arXiv subjects

Luis Fernandez

Publications and source records attributed to Luis Fernandez.

6 recordsLinked to original sources

arXiv2025

The solvable Graph of a finite-dimensional Lie Algebra

We introduce and investigate the solvable graph $\Gamma_\mathfrak{S}(L)$ of a finite-dimensional Lie algebra $L$ over a field $F$. The vertices are the elements outside the solvabilizer $\sol(L)$, and two vertices are adjacent whenever they generate a solvable subalgebra. After developing the basic properties of solvabilizers and $S$-Lie algebras, we establish divisibility conditions, coset decompositions, and degree constraints for solvable graphs. Explicit examples, such as $\mathfrak{sl}_2(\mathbb{F}_3)$, illustrate that solvable graphs may be non-connected, in sharp contrast with the group-theoretic setting. We further determine the degree sequences of $\Gamma_\mathfrak{S}(\mathfrak{gl}_2(\F_q))$ and $\Gamma_\mathfrak{S}(\mathfrak{sl}_2(\F_q))$, highlighting how spectral types of matrices dictate combinatorial patterns. An algorithmic framework based on GAP and SageMath is also provided for practical computations. Our results reveal both analogies and differences with the nilpotent graph of Lie algebras, and suggest that solvable graphs encode structural invariants in a genuinely new way. This work opens the door to a broader graphical approach to solvability in Lie theory.

math.RA↗

arXiv2025

The nilpotent graph of a finite0-dimensional Lie algebra

Let $L$ be a finite-dimensional Lie algebra over a field $F$. In This paper we introduce the \emph{nilpotent graph} $\Gamma_\mathfrak{N}(L)$ as the graph whose vertices are the elements of $L \setminus \nil(L)$, where \[\nil(L) = \{x \in L \mid \langle x, y \rangle \text{ is nilpotent for all } y \in L\},\] and where two vertices $x, y$ are adjacent if the Lie subalgebra they generate is nilpotent. We give some characterizations of $\nil(L)$ and its connection with the hypercenter $Z^*(L)$, for example, they are equal when $F$ has characteristic zero. We prove that the nilpotentizer behaves well under direct sums, allowing a decomposition of $\Gamma_\mathfrak{N}(L)$ between components. The paper also investigates the structural and combinatorial properties of $\Gamma_\mathfrak{N}(L)$, including the conditions under which the graph is connected. We characterize the existence of strongly self-centralizing subalgebras in relation to connectivity and vertex isolation. Explicit computations are carried out for the algebra $\mathfrak{t}(2,\mathbb{F}_q)$, where $\Gamma_\mathfrak{N}(L)$ decomposes into $q+1$ components, each of size $q(q-1)$, forming a $(q^2-q-1)$-regular graph. We conclude with algorithms for constructing $\Gamma_\mathfrak{N}(L)$ in SageMath, and pose open problems concerning bipartiteness, regularity, and structural implications in higher dimensions over finite fields.

math.RA↗

arXiv2024

An algebraic model for the constant loops map

For any simplicial complex $X$ with a total ordering of its vertices, one can construct a chain complex $\mathbb{L}_\bullet(X)$ generated by necklaces of simplices in $X$, which computes the homology of the free loop space of the geometric realization of $X$. Motivated by string topology, we describe two explicit chain maps $C_\bullet(X) \to \mathbb{L}_\bullet(X)$, where $C_\bullet(X)$ denotes the simplicial chains in $X$, lifting the homology map induced by embedding points in $|X|$ into constant loops in the free loop space of $|X|$. One of the maps has a convenient combinatorial description, while the other is described in terms of higher structure on $C_\bullet(X)$.

math.AT↗

arXiv2023

Effects of the two-dimensional Coulomb interaction in both Fermi velocity and energy gap for Dirac-like electrons at finite temperature

We describe both the Fermi velocity and the mass renormalization due to the two-dimensional Coulomb interaction in the presence of a thermal bath. To achieve this, we consider an anisotropic version of pseudo quantum electrodynamics (PQED), within a perturbative approach in the fine-structure constant $\alpha$. Thereafter, we use the so-called imaginary-time formalism for including the thermal bath. In the limit $T\rightarrow 0$, we calculate the renormalized mass $m^R(p)$ and compare this result with the experimental findings for the energy band gap in monolayers of transition metal dichalcogenides, namely, WSe$_2$ and MoS$_2$. In these materials, the quasi-particle excitations behave as a massive Dirac-like particles in the low-energy limit, hence, its mass is related to the energy band gap of the material. In the low-temperature limit $T\ll v_F p $, where $v_F p$ is taken as the Fermi energy, we show that $m^R(p)$ decreases linearly on the temperature, i.e, $m^R(p,T)-m^R(p,T\rightarrow 0)\approx -A_\alpha T +O(T^3)$, where $A_\alpha$ is a positive constant. On the other hand, for the renormalized Fermi velocity, we find that $v^R_F(p,T)-v^R_F(p,T\rightarrow 0)\approx -B_\alpha T^3 +O(T^5)$, where $B_\alpha$ is a positive constant. We also perform numerical tests which confirm our analytical results.

cond-mat.mes-hall↗

arXiv2022

K\"{a}hler identities for almost complex manifolds

We obtain a generalization, for a general compact almost complex manifold, of the well-known K\"{a}hler (or Hodge) identities for K\"{a}hler manifolds involving the commutators of the exterior differential and the Lefschetz operator and its adjoint. The main idea is to study the problem on the Clifford bundle via the Dirac operator, and then translate the results to the exterior bundle.

math.DG↗

arXiv2021

Almost complex manifolds with small Nijenhuis tensor

We give several explicit examples of compact manifolds with a $1$-parameter family of almost complex structures having arbitrarily small Nijenhuis tensor in the $C^0$-norm. The $4$-dimensional examples possess no complex structure, whereas the $6$-dimensional example does not possesses a left invariant complex structure, and whether it possesses a complex structure appears to be unknown.

math.DG↗