Deformations of Azumaya algebras

量子欧几里得 $2n$-空间的 Azumaya 轨迹

Azumaya Loci Of The Quantum Euclidean $2n$-Space

本文研究量子欧几里得 $2n$-空间的 Azumaya 轨迹，构造并刻画其最大维数单模，进而借助中心计算确定该代数的 Azumaya 轨迹。

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…

Braidings on the category of bimodules, Azumaya algebras and epimorphisms of rings

Let $A$ be an algebra over a commutative ring $k$. We prove that braidings on the category of $A$-bimodules are in bijective correspondence to canonical R-matrices, these are elements in $A\ot A\ot A$ satisfying certain axioms. We show that all braidings are symmetries. If $A$ is commutative, then there exists a braiding on ${}_A\Mm_A$ if and only if $k\to A$ is an epimorphism in the category of rings, and then the corresponding $R$-matrix is trivial. If the invariants functor $G = (-)^A:\{}_A\Mm_A\to \Mm_k$ is separable, then $A$ admits a canonical R-matrix; in particular, any Azumaya algebra…

Ideals in deformation quantizations over $\mathbb{Z}/p^n\mathbb{Z}$

Let $X$ be a smooth affine symplectic variety over $\mathbb{Z}/p\mathbb{Z},$ and let $A_1$ be an Azumaya algebra over $X.$ In this note we show that if $A$ is a deformation quantization of $A_1$ over $\mathbb{Z}/p^n\mathbb{Z}$, then any two-sided $\mathbb{Z}/p^n\mathbb{Z}$-flat ideal in $A$ is generated by central elements.

通过PI代数的sheaves范畴研究某些非交换几何

Moduli, Deformations and Algebraic K-Theory of Graded PI-Algebras

本文通过PI代数sheaves的范畴研究非交换几何，引入新的非交换几何结构，并证明了Morita-equivariant Betti/Riemann--Hilbert定理。

多项式恒等式与有理量子球面的Azumaya轨迹

Polynomial identities and Azumaya loci for rational quantum spheres

本文证明了有理量子球面的结构定理：同构分类、PI特征与Azumaya轨迹，并给出中心的分支覆盖实现及有限生成性刻画。

The quantum Frobenius for character varieties and multiplicative quiver varieties

We prove that quantized multiplicative quiver varieties and quantum character varieties define sheaves of Azumaya algebras over the corresponding classical moduli spaces, and we prove that the Azumaya locus of the Kauffman bracket skein algebras contains the smooth locus, proving a strong form of the Unicity Conjecture of Bonahon and Wong. The proofs exploit a strong compatibility between quantum Hamiltonian reduction and the quantum Frobenius homomorphism as it arises in each setting. We therefore introduce the concepts of Frobenius quantum moment maps and their Hamiltonian reduction, and of …

Representation stacks, D-branes and noncommutative geometry

In this note we prove that the spec(C)-points of the representation Artin-stack [rep_n R/PGL_n] of n-dimensional representations of an affine algebra R correspond to algebra morphisms R-->A where A is an Azumaya algebra of degree n over C. We connect this to the study of D-branes and Azumaya noncommutative geometry, developed by Chien-Hao Liu and Shing-Tung Yau in a series of papers from arXiv:0709.1515 to arXiv:1003.1178

Quantum Multiplicative Hypertoric Varieties and Localization

We consider q-deformations of multiplicative hypertoric varieties, for q a non-zero element of an algebraically closed field of characteristic 0. We construct an algebra Dq of q-difference operators as a Heisenberg double in a braided monoidal category. We then focus on the case where q is specialized to a root of unity. In this setting, we use Dq to construct an Azumaya algebra on an l-twist of the multiplicative hypertoric variety, before showing that this algebra splits over the fibers of both the moment and resolution maps. Finally, we sketch a derived localization theorem for these Azumay…

Quantizations of multiplicative hypertoric varieties at a root of unity

We construct quantizations of multiplicative hypertoric varieties using an algebra of q-difference operators on affine space, where q is a root of unity in C. The quantization defines a matrix bundle (i.e. Azumaya algebra) over the multiplicative hypertoric variety and admits an explicit finite étale splitting. The global sections of this Azumaya algebra is a hypertoric quantum group, and we prove a localization theorem. We introduce a general framework of Frobenius quantum moment maps and their Hamiltonian reductions; our results shed light on an instance of this framework.

对称张量范畴中的克利福德代数和外尔代数

Clifford and Weyl algebras in symmetric tensor categories

研究特征≠2的代数闭域上对称张量范畴\(\mathcal C\)中带（斜）对称双线性形式对象的克利福德代数和外尔代数，建立其单性和阿祖马亚性质，计算Verlinde范畴中的此类代数，引入辛维特群并给出特定条件下的表示。

Cherednik algebras and Hilbert schemes in characteristic p (with an appendix by Pavel Etingof)

We prove a localization theorem for the type A rational Cherednik algebra H_c=H_{1,c} over an algebraic closure of the finite field F_p. In the most interesting special case where the parameter c takes values in F_p, we construct an Azumaya algebra A_c on Hilb^n, the Hilbert scheme of n points in the plane, such that the algebra of global sections of A_c is isomorphic to H_c. Our localisation theorem provides an equivalence between the bounded derived categories of H_c-modules and sheaves of coherent A_c-modules on the Hilbert scheme, respectively. Furthermore, we show that the Azumaya algebra…

Quantum Weyl algebras and reflection equation algebras at a root of unity

We compute the center and Azumaya locus in the simplest non-abelian examples of quantized multiplicative quiver varieties at a root of unity: quantum Weyl algebras of rank $N$, and quantum differential operators on the quantum group $\mathrm{GL}_2$. These examples illustrate in elementary terms much more general phenomena explored further in [Ganev-Jordan-Safronov 2019].

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.

霍奇-希尔德上同调与端射的提升

Hochschild cohomology and lifts of endomorphisms

研究代数自同射何时能提升为一阶平坦提升，通过关联一阶平坦提升与自同射的霍奇-希尔德上同调类，证明自同射提升当且仅当中心上的诱导自同射保持提升后的泊松结构。

Deformation quantization of vertex Poisson algebras

We introduce dg Lie algebras controlling the deformations of vertex algebras and vertex Poisson algebras, utilizing the notion of operadic dg Lie algebra and the theory of chiral algebra. In terms of those dg Lie algebras, we formulate the deformation quantization problem of vertex Poisson algebras to vertex algebras.

Leibniz algebra deformations of a Lie algebra

In this note we compute Leibniz algebra deformations of the 3-dimensional nilpotent Lie algebra $\mathfrak{n}_3$ and compare it with its Lie deformations. It turns out that there are 3 extra Leibniz deformations. We also describe the versal Leibniz deformation of $\mathfrak{n}_3$ with the versal base.

Deformation of singularities via $L_\infty$-algebras

This is an addendum to the paper ``Deformation of $L_\infty$-Algebras'' of the same author. We explain in which way the deformation theory of $L_\infty$-algebras extends the deformation theory of singularities. We show that the construction of semi-universal deformations of $L_\infty$-algebras gives explicit formal semi-universal deformations of isolated singularities.