Deformations of type D Kleinian singularities

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.

Isomorphism Problems of Noncommutative Deformations of Type D Kleinian Singularities

We construct all possible noncommutative deformations of a Kleinian singularity ${\mathbb C}^2/Γ$ of type $D_n$ in terms of generators and relations, and solve the problem of when two deformations are isomorphic. We prove that all isomorphisms arise naturally from the action of the normalizer $N_{\SL(2)}(Γ)$ on ${\mathbb C}/Γ$. We deduce that the moduli space of isomorphism classes of noncommutative deformations in type $D_n$ is isomorphic to a vector space of dimension $n$.

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 app…

Algebras Generated by Elements with Given Spectrum and Scalar Sum and Kleinian Singularities

For a certain class of (nonunital) subalgebras of deformed preprojective algebra of affine type we describe their centers as certain deformation of Kleinian singularity and find their PI-degree. These results can be applied to algebras generated by elements with given spectrum and scalar sum.

退化扭曲迹在型A量化克莱因奇点上的研究

Degenerate twisted traces on quantized Kleinian singularities of type A

本文研究型A量化克莱因奇点上的退化扭曲迹，揭示其维度与自同构及量子化参数的关系。

Noncommutative Knörrer periodicity and noncommutative Kleinian singularities

We establish a version of Knörrer's Periodicity Theorem in the context of noncommutative invariant theory. Namely, let $A$ be a left noetherian AS-regular algebra, let $f$ be a normal and regular element of $A$ of positive degree, and take $B=A/(f)$. Then there exists a bijection between the set of isomorphism classes of indecomposable non-free maximal Cohen-Macaulay modules over $B$ and those over (a noncommutative analog of) its second double branched cover $(B^\#)^\#$. Our results use and extend the study of twisted matrix factorizations, which was introduced by the first three authors with…