arXiv subjects

Robert Won

Publications and source records attributed to Robert Won.

At least 19 recordsLinked to original sources

arXiv2026

Non-attacking rook placements on crossword grids

We introduce the notion of a non-attacking rook placement on a crossword grid. A crossword grid is a collection of white squares (which comprise across and down words) and black squares (which separate the words), and a complete non-attacking rook placement on such a grid is a subset of white squares which intersects every across and every down word exactly once. We prove an upper bound on the number of rook placements that a general grid can admit. We then study sparse grids in which no two black squares share an edge and show that rook placements on certain sparse grids correspond bijectively to alternating sign matrices with prescribed $-1$ entries. Specializing further to permutation grids, we prove that every permutation grid admits at least one rook placement, and characterize the permutations whose grids admit exactly one placement in terms of the Robinson--Schensted correspondence. Throughout, we pose a variety of conjectures and open questions.

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arXiv2025

Log-ozone groups and centers of polynomial Poisson algebras

In previous work, the authors introduced the ozone group of an associative algebra as the subgroup of automorphisms which fix the center pointwise. The authors studied PI skew polynomial algebras, using the ozone group to understand their centers and to characterize them among graded algebras. In this work, we introduce and study the log-ozone group of a Poisson algebra over a field of positive characteristic. The log-ozone group is then used to characterize polynomial Poisson algebras with skew symmetric structure. We prove that unimodular Poisson algebras with skew symmetric structure have Gorenstein centers. A related result is proved for graded polynomial Poisson algebras of dimension three.

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arXiv2025

Higher rank Bell--Rogalski algebras

We generalize a construction of Bell and Rogalski to realize new examples of $\mathbb{Z}^n$-graded simple rings. This construction also generalizes TGWAs of type $(A_1)^n$. In addition to considering basic properties of these algebras, we provide a classification of weight modules in the setting of torsion-free orbits, study their (twisted) tensor products, and provide a simplicity criterion.

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arXiv2025

Homological Integrals for Weak Hopf Algebras

We introduce the notion of a homological integral for an infinite-dimensional weak Hopf algebra and use the homological integral to prove several structure theorems. For example, we prove that the Artin--Schelter property and the Van den Bergh condition are equivalent for a noetherian weak Hopf algebra, and that the antipode is automatically invertible in this case. We also prove a decomposition theorem that states that any weak Hopf algebra finite over an affine center is a direct sum of Artin--Schelter Gorenstein, Cohen--Macaulay, GK dimension homogeneous weak Hopf algebras.

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arXiv2024

Actions of Taft Algebras on Noetherian Down-Up Algebras

We consider actions of Taft algebras on noetherian graded down-up algebras. We classify all such actions and determine properties of the corresponding invariant rings $A^T$. We identify precisely when $A^T$ is commutative, when it is Artin-Schelter regular, and give sufficient conditions for it to be Artin-Schelter Gorenstein. Our results show that many results and conjectures in the literature concerning actions of semisimple Hopf algebras on Artin-Schelter regular algebras can fail when the semisimple hypothesis is omitted.

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arXiv2023

Ozone groups of Artin--Schelter regular algebras satisfying a polynomial identity

We study the ozone group of noetherian Artin--Schelter regular algebras satisfying a polynomial identity (or PI for short). The ozone group was shown in previous work by the authors to be an important invariant in the study of PI skew polynomial rings and their centers. In this paper, we show that skew polynomial rings are in fact characterized as those algebras with maximal rank ozone groups. We also classify those with trivial ozone groups, which must necessarily be Calabi--Yau. This class includes most three-dimensional PI Sklyanin algebras. Further examples and applications are given, including applications to the Zariski Cancellation Problem.

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arXiv2023

Ozone groups and centers of skew polynomial rings

We introduce the ozone group of a noncommutative algebra $A$, defined as the group of automorphisms of $A$ which fix every element of its center. In order to initiate the study of ozone groups, we study PI skew polynomial rings, which have long proved to be a fertile testing ground in noncommutative algebra. Using the ozone group and other invariants defined herein, we give explicit conditions for the center of a PI skew polynomial to be Gorenstein (resp. regular) in low dimension.

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arXiv2023

Weight modules over Bell--Rogalski algebras

We study a class of $\mathbb{Z}$-graded algebras introduced by Bell and Rogalski. Their construction generalizes in large part that of rank one generalized Weyl algebras (GWAs). We establish certain ring-theoretic properties of these algebras and study their connection to GWAs. We classify the simple weight modules in the infinite orbit case and provide a partial classification in the case of orbits of finite order.

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arXiv2023

Weighted homological regularities

Let $A$ be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the homological and internal degrees of cochain complexes of graded $A$-modules, providing weighted versions of Castelnuovo--Mumford regularity, Tor-regularity, Artin--Schelter regularity, and concavity. In some cases an invariant (such as Tor-regularity) that is infinite can be replaced with a weighted invariant that is finite, and several homological invariants of complexes can be expressed as weighted homological regularities. We prove a few weighted homological identities some of which unify different classical homological identities and produce interesting new ones.

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arXiv2022

Symmetries of algebras captured by actions of weak Hopf algebras

In this paper, we present a generalization of well-established results regarding symmetries of $\Bbbk$-algebras, where $\Bbbk$ is a field. Traditionally, for a $\Bbbk$-algebra $A$, the group $\Bbbk$-algebra automorphisms of $A$ captures the symmetries of $A$ via group actions. Similarly, the Lie algebra of derivations of $A$ captures the symmetries of $A$ via Lie algebra actions. In this paper, given a category $\mathcal{C}$ whose objects possess $\Bbbk$-linear monoidal categories of modules, we introduce an object $\operatorname{Sym}_{\mathcal{C}}(A)$ that captures the symmetries of $A$ via actions of objects in $\mathcal{C}$. Our study encompasses various categories whose objects include groupoids, Lie algebroids, and more generally, cocommutative weak Hopf algebras. Notably, we demonstrate that for a positively graded non-connected $\Bbbk$-algebra $A$, some of its symmetries are naturally captured within the weak Hopf framework.

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arXiv2022

Universal quantum semigroupoids

We introduce the concept of a universal quantum linear semigroupoid (UQSGd), which is a weak bialgebra that coacts on a (not necessarily connected) graded algebra $A$ universally while preserving grading. We restrict our attention to algebraic structures with a commutative base so that the UQSGds under investigation are face algebras (due to Hayashi). The UQSGd construction generalizes the universal quantum linear semigroups introduced by Manin in 1988, which are bialgebras that coact on a connected graded algebra universally while preserving grading. Our main result is that when $A$ is the path algebra $\Bbbk Q$ of a finite quiver $Q$, each of the various UQSGds introduced here is isomorphic to the face algebra attached to $Q$. The UQSGds of preprojective algebras and of other algebras attached to quivers are also investigated.

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arXiv2022

Pointed Hopf actions on quantum generalized Weyl algebras

We study actions of pointed Hopf algebras in the $\ZZ$-graded setting. Our main result classifies inner-faithful actions of generalized Taft algebras on quantum generalized Weyl algebras which respect the $\ZZ$-grading. We also show that generically the invariant rings of Taft actions on quantum generalized Weyl algebras are commutative Kleinian singularities.

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arXiv2022

Reflexive hull discriminants and applications

We introduce the reflexive hull discriminant as a tool to study noncommutative algebras that are finitely generated, but not necessarily free, over their centers. As an example, we compute the reflexive hull discriminants for quantum generalized Weyl algebras and use them to determine automorphism groups and other properties, recovering results of Su{á}rez-Alvarez, Vivas, and others.

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arXiv2021

Algebraic structures in comodule categories over weak bialgebras

For a bialgebra $L$ coacting on a $\Bbbk$-algebra $A$, a classical result states that $A$ is a right $L$-comodule algebra if and only if $A$ is an algebra in the monoidal category $\mathcal{M}^{L}$ of right $L$-comodules; the former notion is formulaic while the latter is categorical. We generalize this result to the setting of weak bialgebras $H$. The category $\mathcal{M}^H$ admits a monoidal structure by work of Nill and Böhm-Caenepeel-Janssen, but the algebras in $\mathcal{M}^H$ are not canonically $\Bbbk$-algebras. Nevertheless, we prove that there is an isomorphism between the category of right $H$-comodule algebras and the category of algebras in $\mathcal{M}^H$. We also recall and introduce the formulaic notion of $H$ coacting on a $\Bbbk$-coalgebra and on a Frobenius $\Bbbk$-algebra, respectively, and prove analogous category isomorphism results. Our work is inspired by the physical applications of Frobenius algebras in tensor categories and by symmetries of algebras with a base algebra larger than the ground field (e.g. path algebras). We produce examples of the latter by constructing a monoidal functor from a certain corepresentation category of a bialgebra $L$ to the corepresentation category of a weak bialgebra built from $L$ (a "quantum transformation groupoid"), thereby creating weak quantum symmetries from ordinary quantum symmetries.

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arXiv2021

Degree bounds for Hopf actions on Artin-Schelter regular algebras

We study semisimple Hopf algebra actions on Artin-Schelter regular algebras and prove several upper bounds on the degrees of the minimal generators of the invariant subring, and on the degrees of syzygies of modules over the invariant subring. These results are analogues of results for group actions on commutative polynomial rings proved by Noether, Fogarty, Fleischmann, Derksen, Sidman, Chardin, and Symonds.

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arXiv2021

Homological regularities and concavities

This paper concerns homological notions of regularity for noncommutative algebras. Properties of an algebra $A$ are reflected in the regularities of certain (complexes of) $A$-modules. We study the classical Tor-regularity and Castelnuovo-Mumford regularity, which were generalized from the commutative setting to the noncommutative setting by J{\o}rgensen and Dong-Wu. We also introduce two new numerical homological invariants: concavity and Artin-Schelter regularity. Artin-Schelter regular algebras occupy a central position in noncommutative algebra and noncommutative algebraic geometry, and we use these invariants to establish criteria which can be used to determine whether a noetherian connected graded algebra is Artin-Schelter regular.

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arXiv2021

Improved bounds on sizes of generalized caps in $AG(n,q)$

An $m$-general set in $AG(n,q)$ is a set of points such that any subset of size $m$ is in general position. A $3$-general set is often called a capset. In this paper, we study the maximum size of an $m$-general set in $AG(n,q)$, significantly improving previous results. When $m=4$ and $q=2$ we give a precise estimate, solving a problem raised by Bennett.

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arXiv2021

Semisimple Reflection Hopf Algebras of Dimension Sixteen

For each nontrivial semisimple Hopf algebra $H$ of dimension sixteen over $\mathbb{C}$, the smallest dimension inner-faithful representation of $H$ acting on a quadratic AS regular algebra $A$ of dimension 2 or 3, homogeneously and preserving the grading, is determined. Each invariant subring $A^H$ is determined. When $A^H$ is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that $H$ is a reflection Hopf algebra for $A$.

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