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Rafael Reno S. Cantuba

Publications and source records attributed to Rafael Reno S. Cantuba.

16 recordsLinked to original sources
arXiv2026

Zero Harmonic Density implies Zero Density

A set of integers has zero harmonic density if its elements, viewed as frequencies of continuous characters of the unit circle as a compact group, are measure-theoretically sparse enough that the values of Fourier transforms of Radon measures supported on small open arcs can interpolate those of any global Radon measure. In contrast, zero density quantifies the proportional size of a set of integers relative to expanding intervals on the real line. Although these notions arise from abstract harmonic analysis and elementary number theory, respectively, zero harmonic density is proven as a sufficient condition for zero density, resolving what has been stated in the literature as an open problem concerning the relation between these two notions. The proof makes use of classical gap theorems from complex analysis.

math.GM↗
arXiv2026

Extension Theorems in Measure and Integration Theory as Completeness Axioms

In measure and integration theory, the themes of extending premeasures to outer measures and extending Daniell linear functionals to Daniell integrals are governed by the foundational extension theorems of Caratheodory and Daniell. These are proven to be equivalent to Littlewood's Three Principles, which are known to be equivalent alternatives to a completeness axiom for an ordered field. The continuity theorems of measure and integration theory are consequently shown to be included in the list of equivalences.

math.GM↗
arXiv2026

Sufficient Conditions for Strong Natural Boundaries involving Lacunary Sequences

The notion of a regular point is extended to that of a weak regular point, and so the notion of a strong natural boundary may be described locally as the absence of weak regular points, just as a natural boundary is locally the absence of regular points. The main result is a sufficient condition for a strong natural boundary, which is the conjunction of the unboundedness of the coefficients of the underlying power series with either of two specific gap density conditions imposed on the exponents that yields a lacunary sequence.

math.CA↗
arXiv2026

A Kronecker-Weyl Theorem from Interpolation Sets

An analog of the classical Kronecker-Weyl theorem for weak uniform distribution is obtained for an arbitrary countably infinite independent subset of a discrete abelian group using a type of interpolation sets called epsilon-Kronecker sets. Specifically, the topological size of the set of homomorphisms inducing weakly uniformly distributed sequences is established by showing that it forms a dense G-delta set.

math.FA↗
arXiv2025

Lie polynomials in a $q$-deformed universal enveloping algebra of a low-dimensional Lie algebra

The nonabelian two-dimensional Lie algebra over a field $\mathbb{F}$ has a presentation by generators $A$, $B$ and relation $\left[ A,B\right]=A$, with the universal enveloping algebra having a presentation by generators $A$, $B$ and relation $AB-BA=A$. A well-known fact is that the said Lie algebra is isomorphic to that which has a universal enveloping algebra that has a presentation by generators $A$, $B$ and relation $AB-BA=B$. Given $q,r,s\in\mathbb{F}$, solutions to the Lie polynomial characterization problems in the corresponding $q$-deformed universal enveloping algebras, with generalized relations $AB-qBA=rA$ and $AB-qBA=sB$, respectively, are presented.

math.RA↗
arXiv2023

Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra

Let $q$ be a nonzero complex number that is not a root of unity. In the $q$-oscillator with commutation relation $ a a^+-qa^+ a =1$, it is known that the smallest commutator algebra of operators containing the creation and annihilation operators $a^+$ and $ a $ is the linear span of $a^+$ and $ a $, together with all operators of the form ${a^+}^l{\left[a,a^+\right]}^k$, and ${\left[a,a^+\right]}^k a ^l$, where $l$ is a nonnegative integer and $k$ is a positive integer. That is, linear combinations of operators of the form $ a ^h$ or $(a^+)^h$ with $h\geq 2$ or $h=0$ are outside the commutator algebra generated by $ a $ and $a^+$. This is a solution to the Lie polynomial characterization problem for the associative algebra generated by $a^+$ and $ a $. In this work, we extend the Lie polynomial characterization into the associative algebra $\mathcal{P}=\mathcal{P}(q)$ generated by $ a $, $a^+$, and the operator $e^{ωN}$ for some nonzero real parameter $ω$, where $N$ is the number operator, and we relate this to a $q$-oscillator representation of the Askey-Wilson algebra $AW(3)$.

math.RA↗
arXiv2022

Lie structure of the Heisenberg-Weyl algebra

As an associative algebra, the Heisenberg-Weyl algebra $\mathcal{H}$ is generated by two elements $A$, $B$ subject to the relation $AB-BA=1$. As a Lie algebra, however, where the usual commutator serves as Lie bracket, the elements $A$ and $B$ are not able to generate the whole space $\mathcal{H}$. We identify a non-nilpotent but solvable Lie subalgebra $\mathfrak{g}$ of $\mathcal{H}$, for which, using some facts from the theory of bases for free Lie algebras, we give a presentation by generators and relations. Under this presentation, we show that, for some algebra isomorphism $φ:\mathcal{H}\longrightarrow\mathcal{H}$, the Lie algebra $\mathcal{H}$ is generated by the generators of $\mathfrak{g}$, together with their images under $φ$, and that $\mathcal{H}$ is the sum of $\mathfrak{g}$, $φ(\mathfrak{g})$ and $\left[ \mathfrak{g},φ(\mathfrak{g})\right]$.

math.RA↗
arXiv2019

Ubiquitous Lie polynomials in a two-generator universal enveloping algebra

The universal enveloping algebra $\mathscr{U}$ of a two-dimensional nonabelian Lie algebra $L$ is a Lie algebra itself with the commutator as Lie bracket. There exists a presentation of $\mathscr{U}$ with generators $x,y$ and relation $xy-yx=x$ such that the Lie subalgebra of $\mathscr{U}$ generated by $x,y$ is isomorphic to $L$, which is only a two-dimensional vector subspace of the infinite-dimensional $\mathscr{U}$. Much then of the Lie structure of $\mathscr{U}$ is ubiquitous, yet unexamined when the characteristic of the scalar field is zero. In such a case, we show that there exists a linear complement of $L$ in $\mathscr{U}$ that contains an infinite-dimensional Lie subalgebra of $\mathscr{U}$ for which we give a presentation by generators and relations. We extend this Lie subalgebra into a filtration of $\mathscr{U}$.

math.RA↗
arXiv2019

A $q$-deformed Heisenberg algebra as a normed space

Given a real number $q$ such that $0<q<1$, the natural setting for the mathematics of a $q$-oscillator is an infinite-dimensional, separable Hilbert space that is said to provide an interpolation between the Bargmann-Segal space of holomorphic functions and the Hardy-Lebesgue space of analytic functions. The traditional basis states are interrelated by the creation and annihilation operators. Since the commutation relation is $q$-deformed, the commutator algebra for the creation and annihilation operators is not a low-dimensional Lie algebra like that for the canonical commutation relation. In this study, a characterization of the elements of the said commutator algebra is obtained using spectral properties of the creation and annihilation operators as these faithfully represent the generators of a $q$-deformed Heisenberg algebra. The derived algebra of the commutator algebra is precisely the set of all compact operators, and the resulting Calkin algebra is algebraically isomorphic to the complex algebra of Laurent polynomials in one indeterminate. As for any operator that is not in the commutator algebra, the action of such an operator on an arbitrary basis state can be approximated by a Lie series of elements from the commutator algebra.

math.OA↗
arXiv2019

Torsion-type $q$-deformed Heisenberg algebra and its Lie polynomials

Given a scalar parameter $q$, the $q$-deformed Heisenberg algebra $\mathcal{H}(q)$ is the unital associative algebra with two generators $A,B$ that satisfy the $q$-deformed commutation relation $AB-qBA= I$, where $I$ is the multiplicative identity. For $\mathcal{H}(q)$ of torsion-type, that is if $q$ is a root of unity, characterization is obtained for all the Lie polynomials in $A,B$ and basis and graded structure and commutation relations for associated Lie algebras are studied.

math.RA↗
arXiv2018

An extension of a $q$-deformed Heisenberg algebra and its Lie polynomials

Let $\mathbb{F}$ be a field, and fix a $q\in\mathbb{F}$. The $q$-deformed Heisenberg algebra $\mathcal{H}(q)$ is the unital associative algebra over $\mathbb{F}$ with generators $A$, $B$ and a relation which asserts that $AB - qBA$ is the multiplicative identity in $\mathcal{H}(q)$. We extend $\mathcal{H}(q)$ into an algebra $\mathcal{R}(q)$ defined by generators $A$, $B$ and a relation which asserts that $AB-qBA$ is central in $\mathcal{R}(q)$. We identify all elements of $\mathcal{R}(q)$ that are Lie polynomials in $A$, $B$.

math.RA↗
arXiv2018

Lie polynomials in an algebra defined by a linearly twisted commutation relation

We present an elementary approach in characterizing Lie polynomials in the generators $A,B$ of an algebra with a defining relation that is in the form of a deformed or twisted commutation relation $AB=σ(BA)$ where the deformation or twisting map $σ$ is a linear polynomial with a slope parameter that is not a root of unity. The class of algebras defined as such encompasses $q$-deformed Heisenberg algebras, rotation algebras, and some types of $q$-oscillator algebras whose deformation parameters are not roots of unity, and so we have a general solution for the Lie polynomial characterization problem for these algebras.

math.RA↗
arXiv2018

A Casimir element inexpressible as a Lie polynomial

Let $q$ be a scalar that is not a root of unity. We show that any polynomial in the Casimir element of the Fairlie-Odesskii algebra $U_q'(\mathfrak{so}_3)$ cannot be expressed in terms of only Lie algebra operations performed on the generators $I_1,I_2,I_3$ in the usual presentation of $U_q'(\mathfrak{so}_3)$. Hence, the vector space sum of the center of $U_q'(\mathfrak{so}_3)$ and the Lie subalgebra of $U_q'(\mathfrak{so}_3)$ generated by $I_1,I_2,I_3$ is direct.

math.RA↗
arXiv2017

Lie polynomials in $q$-deformed Heisenberg algebras

Let $\mathbb{F}$ be a field, and let $q\in\mathbb{F}$. The $q$-deformed Heisenberg algebra is the unital associative $\mathbb{F}$-algebra $\mathcal{H}(q)$ with generators $A,B$ and relation $AB-qBA=I$, where $I$ is the multiplicative identity in $\mathcal{H}(q)$. The set of all Lie polynomials in $A,B$ is the Lie subalgebra $\mathcal{L}(q)$ of $\mathcal{H}(q)$ generated by $A,B$. If $q\neq 1$ or the characteristic of $\mathbb{F}$ is not $2$, then the equation $AB-qBA=I$ cannot be expressed in terms of Lie algebra operations only, yet this equation still has consequences on the Lie algebra structure of $\mathcal{L}(q)$, which we investigate. We show that if $q$ is not a root of unity, then $\mathcal{L}(q)$ is a Lie ideal of $\mathcal{H}(q)$, and the resulting quotient Lie algebra is infinite-dimensional and one-step nilpotent.

math.RA↗
arXiv2016

A Lie algebra related to the universal Askey-Wilson algebra

Let $\mathbb{F}$ denote an algebraically closed field. Denote the three-element set by $\mathcal{X}=\{A,B,C\}$, and let $\mathbb{F}\left<\mathcal{X}\right>$ denote the free unital associative $\mathbb{F}$-algebra on $\mathcal{X}$. Fix a nonzero $q\in\mathbb{F}$ such that $q^4\neq 1$. The universal Askey-Wilson algebra $Δ$ is the quotient space $\mathbb{F}\left<\mathcal{X}\right>/\mathbb{I}$, where $\mathbb{I}$ is the two-sided ideal of $\mathbb{F}\left<\mathcal{X}\right>$ generated by the nine elements $UV-VU$, where $U$ is one of $A,B,C$, and $V$ is one of \begin{equation} (q+q^{-1}) A+\frac{qBC-q^{-1}CB}{q-q^{-1}},\nonumber \end{equation} \begin{equation} (q+q^{-1}) B+\frac{qCA-q^{-1}AC}{q-q^{-1}},\nonumber \end{equation} \begin{equation} (q+q^{-1}) C+\frac{qAB-q^{-1}BA}{q-q^{-1}}.\nonumber \end{equation} Turn $\mathbb{F}\left<\mathcal{X}\right>$ into a Lie algebra with Lie bracket $\left[ X,Y\right] = XY-YX$ for all $X,Y\in\mathbb{F}\left<\mathcal{X}\right>$. Let $\mathcal{L}$ denote the Lie subalgebra of $\mathbb{F}\left<\mathcal{X}\right>$ generated by $\mathcal{X}$, which is also the free Lie algebra on $\mathcal{X}$. Let $L$ denote the Lie subalgebra of $Δ$ generated by $A,B,C$. Since the given set of defining relations of $Δ$ are not in $\mathcal{L}$, it is natural to conjecture that $L$ is freely generated by $A,B,C$. We give an answer in the negative by showing that the kernel of the canonical map $\mathbb{F}\left<\mathcal{X}\right>\rightarrowΔ$ has a nonzero intersection with $\mathcal{L}$. Denote the span of all Hall basis elements of $\mathcal{L}$ of length $n$ by $\mathcal{L}_n$, and denote the image of $\sum_{i=1}^n\mathcal{L}_i$ under the canonical map $\mathcal{L}\rightarrow L$ by $L_n$. We study some properties of $L_4$ and $L_5$.

math.RA↗