arXiv subjects

João Schwarz

Publications and source records attributed to João Schwarz.

13 recordsLinked to original sources

arXiv2026

Growth functions of algebras and an application to Leavitt path algebras

In this paper, we consider three growth functions of algebras: the Gelfand-Kirillov dimension, superdimension, and entropy -- as well as a variation of the latter successfully used in the study of Leavitt path algebras. We prove results that make precise the heuristic fact that the Gelfand-Kirillov dimension is suitable for the study of algebras with polynomial growth, the superdimension for algebras with subexponential growth, and the entropy for algebras with exponential growth. We introduce a definition of entropy for graded modules motivated by the entropy for graded algebras. As an application of our study, we give a new criterion for Leavitt path algebras to be PI.

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arXiv2026

Galois Rings: Ring-Theoretic Properties and Applications to Coulomb Branches and Affine Hecke Algebras

Galois rings and Galois orders, introduced by Futorny and Ovsienko, are realized as subrings of fixed subrings of skew group (or monoid) rings and have numerous applications in the structure and representation theory of associative algebras. This paper consists of two parts. The first parte investigates ring-theoretic properties that follow from the Galois ring structure alone. In particular, we estabilish natural conditions under which Galois are Ore domains or (semi)prime Goldie rings. We also study several ring-theoretic dimensions and combine the theories of Galois rings and PI-algebras to obtain new structural results. In the second part, we apply these results, together with general techniques from ring theory, to affine Hecke algebras in the sense of Ginzburg, Kapranov, and Vasserot, as well as to spherical Coulomb branch algebras. In particular, we prove that these algebras are Jacobson semiprimitive, satisfy the Nullstellensatz, and determine several of their ring-theoretic dimensions. For affine Hecke algebras, we further prove that they satisfy the maximal Nullstellensatz, are integral over their centers, and determine their T-ideals of polynomial identities, PI-degree, and PI-exponents. For spherical Coulomb branch algebras, we compute the Krull dimension, estabilish that they satisfy the Gelfand-Kirillov conjecture, and prove that they are not PI-algebras

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arXiv2026

Irreducible non-holonomic modules for rational Cherednik algebras

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and let $A_n(\mathbb{K})$ be the $n$-th Weyl algebra. We prove that for every complex reflection group $G$ and every $n \geq 2$ the ring of invariant differential operators $A_n(\mathbb{K})^G$ has irreducible non-holonomic modules of Gelfand-Kirillov dimension $2n-1$, and we exhibit such modules explicitly. Through the Morita equivalence between $A_n(\mathbb{K})^G$ and the rational Cherednik algebra $H_{\mathfrak c}$ at an integral parameter, we deduce that $H_{\mathfrak c}$ has irreducible non-holonomic modules of Gelfand-Kirillov dimension $2n-1$ for $n \geq 2$. In an appendix, following a proof kindly shared by O. Mathieu, we show that $A_n(\mathbb{K})$ has irreducible modules of every Gelfand-Kirillov dimension in the interval $[n, 2n-1]$.

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arXiv2025

Harish-Chandra modules and Galois orders revisited

The main subject of study of this paper are general properties of HarishChandra algebras and modules with respect wito a pair of algebra and subalgebra, with special focus on the transfer properties to a "spherical subalgebra". We also discuss general properties of Galois rings and algebras, where the former discussion is specialized, and we obtain an important link between different approaches to it in the literature. Then we focus our study into finite multiplicative invariants on the ring of differential operators on the torus and fixed rings under the action of a finite group of algebra automorphisms of generalized Weyl algebras. We study freeness over the Harish-Chandra subalgebra and the Gelfand-Kirillov Conjecture for them. Our last section construction some concrete irreducible Harish-Chandra modules. This paper also introduces the notion of an infinite rank generalized Weyl algebra.

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arXiv2025

A note on the definition of derived functors

The purpose of this note is to consider in detail the construction of derived functors. The classical construction, such as in Cartan-Eilenberg or Grothendieck, is clarified, and it is shown, at the same time, that everything can be formalized in ZFC, unlike the approach using derived categories. Our work is done in a more general context in which the codomain of our functors is any Grothendieck category, not necessarily abelian groups.

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arXiv2024

Gelfand-Kirillov conjecture as a first-order formula

Let $Σ$ be a (reduced) root system. Let $\mathsf{k}$ be an algebraically closed field of zero characteristic, and consider the corresponding semisimple Lie algebra $\mathfrak{g}_{\mathsf{k}, Σ}$. Then there is a first-order sentence $ϕ_Σ$ in the language $\mathcal{L}=(1,0,+,*,-)$ of rings such that, for any algebraically closed field $\mathsf{k}$ of characteristic 0, the validity of the Gelfand-Kirillov Conjecture for $\mathfrak{g}_{\mathsf{k}, Σ}$ is equivalent to $ACF_0 \vdash ϕ_Σ$.

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arXiv2024

Hilbert-Samuel Polynomials for Algebras with Special Filtrations

The notion of multiplicity of a module first arose as consequence of Hilbert's work on commutative algebra, relating the dimension of rings with the degree of certain polynomials. For noncommutative rings, the notion of multiplicity first appeared in the context of modules for the Weyl algebra in Bernstein's solution of the problem of analytic continuation posed by I. Gelfand. The notion was shown to be useful to many more noncommutative rings, especially enveloping algebras, rings of differential operators, and quantum groups. In all these cases, the existence of multiplicity is related to the existence of Hilbert-Samuel polynomials. In this work we give an axiomatic definition of algebras with a notion of multiplicity, which we call very nice and modest algebras. We show, in an abstract setting, how the existence of Hilbert-Samuel polynomials implies the existence of a notion of multiplicity. We apply our results for the category of min-holonomic modules -- a notion which coincides with holonomic modules for simple algebras -- and that shares many similarities with it. In particular, we generalize the usual results in the literature that are stated for Ore domains, in the more general context of prime algebras, and we show that rational Cherednik algebras admit a notion of multiplicity.

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arXiv2024

Generalizations of noncommutative Noether's problem

Noether's problem is classical and very important problem in algebra. It is an intrinsically interesting problem in invariant theory, but with far reaching applications in the sutdy of moduli spaces, PI-algebras, and the Inverse problem of Galois theory, among others. To obtain an noncommutative analogue of Noether's problem, one would need a significant skew field that shares a role similar to the field of ratioal functions. Given the importance of the Weyl fields due to Gelfand-Kirillov's Conjecture, in 2006 J. Alev and F. Dumas introduced what is nowdays called the noncommutative Noether's problem. Many papers in recent years \cite{FMO}, \cite{EFOS}, \cite{FS}, \cite{Tikaradze} have been dedicated to the subject. The aim of this article is to generalize the main result of \cite{FS} for more general versions of Noether's problem; and consider its analogue in prime characteristic.

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arXiv2021

Holonomic modules for rings of invariant differential operators

We study holonomic modules for the rings of invariant differential operators on affine commutative domains with finite Krull dimension with respect to arbitrary actions of finite groups. We prove the Bernstein inequality for these rings. Our main tool is the filter dimension introduced by Bavula. We extend the results for the invariants of the Weyl algebra with respect to the symplectic action of a finite group, for the rings of invariant differential operators on quotient varieties, and invariants of certain generalized Weyl algebras under the linear actions. We show that the filter dimension of all above mentioned algebras equals 1.

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arXiv2020

On rational twisted generalized Weyl algebra

The aim of this work is to investigate the structure of some skew twisted algebras, when the coefficient ring is a localization of the polynomial ring over the field of characteristic zero, and an involution is provided. A parallel construction of the rational twisted generalized Weyl algebras is given. We propose a method and explicit formulas for a constructive description of these algebras and their involution-symmetric invariant subalgebras based on the Gelfand-Zeitlin realization of the universal enveloping algebra of some complex Lie algebras. As concrete examples we discuss special unitary and orthogonal algebras of rank three.

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arXiv2020

LD-stability for Goldie rings

The lower transcendence degree, introduced by J. J Zhang, is an important non-commutative invariant in ring theory and non-commutative geometry strongly connected to the classical Gelfand-Kirillov transcendence degree. For LD-stable algebras, the lower transcendence degree coincides with the Gelfand-Kirillov dimension. We show that the following algebras are LD-stable and compute their lower transcendence degrees: rings of differential operators of affine domains, universal enveloping algebras of finite dimensional Lie superalgebras, symplectic reflection algebras and their spherical subalgebras, finite $W$-algebras of type $A$, generalized Weyl algebras over Noetherian domain (under a mild condition), some quantum groups. We show that the lower transcendence degree behaves well with respect to the invariants by finite groups, and with respect to the Morita equivalence. Applications of these results are given.

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arXiv2020

A Poisson Analogue of Noether's Problem

In this paper we show that the Poisson analogue of the Noether's Problem has a positive solution for essentially all finite symplectic reflection groups - the analogue of complex reflection groups in the symplectic world. Our proofs are constructive, and generalize and refines previously known results. As an interesting consequence of the solution of this problem for complex reflection groups, we obtain the Poisson rationality of the Calogero-Moser spaces associated to any complex reflection group. The results of this paper can be thought as analogues of the Noncommutative Noether Problem and the Gelfand-Kirillov Conjecture for rational Cherednik algebras in the 'quasi-classical limit'. In the second half of the paper, an abstract framework to understand these results is introduced, and it is shown that every Coloumb branch of a $3d \, \mathcal{N}=4$ gauge theory is Poisson rational as an application. We also obtain the Gelfand-Kirillov Conjecture for trigonometric Cherednik algebras and the Poisson rationality of their trigonometric Calogero-Moser spaces at the same time.

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arXiv2018

Noncommutative Noether's Problem vs Classical Noether's Problem

We address the Noncommutative Noether's Problem on the invariants of Weyl fields for linear actions of finite groups. We prove that if the variety An(k)/G is rational then the Noncommutative Noether's Problem is positively solved for G and any field k of characteristic zero. In particular, this gives positive solution for all pseudo-reflections groups, for the alternating groups (n = 3,4,5) and for any finite group when n = 3. Alternative proofs are given for the complex field and for all pseudo-reflections groups. In the later case an effective algorithm of finding the Weyl generators is described. We also study birational equivalence for the rings of invariant differential operators on complex affine irreducible varieties.

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