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Robert Won

Publications and source records attributed to Robert Won.

29 records · Page 2Linked to original sources
arXiv2020

Simple $\mathbb{Z}$-graded domains of Gelfand-Kirillov dimension two

Let $k$ be an algebraically closed field and $A$ a $\mathbb{Z}$-graded finitely generated simple $k$-algebra which is a domain of Gelfand-Kirillov dimension 2. We show that the category of $\mathbb{Z}$-graded right $A$-modules is equivalent to the category of quasicoherent sheaves on a certain quotient stack. The theory of these simple algebras is closely related to that of a class of generalized Weyl algebras (GWAs). We prove a translation principle for the noncommutative schemes of these GWAs, shedding new light on the classical translation principle for the infinite-dimensional primitive quotients of $U(\mathfrak{sl}_2)$.

math.RA↗
arXiv2019

Three infinite families of reflection Hopf algebras

Let $H$ be a semisimple Hopf algebra acting on an Artin-Schelter regular algebra $A$, homogeneously, inner-faithfully, preserving the grading on $A$, and so that $A$ is an $H$-module algebra. When the fixed subring $A^H$ is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that $H$ is a reflection Hopf algebra for $A$. We show that each of the semisimple Hopf algebras $H_{2n^2}$ of Pansera, and $\mathcal{A}_{4m}$ and $\mathcal{B}_{4m}$ of Masuoka is a reflection Hopf algebra for an AS regular algebra of dimension 2 or 3.

math.RA↗
arXiv2019

Sidon sets and 2-caps in $\mathbb{F}_3^n$

For each natural number $d$, we introduce the concept of a $d$-cap in $\mathbb{F}_3^n$. A subset of $\mathbb{F}_3^n$ is called a $d$-cap if, for each $k = 1, 2, \dots, d$, no $k+2$ of the points lie on a $k$-dimensional flat. This generalizes the notion of a cap in $\mathbb{F}_3^n$. We prove that the $2$-caps in $\mathbb{F}_3^n$ are exactly the Sidon sets in $\mathbb{F}_3^n$ and study the problem of determining the size of the largest $2$-cap in $\mathbb{F}_3^n$.

math.CO↗
arXiv2018

Auslander's Theorem for permutation actions on noncommutative algebras

When $A = \mathbb{k}[x_1, \ldots, x_n]$ and $G$ is a small subgroup of $\operatorname{GL}_n(\mathbb{k})$, Auslander's Theorem says that the skew group algebra $A \# G$ is isomorphic to $\operatorname{End}_{A^G}(A)$ as graded algebras. We prove a generalization of Auslander's Theorem for permutation actions on $(-1)$-skew polynomial rings, $(-1)$-quantum Weyl algebras, three-dimensional Sklyanin algebras, and a certain graded down-up algebra. We also show that certain fixed rings $A^G$ are graded isolated singularities in the sense of Ueyama.

math.RA↗
arXiv2018

Fixed rings of generalized Weyl algebras

We study actions by filtered automorphisms on classical generalized Weyl algebras (GWAs). In the case of a defining polynomial of degree two, we prove that the fixed ring under the action of a finite cyclic group of filtered automorphisms is again a classical GWA, extending a result of Jordan and Wells. Partial results are provided for the case of higher degree polynomials. In addition, we establish a version of Auslander's theorem for finite cyclic groups of filtered automorphisms acting on classical GWAs.

math.RA↗
arXiv2018

Discriminants of Taft algebra smash products and applications

A general criterion is given for when the center of a Taft algebra smash product is the fixed ring. This is applied to the study of the noncommutative discriminant. Our method relies on the Poisson methods of Nguyen, Trampel, and Yakimov, but also makes use of Poisson Ore extensions. Specifically, we fully determine the inner faithful actions of Taft algebras on quantum planes and quantum Weyl algebras. We compute the discriminant of the corresponding smash product and apply it to compute the Azumaya locus and restricted automorphism group.

math.RA↗
arXiv2018

The noncommutative schemes of generalized Weyl algebras

The first Weyl algebra over $k$, $A_1 = k \langle x, y\rangle/(xy-yx - 1)$ admits a natural $\mathbb{Z}$-grading by letting $\operatorname{deg} x = 1$ and $\operatorname{deg} y = -1$. Paul Smith showed that $\operatorname{gr}- A_1$ is equivalent to the category of quasicoherent sheaves on a certain quotient stack. Using autoequivalences of $\operatorname{gr}- A_1$, Smith constructed a commutative ring $C$, graded by finite subsets of the integers. He then showed $\operatorname{gr}- A_1 \equiv \operatorname{gr}- (C, \mathbb{Z}_{\mathrm{fin}})$. In this paper, we generalize results of Smith by using autoequivalences of a graded module category to construct rings with equivalent graded module categories. For certain generalized Weyl algebras, we use autoequivalences defined in a companion paper so that these constructions yield commutative rings.

math.RA↗
arXiv2017

A structure theorem for product sets in extra special groups

Hegyvári and Hennecart showed that if $B$ is a sufficiently large brick of a Heisenberg group, then the product set $B\cdot B$ contains many cosets of the center of the group. We give a new, robust proof of this theorem that extends to all extra special groups as well as to a large family of quasigroups.

math.NT↗
arXiv2016

The Picard group of the graded module category of a generalized Weyl algebra

The first Weyl algebra, $A_1 = k \langle x, y\rangle/(xy-yx - 1)$ is naturally $\mathbb{Z}$-graded by letting $\operatorname{deg} x = 1$ and $\operatorname{deg} y = -1$. Sue Sierra studied $\operatorname{gr}- A_1$, category of graded right $A_1$-modules, computing its Picard group and classifying all rings graded equivalent to $A_1$. In this paper, we generalize these results by studying the graded module category of certain generalized Weyl algebras. We show that for a generalized Weyl algebra $A(f)$ with base ring $k[z]$ defined by a quadratic polynomial $f$, the Picard group of $\operatorname{gr}- A(f)$ is isomorphic to the Picard group of $\operatorname{gr}- A_1$. In a companion paper, we use these results to construct commutative rings which are graded equivalent to generalized Weyl algebras.

math.RA↗
arXiv2013

Partitions of AG(4,3) into Maximal Caps

In a geometry, a maximal cap is a collection of points of largest size containing no lines. In AG(4,3), maximal caps contain 20 points. The 81 points of AG(4,3) can be partitioned into 4 mutually disjoint maximal caps together with a single point P, where every pair of points that makes a line with P lies entirely inside one of those caps. The caps in a partition can be paired up so that both pairs are either in exactly one partition or they are both in two different partitions. This difference determines the two equivalence classes of partitions of AG(4,3) under the action by affine transformations.

math.CO↗