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Samuel A. Lopes

Publications and source records attributed to Samuel A. Lopes.

At least 19 recordsLinked to original sources
arXiv2026

Representations of compatible Lie algebras

We study compatible Lie algebras from algebraic and representation-theoretic points of view, obtaining counterexamples to some fundamental theorems from classical Lie algebra theory, namely the theorems of Lie, Weyl and Levi. We also classify the two-dimensional compatible Lie algebras up to isomorphism and explore their representation theory, presenting families of indecomposable non-semisimple representations, showing that the solvable two-dimensional compatible Lie algebras have wild representation type, and classifying all irreducible finite-dimensional line representations. Finally, we prove a Clebsch-Gordan decomposition for tensor products of finite-dimensional irreducible line representations.

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arXiv2025

Quantum upper triangular matrix algebras

Following the ideas in~\cite{yM88}, \cite{T90} and inspiration from~\cite{KO24}, we construct a bialgebra $T_q(n)$ and a pointed Hopf algebra $UT_q(n)$ which quantize the coordinate rings of the algebra of upper triangular matrices and of the group of invertible upper triangular matrices of size $n\geq 2$, respectively, where $q$ is a nonzero parameter. The resulting structure on $UT_q(n)$ is neither commutative nor cocommutative and it can be seen as a Hopf quotient of the Takeuchi's two-parameter quantization~\cite{T90} of ${\rm GL}(n)$ corresponding to a specific choice of parameters. The motivation comes from the idea of quantizing the incidence algebra of a finite poset, as the latter can be embedded as a subalgebra of the algebra of upper triangular matrices. We further study and compare the Lie algebras of derivations, the automorphism groups and the low degree Hochschild cohomology of these algebras in case $n=2$.

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arXiv2025

Lifting free modules to generalized Weyl algebras

We study modules over a generalized Weyl algebra $R(\sigma,a)$ which are free when restricted to the base ring $R$. When $R$ is an integral domain, we construct all such finite-rank modules up to isomorphism, leading to new simple modules over a variety of algebras. In particular, we show that free modules that have rank $1$ over $R$ can be parametrized as $V_{\mathsf{p}}$ where $\mathsf{p}$ is a divisor of $a$. We give simplicity criteria for $V_{\mathsf{p}}$ and, additionally, when $R$ is a PID, provide a complete combinatorial description of the submodule structure of $V_{\mathsf{p}}$ and of the weight modules occurring as subquotients. We also show that, under some mild conditions on $R(\sigma,a)$, there exist simple $R$-free modules of arbitrary finite rank. We apply our results to $\mathfrak{sl}_2$ in order to construct new families of simple Cartan-free modules of all finite ranks.

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arXiv2025

Derivations and Hochschild cohomology of quantum nilpotent algebras

We compute the derivations of Quantum Nilpotent Algebras under a technical (but necessary) assumption on the center. As a consequence, we give an explicit description of the first Hochschild cohomology group of $U_q^+(\mathfrak{g})$, the positive part of the quantized enveloping algebra of a finite-dimensional complex simple Lie algebra $\mathfrak{g}$. Our results are obtained leveraging an initial cluster constructed by Goodearl and Yakimov.

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arXiv2025

Homological properties of some quantum Heisenberg algebras

In this paper we study the properties Koszul, Artin-Schelter regular and (skew) Calabi-Yau of some special types of quantum and generalized Heisenberg algebras and also analyze relations between these algebras, (graded) iterated Ore extensions and (graded) skew PBW extensions. The first-named author and Razavinia introduced the quantum generalized Heisenberg algebras, which depend on a parameter $q$ and two polynomials $f, g\in K[t]$. We prove that under certain conditions for $f$, $g$ these algebras are Koszul, Artin-Shelter regular, Calabi-Yau and graded Calabi-Yau.

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arXiv2024

A classification of nilpotent compatible Lie algebras

Working over an arbitrary field of characteristic different from $2$, we extend the Skjelbred-Sund method to compatible Lie algebras and give a full classification of nilpotent compatible Lie algebras up to dimension $4$. In case the base field is cubically closed, we find that there are three isomorphism classes and a one-parameter family in dimension $3$, and $12$ isomorphism classes, $6$ one-parameter families and two $2$-parameter families in dimension $4$

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arXiv2023

Representations of Smith algebras which are free over the Cartan subalgebra

In this paper, we study the category of modules over the Smith algebra which are free of finite rank over the unital polynomial subalgebra generated by the Cartan element $h$ and obtain families of such simple modules of arbitrary rank. In the case of rank one we obtain a full description of the isomorphism classes, a simplicity criterion, and an algorithm to produce all composition series. We show that all such modules have finite length and describe the composition factors and their multiplicity.

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arXiv2023

A Deleting Derivations Algorithm for Quantum Nilpotent Algebras at Roots of Unity

This paper extends an algorithm and canonical embedding by Cauchon to a large class of quantum algebras. It applies to iterated Ore extensions over a field satisfying some suitable assumptions which cover those of Cauchon's original setting but also allows for roots of unity. The extended algorithm constructs a quantum affine space $A'$ from the original quantum algebra $A$ via a series of change of variables within the division ring of fractions $\mathrm{Frac}(A)$. The canonical embedding takes a completely prime ideal $P\lhd A$ to a completely prime ideal $Q\lhd A'$ such that when $A$ is a PI algebra, ${\rm PI}\text{-}{\rm deg}(A/P) = {\rm PI}\text{-}{\rm deg}(A'/Q)$. When the quantum parameter is a root of unity we can state an explicit formula for the PI degree of completely prime quotient algebras. This paper ends with a method to construct a maximum dimensional irreducible representation of $A/P$ given a suitable irreducible representation of $A'/Q$ when $A$ is PI.

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arXiv2021

The geometric classification of nilpotent algebras

We give a geometric classification of $n$-dimensional nilpotent, commutative nilpotent and anticommutative nilpotent algebras. We prove that the corresponding geometric varieties are irreducible, find their dimensions and describe explicit generic families of algebras which define each of these varieties. We show some applications of these results in the study of the length of anticommutative algebras.

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arXiv2020

The algebraic classification of nilpotent algebras

We give the complete algebraic classification of all complex 4-dimensional nilpotent algebras. The final list has 234 (parametric families of) isomorphism classes of algebras, 66 of which are new in the literature.

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arXiv2020

Structure and isomorphisms of quantum generalized Heisenberg algebras

In [14] we introduced a new class of algebras, which we named \textit{quantum generalized Heisenberg algebras} and which depend on a parameter $q$ and two polynomials $f,g$. We have shown that this class includes all generalized Heisenberg algebras (as defined in [8] and [16]) as well as generalized down-up algebras (as defined in [3] and [7]), but the parameters of freedom we allow give rise to many algebras which are in neither one of these two classes (if $q\neq 1$ and $\, \mathsf{deg}\, f>1$). Having classified their finite-dimensional irreducible representations in [14], in this paper we turn to their classification by isomorphism, the description of their automorphism groups and the study of ring-theoretical properties like Gelfand-Kirillov dimension and being Noetherian.

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arXiv2020

Quantum generalized Heisenberg algebras and their representations

We introduce and study a new class of algebras, which we name \textit{quantum generalized Heisenberg algebras} and denote by $\mathcal{H}_q (f,g)$, related to generalized Heisenberg algebras, but allowing more parameters of freedom, so as to encompass a wider range of applications and include previously studied algebras, such as (generalized) down-up algebras. In particular, our class now includes the enveloping algebra of the $3$-dimensional Heisenberg Lie algebra and its $q$-deformation, neither of which can be realized as a generalized Heisenberg algebra. This paper focuses mostly on the classification of finite-dimensional irreducible representations of quantum generalized Heisenberg algebras, a study which reveals their rich structure. Although these algebras are not in general noetherian, their representations still retain some Lie-theoretic flavor. We work over a field of arbitrary characteristic, although our results on the representations require that it be algebraically closed.

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arXiv2020

The algebraic and geometric classification of nilpotent anticommutative algebras

We give algebraic and geometric classifications of $6$-dimensional complex nilpotent anticommutative algebras. Specifically, we find that, up to isomorphism, there are $14$ one-parameter families of $6$-dimensional nilpotent anticommutative algebras, complemented by $130$ additional isomorphism classes. The corresponding geometric variety is irreducible and determined by the Zariski closure of a one-parameter family of algebras. In particular, there are no rigid $6$-dimensional complex nilpotent anticommutative algebras.

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arXiv2019

Lie structure on the Hochschild cohomology of a family of subalgebras of the Weyl algebra

For each nonzero $h\in \mathbb{F}[x]$, where $\mathbb{F}$ is a field, let $\mathsf{A}_h$ be the unital associative algebra generated by elements $x,y$, satisfying the relation $yx-xy = h$. This gives a parametric family of subalgebras of the Weyl algebra $\mathsf{A}_1$, containing many well-known algebras which have previously been studied independently. In this paper, we give a full description the Hochschild cohomology $\mathsf{HH}^\bullet(\mathsf{A}_h)$ over a field of arbitrary characteristic. In case $\mathbb{F}$ has positive characteristic, the center of $\mathsf{A}_h$ is nontrivial and we describe $\mathsf{HH}^\bullet(\mathsf{A}_h)$ as a module over its center. The most interesting results occur when $\mathbb{F}$ has characteristic $0$. In this case, we describe $\mathsf{HH}^\bullet(\mathsf{A}_h)$ as a module over the Lie algebra $\mathsf{HH}^1(\mathsf{A}_h)$ and find that this action is closely related to the intermediate series modules over the Virasoro algebra. We also determine when $\mathsf{HH}^\bullet(\mathsf{A}_h)$ is a semisimple $\mathsf{HH}^1(\mathsf{A})$-module.

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