量子欧几里得 $2n$-空间的 Azumaya 轨迹

Centers and Azumaya loci of finite $W$-algebras

In this paper, we study the center $Z$ of the finite $W$-algebra $\mathcal{T}(\mathfrak{g},e)$ associated with a semi-simple Lie algebra $\mathfrak{g}$ over an algebraically closed field $\mathds{k}$ of characteristic $p\gg0$, and an arbitrarily given nilpotent element $e\in\mathfrak{g}$. We obtain an analogue of Veldkamp's theorem on the center. For the maximal spectrum $\text{Specm}(Z)$, we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $\mathcal{T}(\mathfrak{g},e)$.

On the Azumaya locus of almost commutative algebras

We prove a general statement which implies the coincidence of the Azumaya and smooth loci of the center of an algebra in positive characteristic, provided that the spectrum of its associated graded algebra has a large symplectic leaf. In particular, we show that for a symplectic reflection algebra smooth and the Azumaya loci coincide.

PI量子Weyl代数上的单模与Azumaya轨迹

Simple Modules and Azumaya Loci over the PI quantized Weyl Algebras

该研究针对单位根下的两类多参数量子Weyl代数，完成了其中心、PI次数、极大维单模及Azumaya轨迹的显式计算，丰富了PI量子代数的相关结构理论。

Morita equivalences and Azumaya loci from Higgsing dimer algebras

Let $ψ: A \to A'$ be a cyclic contraction of dimer algebras, with $A$ non-cancellative and $A'$ cancellative. $A'$ is then prime, noetherian, and a finitely generated module over its center. In contrast, $A$ is often not prime, nonnoetherian, and an infinitely generated module over its center. We present certain Morita equivalences that relate the representation theory of $A$ with that of $A'$. We then characterize the Azumaya locus of $A$ in terms of the Azumaya locus of $A'$, and give an explicit classification of the simple $A$-modules parameterized by the Azumaya locus. Furthermore, we sho…

Poisson geometry and Azumaya loci of cluster algebras

There are two main types of objects in the theory of cluster algebras: the upper cluster algebras ${\boldsymbol{\mathsf U}}$ with their Gekhtman-Shapiro-Vainshtein Poisson brackets and their root of unity quantizations ${\boldsymbol{\mathsf U}}_\varepsilon$. On the Poisson side, we prove that (without any assumptions) the spectrum of every finitely generated upper cluster algebra ${\boldsymbol{\mathsf U}}$ with its GSV Poisson structure always has a Zariski open orbit of symplectic leaves and give an explicit description of it. On the quantum side, we describe the fully Azumaya loci of the qua…

Azumaya loci and discriminant ideals of PI algebras

We prove that, under mild assumptions, for all positive integers $\ell$, the zero set of the discriminant ideal $D_{\ell}(R/Z(R); tr)$ of a prime polynomial identity (PI) algebra $R$ coincides with the zero set of the modified discriminant ideal $MD_{\ell}(R/Z(R); tr)$ of $R$. Furthermore, we prove that, when $\ell$ is the square of the PI-degree of $R$, this zero set is precisely the complement of the Azumaya locus of $R$. This description is used to classify the Azumaya loci of the mutiparameter quantized Weyl algebras at roots of unity. As another application, we prove that the zero set of …