arXiv subjects

Simon Crawford

Publications and source records attributed to Simon Crawford.

7 recordsLinked to original sources

arXiv2025

Resolutions of Type $\mathbb{A}$ Quantum Surface Singularities

Let $B = \Bbbk_q[u,v]^{C_{n+1}}$ be a Type $\mathbb{A}_n$ quantum Kleinian singularity, which is an example of a noncommutative surface singularity. This singularity is known to have a noncommutative quasi-crepant resolution $Λ$, which is an "algebraic" resolution of $B$. We construct a category $\mathcal{X}$ which serves as a "geometric" resolution of $B$ by adapting techniques from quiver GIT and show that $\mathcal{X}$ and $\text{mod-}Λ$ are derived equivalent. Furthermore, we show that the intersection arrangement of lines in the exceptional locus of $\mathcal{X}$ corresponds to a Type $\mathbb{A}_n$ Dynkin diagram. This generalises the geometric McKay correspondence for classical Kleinian singularities.

math.RA↗

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.

math.RA↗

arXiv2023

Group coactions on two-dimensional Artin-Schelter regular algebras

We describe all possible coactions of finite groups (equivalently, all group gradings) on two-dimensional Artin-Schelter regular algebras. We give necessary and sufficient conditions for the associated Auslander map to be an isomorphism, and determine precisely when the invariant ring for the coaction is Artin-Schelter regular. The proofs of our results are combinatorial and exploit the structure of the McKay quiver associated to the coaction.

math.RA↗

arXiv2021

Superpotentials and Quiver Algebras for Semisimple Hopf Actions

We consider the action of a semisimple Hopf algebra $H$ on an $m$-Koszul Artin-Schelter regular algebra $A$. Such an algebra $A$ is a derivation-quotient algebra for some twisted superpotential $\mathsf{w}$, and we show that the homological determinant of the action of $H$ on $A$ can be easily calculated using $\mathsf{w}$. Using this, we show that the smash product $A\,\#\,H$ is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra $Λ$ to which $A\,\#\,H$ is Morita equivalent, generalising a result of Bocklandt-Schedler-Wemyss. We also show how $Λ$ can be used to determine whether the Auslander map is an isomorphism. We compute a number of examples, and show how several results for the quantum Kleinian singularities studied by Chan-Kirkman-Walton-Zhang follow using our techniques.

math.RA↗

arXiv2019

Actions of Small Groups on Two-Dimensional Artin-Schelter Regular Algebras

In commutative invariant theory, a classical result due to Auslander says that if $R = \Bbbk[x_1, \dots, x_n]$ and $G$ is a finite subgroup of $\text{Aut}_{\text{gr}}(R) \cong \text{GL}(n,\Bbbk)$ which contains no reflections, then there is a natural graded isomorphism $R \hspace{1pt} \# \hspace{1pt} G \cong \text{End}_{R^G}(R)$. In this paper, we show that a version of Auslander's Theorem holds if we replace $R$ by an Artin-Schelter regular algebra $A$ of global dimension 2, and $G$ by a finite subgroup of $\text{Aut}_{\text{gr}}(A)$ which contains no quasi-reflections. This extends work of Chan-Kirkman-Walton-Zhang. As part of the proof, we classify all such pairs $(A,G)$, up to conjugation of $G$ by an element of $\text{Aut}_{\text{gr}}(A)$. In all but one case, we also write down explicit presentations for the invariant rings $A^G$, and show that they are isomorphic to factors of AS regular algebras.

math.RA↗

arXiv2019

Singularity categories of deformations of Kleinian singularities

Let $G$ be a finite subgroup of $\text{SL}(2,\Bbbk)$ and let $R = \Bbbk[x,y]^G$ be the coordinate ring of the corresponding Kleinian singularity. In 1998, Crawley-Boevey and Holland defined deformations $\mathcal{O}^λ$ of $R$ parametrised by weights $λ$. In this paper, we determine the singularity categories $\mathcal{D}_{\text{sg}}(\mathcal{O}^λ)$ of these deformations, and show that they correspond to subgraphs of the Dynkin graph associated to $R$. This generalises known results on the structure of $\mathcal{D}_{\text{sg}}(R)$. We also provide a generalisation of the intersection theory appearing in the geometric McKay correspondence to a noncommutative setting.

math.RA↗

arXiv2017

Azumaya skew group algebras and an application to quantum Kleinian singularities

We provide easily-verified necessary and sufficient conditions for a skew group ring, or more generally, a crossed product ring, to be an Azumaya algebra. We use our results to show that (suitable localisations of) skew group rings associated to the quantum Kleinian singularities introduced by Chan, Kirkman, Walton, and Zhang in [CKWZ16a] are Azumaya, and use this to show that these algebras are maximal orders. We also give a new proof of Auslander's Theorem for quantum Kleinian singularities.

math.RA↗