthermore, we completely solve this decomposition problem for $k=2$ and nilpotent matrices over arbitrary fields (even over division rings). This somewhat continues our recent publications in Lin. \& Multilin. Algebra (2023) and Internat. J. Algebra \& Computat. (2022) as well as it strengthens results due to Calugareanu-Lam in J. Algebra \& Appl. (2016).

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