arXiv subjects

K. R. Goodearl

Publications and source records attributed to K. R. Goodearl.

At least 19 recordsLinked to original sources

arXiv2026

Skew fields not finitely generated as algebras

An associative division algebra D is said to be _affine_ over a central subfield k if D is finitely generated as a k-algebra. In 1956 Amitsur famously proved that, when k is uncountable, D cannot be k-affine unless D is algebraic over k. In this paper we consider affineness -- and nonaffineness -- for certain naturally occurring classes of division algebras over arbitrary fields. The primary applications are to division algebras of fractions of suitably conditioned iterated skew polynomial rings over k, including many examples naturally arising in Lie theoretic and quantum group settings. Many transcendental division algebras are thus verified to be nonaffine over k. Division algebras of fractions of Weyl algebras and quantum affine spaces are determined to be affine over their centers exactly when they are finite dimensional over their centers.

math.RA↗

arXiv2025

Extensions of locally matricial and locally semisimple algebras

Two extension problems are solved. First, the class of locally matricial algebras over an arbitrary field is closed under extensions. Second, the class of locally finite dimensional semisimple algebras over a fixed field is closed under extensions if and only if the base field is perfect. Regardless of the base field, extensions of the latter type are always locally unit-regular.

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arXiv2024

Finite Dimensional Representations of Quivers with Oriented Cycles

Let $K$ be a field, $Q$ a quiver, and $\mathcal{A}$ the ideal of the path algebra $KQ$ that is generated by the arrows of $Q$. We present old and new results about the representation theories of the truncations $KQ/\mathcal{A}^L$, $L \in \mathbb{N}$, tracking their development as $L$ goes to infinity. The goal is to gain a better understanding of the category of those finite dimensional $KQ$-modules which arise as finitely generated modules over admissible quotients of $KQ$.

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arXiv2024

Binomial ideals in quantum tori and quantum affine spaces

The article targets binomial ideals in quantum tori and quantum affine spaces. First, noncommutative analogs of known results for commutative (Laurent) polynomial rings are obtained, including the following: Under the assumption of an algebraically closed base field, it is proved that primitive ideals are binomial, as are radicals of binomial ideals and prime ideals minimal over binomial ideals. In the case of a quantum torus $\mathcal{T}_{\bf{q}}$, the results are strongest: In this situation, the binomial ideals are parametrized by characters on sublattices of the free abelian group whose group algebra is the center of $\mathcal{T}_{\bf{q}}$; the sublattice-character pairs corresponding to primitive ideals as well as to radicals and minimal primes of binomial ideals are determined. As for occurrences of binomial ideals in quantum algebras: It is shown that cocycle-twisted group algebras of finitely generated abelian groups are quotients of quantum tori modulo binomial ideals. Another appearance is as follows: Cocycle-twisted semigroup algebras of finitely generated commutative monoids, as well as quantum affine toric varieties, are quotients of quantum affine spaces modulo certain types of binomial ideals.

math.QA↗

arXiv2022

Spectra of quantum algebras

This is a survey of what is known and/or conjectured about the prime and primitive spectra of quantum algebras, of quantized coordinate rings in particular. The topological structure of these spectra, their relations to classical affine algebraic varieties, and their relations to each other are discussed.

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arXiv2021

Poisson catenarity in Poisson nilpotent algebras

We prove that for the iterated Poisson polynomial rings known as Poisson nilpotent algebras (or Poisson-CGL extensions), the Poisson prime spectrum is catenary, i.e., all saturated chains of inclusions of Poisson prime ideals between any two given Poisson prime ideals have the same length.

math.AC↗

arXiv2020

Catenarity in quantum nilpotent algebras

In this paper, it is established that quantum nilpotent algebras (also known as CGL extensions) are catenary, i.e., all saturated chains of inclusions of prime ideals between any two given prime ideals $P \subsetneq Q$ have the same length. This is achieved by proving that the prime spectra of these algebras have normal separation, and then establishing the mild homological conditions necessary to apply a result of Lenagan and the first author. The work also recovers the Tauvel height formula for quantum nilpotent algebras, a result that was first obtained by Lenagan and the authors through a different approach.

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arXiv2020

Integral quantum cluster structures

We prove a general theorem for constructing integral quantum cluster algebras over ${\mathbb{Z}}[q^{\pm 1/2}]$, namely that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures. These algebras are then shown to be isomorphic to the corresponding upper quantum cluster algebras, again defined over ${\mathbb{Z}}[q^{\pm 1/2}]$. Previously, this was only known for acyclic quantum cluster algebras. The theorem is applied to prove that for every symmetrizable Kac-Moody algebra ${\mathfrak{g}}$ and Weyl group element $w$, the dual canonical form $A_q({\mathfrak{n}}_+(w))_{\mathbb{Z}[q^{\pm 1}]}$ of the corresponding quantum unipotent cell has the property that $A_q( {\mathfrak{n}}_+(w))_{\mathbb{Z}[q^{\pm 1}]} \otimes_{\mathbb{Z}[q^{ \pm 1}]} {\mathbb{Z}}[ q^{\pm 1/2}]$ is isomorphic to a quantum cluster algebra over ${\mathbb{Z}}[q^{\pm 1/2}]$ and to the corresponding upper quantum cluster algebra over ${\mathbb{Z}}[q^{\pm 1/2}]$.

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arXiv2018

The Berenstein-Zelevinsky quantum cluster algebra conjecture

We prove the Berenstein-Zelevinsky conjecture that the quantized coordinate rings of the double Bruhat cells of all finite dimensional simple algebraic groups admit quantum cluster algebra structures with initial seeds as specified by [4]. We furthermore prove that the corresponding upper quantum cluster algebras coincide with the constructed quantum cluster algebras and exhibit a large number of explicit quantum seeds. Along the way a detailed study of the properties of quantum double Bruhat cells from the viewpoint of noncommutative UFDs is carried out and a quantum analog of the Fomin-Zelevinsky twist map is constructed and investigated for all double Bruhat cells. The results are valid over base fields of arbitrary characteristic and the deformation parameter is only assumed to be a non-root of unity.

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arXiv2018

Closures in varieties of representations and irreducible components

For any truncated path algebra $Λ$ of a quiver, we classify, by way of representation-theoretic invariants, the irreducible components of the parametrizing varieties $\mathbf{Rep}_{\mathbf{d}}(Λ)$ of the $Λ$-modules with fixed dimension vector $\mathbf{d}$. In this situation, the components of $\mathbf{Rep}_{\mathbf{d}}(Λ)$ are always among the closures $\overline{\mathbf{Rep}\,\mathbb{S}}$, where $\mathbb{S}$ traces the semisimple sequences with dimension vector $\mathbf{d}$, and hence the key to the classification problem lies in a characterization of these closures. Our first result concerning closures actually addresses arbitrary basic finite dimensional algebras over an algebraically closed field. In the general case, it corners the closures $\overline{\mathbf{Rep}\,\mathbb{S}}$ by means of module filtrations "governed by $\mathbb{S}$", in case $Λ$ is truncated, it pins down the $\overline{\mathbf{Rep}\,\mathbb{S}}$ completely. The analysis of the varieties $\overline{\mathbf{Rep}\,\mathbb{S}}$ leads to a novel upper semicontinuous module invariant which provides an effective tool towards the detection of components of $\mathbf{Rep}_{\mathbf{d}}(Λ)$ in general. It detects all components when $Λ$ is truncated.

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arXiv2018

Understanding finite dimensional representations generically

We survey the development and status quo of a subject best described as "generic representation theory of finite dimensional algebras", which started taking shape in the early 1980s. Let $Λ$ be a finite dimensional algebra over an algebraically closed field. Roughly, the theory aims at (a) pinning down the irreducible components of the standard parametrizing varieties for the $Λ$-modules with a fixed dimension vector, and (b) assembling generic information on the modules in each individual component, that is, assembling data shared by all modules in a dense open subset of that component. We present an overview of results spanning the spectrum from hereditary algebras through the tame non-hereditary case to wild non-hereditary algebras.

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arXiv2018

Cluster algebra structures on Poisson nilpotent algebras

Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac--Moody groups. We prove that every Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements.

math.AC↗

arXiv2017

Twist invariants of graded algebras

We define two invariants for (semiprime right Goldie) algebras, one for algebras graded by arbitrary abelian groups, which is unchanged under twists by $2$-cocycles on the grading group, and one for $\mathbb Z$-graded or $\mathbb Z_{\ge 0}$-filtered algebras. The first invariant distinguishes quantum algebras which are "truly multiparameter" apart from ones that are "essentially uniparameter", meaning cocycle twists of uniparameter algebras. We prove that both invariants are stable under adjunction of polynomial variables. Methods for computing these invariants for large families of algebras are given, including quantum nilpotent algebras and algebras admitting one quantum cluster, and applications to non-isomorphism theorems are obtained.

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arXiv2016

Non-affine Hopf algebra domains of Gelfand-Kirillov dimension two

We classify all non-affine Hopf algebras $H$ over an algebraically closed field $k$ of characteristic zero that are integral domains of Gelfand-Kirillov dimension two and satisfy the condition $\text{Ext}^1_H(k, k) \neq 0$. The affine ones were classified by the authors in 2010.

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arXiv2015

The realization problem for some wild monoids and the Atiyah problem

The Realization Problem for (von Neumann) regular rings asks what are the conical refinement monoids which can be obtained as the monoids of isomorphism classes of finitely generated projective modules over a regular ring. The analogous realization question for the larger class of exchange rings is also of interest. A refinement monoid is said to be wild if it cannot be expressed as a direct limit of finitely generated refinement monoids. In this paper, we consider the problem of realizing some concrete wild refinement monoids by regular rings and by exchange rings. The most interesting monoid we consider is the monoid M obtained by successive refinements of the identity x_0+y_0=x_0+z_0. This monoid is known to be realizable by the algebra A = K[F] of the monogenic free inverse monoid F, for any choice of field K, but A is not an exchange ring. We show that, for any uncountable field K, M is not realizable by a regular K-algebra, but that a suitable universal localization of A provides an exchange, non-regular, K-algebra realizing M. For any countable field F, we show that a skew version of the above construction gives a regular F-algebra realizing M. Finally, we develop some connections with the Atiyah Problem for the lamplighter group G (the wreath product of Z/2Z by Z). We prove that the algebra A can be naturally seen as a *-subalgebra of the group algebra KG, for any complex subfield K closed under conjugation, and we determine the structure of the *-regular closure of A in the regular ring of G. Using this, we show that the subgroup of R generated by the von Neumann dimensions of matrices over KG contains the rational field.

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arXiv2015

From quantum Ore extensions to quantum tori via noncommutative UFDs

All iterated skew polynomial extensions arising from quantized universal enveloping algebras of Kac-Moody algebras are special examples of a very large, axiomatically defined class of algebras, called CGL extensions. For the purposes of constructing initial clusters for quantum cluster algebra structures on an algebra R, and classification of the automorphisms of R, one needs embeddings of R into quantum tori T which have the property that R contains the corresponding quantum affine space algebra A. We explicitly construct such an embedding A \subseteq R \subset T for each CGL extension R using the methods of noncommutative noetherian unique factorization domains and running a Gelfand-Tsetlin type procedure with normal, instead of central elements. Along the way we classify the homogeneous prime elements of all CGL extensions and we prove that each CGL extension R has an associated maximal torus which covers the automorphisms of R corresponding to all normal elements. For symmetric CGL extensions, we describe the relationship between our quantum affine space algebra A and Cauchon's quantum affine space algebra generated by elements obtained via deleting derivations.

math.RA↗

arXiv2015

Quantum cluster algebra structures on quantum nilpotent algebras

All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always equal the corresponding upper quantum cluster algebras. Previous approaches to these problems for the construction of (quantum) cluster algebra structures on (quantized) coordinate rings arising in Lie theory were done on a case by case basis relying on the combinatorics of each concrete family. The results of the paper have a broad range of applications to these problems, including the construction of quantum cluster algebra structures on quantum unipotent groups and quantum double Bruhat cells (the Berenstein-Zelevinsky conjecture), and treat these problems from a unified perspective. All such applications also establish equality between the constructed quantum cluster algebras and their upper counterparts. The proofs rely on Chatters' notion of noncommutative unique factorization domains. Toric frames are constructed by considering sequences of homogeneous prime elements of chains of noncommutative UFDs (a generalization of the construction of Gelfand-Tsetlin subalgebras) and mutations are obtained by altering chains of noncommutative UFDs. Along the way, an intricate (and unified) combinatorial model for the homogeneous prime elements in chains of noncommutative UFDs and their alterations is developed. When applied to special families, this recovers the combinatorics of Weyl groups and double Weyl groups previously used in the construction and categorification of cluster algebras. It is expected that this combinatorial model of sequences of homogeneous prime elements will have applications to the unified categorification of quantum nilpotent algebras.

math.QA↗

arXiv2014

Tame and wild refinement monoids

The class of refinement monoids (abelian monoids satisfying the Riesz refinement property) is subdivided into those which are tame, defined as being an inductive limit of finitely generated refinement monoids, and those which are wild, i.e., not tame. It is shown that tame refinement monoids enjoy many positive properties, including separative cancellation ($2x=2y=x+y \implies x=y$) and multiplicative cancellation with respect to the algebraic ordering ($mx\le my \implies x\le y$). In contrast, examples are constructed to exhibit refinement monoids which enjoy all the mentioned good properties but are nonetheless wild.

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