固定K <x，y，z>的一个变量的自同构

二元自由Poisson代数的自同构与导子

Automorphisms and derivations of free Poisson algebras in two variables

设P是特征为零的域上的二元自由Poisson代数。我们证明了P的自同构是驯服的，P的局部幂零导子是可三角化的。

具有可分变量的刚性超曲面的自同构群

Automorphisms of rigid hypersurfaces with separable variables

本研究针对具有可分变量的刚性超曲面，刻画其自同构群的结构，证明该自同构群是其唯一极大环面的有限扩张，丰富了刚性代数簇自同构群的相关理论。

四变量Anick自同构的显式 tame 分解

An explicit tame factorization of the four-variable Anick automorphism

本文给出四变量多项式环中Anick自同构的显式 tame 分解，含16个固定z的初等自同构，定义在ℤ[1/2]上，未涉及特征2情形。

A wild automorphism of a free Novikov algebra

An example of a non-triangulable locally nilpotent derivation and an example of a wild exponential automorphism of the free Novikov algebra $N\left\langle x,y,z\right\rangle$ in three variables $x,y,z$ over a field of characteristic zero are constructed.

Tame and wild automorphisms of differential polynomial algebras of rank 2

It is proved that the tame automorphism group of a differential polynomial algebra $k\{x,y\}$ over a field $k$ of characteristic $0$ in two variables $x,y$ with $m$ commuting derivations $δ_1, \ldots, δ_m$ is a free product with amalgamation. An example of a wild automorphism of the algebra $k\{x,y\}$ in the case of $m\geq 2$ derivations is constructed.

Automorphisms of Weyl Algebra and a Conjecture of Kontsevich

We outline the proof of a conjecture of Kontsevich on the isomorphism between the group of polynomial symplectomorphisms in $2n$ variables and the group of automorphisms of the $n$-th Weyl algebra over complex numbers. Our proof uses lifting of polynomial symplectomorphisms to Weyl algebra automorphisms by means of approximation by tame symplectomorphisms and gauging of the lifted morphism. Approximation by tame symplectomorphisms is the symplectic version of the well-known theorem of D. Anick and is a result of our prior work.

Nakayama automorphisms of Ore extensions over polynomial algebras

Nakayama automorphisms play an important role in several mathematical branches, which are known to be tough to compute in general. We compute the Nakayama automorphism $ν$ of any Ore extension $R[x; σ, δ]$ over a polynomial algebra $R$ in $n$ variables for an arbitrary $n$. The formula of $ν$ is obtained explicitly. When $σ$ is not the identity map, the invariant $E^G$ is also investigated in term of Zhang's twist, where $G$ is a cyclic group sharing the same order with $σ$.

Gradings allowing wild automorphisms

In 2004 Shestakov and Umirbaev proved that the Nagata automorphism of the polynomial algebra in three variables is wild. We fix a Z-grading on this algebra and consider graded-wild automorphisms, i.e. such automorphisms that can not be decomposed onto elementary automorphisms respecting the grading. We describe all gradings allowing graded-wild automorphisms.

${\rm K}_1(\mS_1)$与二元多项式代数的单侧逆代数$\mS_2$的自同构群

${\rm K}_1(\mS_1)$ and the group of automorphisms of the algebra $\mS_2$ of one-sided inverses of a polynomial algebra in two variables

在特征为零的域上的二元多项式代数的单侧逆的代数$\mS_2$的自同构群$G_2$中找到了显式生成元。本文还证明了$$ G_2\simeq S_2\ltimes \mT^2\ltimes \Z\ltimes ((K^*\ltimes E_\infty (\mS_1))\boxtimes_{\GL_\infty (K)}(K^*\ltimes E_\infty (\mS_1)))$$，其中$S_2$是对称群，$\mT^2$是二维环面，$E_\infty (\mS_1)$是由初等矩阵生成的$\GL_\infty (\mS_1)$的子群.在证明中，我们利用并证明了关于算子指数的几个结果，证明中的最后一个论点是${\rm K}_1 (\mS_1) \simeq K^*$在文中证明的事实。代数$\mS_1$和$\mS_2$是非交换的，非诺特的，不是域。代数$\mS_2$的单位群被发现（它是巨大的）。

多项式代数的单侧逆代数II的自同构群

The group of automorphisms of the algebra of one-sided inverses of a polynomial algebra, II

给出了以$n$为变量的多项式代数的单侧逆的自同构自同构群的显式生成元。一个类似的多项式雅可比同态被发现。

椭圆Poisson代数的自同构

Automorphisms of elliptic Poisson algebras

本文描述了三元多项式代数上椭圆Poisson代数的自同构群，给出了这类自同构群的生成元和定义关系。

秩为3的多项式与自由结合代数的坐标与自同构

Coordinates and Automorphisms of Polynomial and Free Associative Algebras of Rank Three

研究了域K上多项式代数K[x，y，z]和自由结合代数K<x，y，z>的z-自同构，即，固定变量z的自同构。我们调查最近的一些结果，这样的自同构和相应的坐标。对于K<x，y，z>，我们还包括了关于z-tame自同构的结构的结果和识别z-tame自同构和z-tame坐标的算法。

Cluster automorphisms and quasi-automorphisms

We study the relation between the cluster automorphisms and the quasi-automorphisms of a cluster algebra $\mathcal{A}$. We proof that under some mild condition, satisfied for example by every skew-symmetric cluster algebra, the quasi-automorphism group of $\mathcal{A}$ is isomorphic to a subgroup of the cluster automorphism group of $\mathcal{A}_{triv}$, and the two groups are isomorphic if $\mathcal{A}$ has principal or universal coefficients; here $\mathcal{A}_{triv}$ is the cluster algebra with trivial coefficients obtained from $\mathcal{A}$ by setting all frozen variables equal to the int…

映射类群的展示及其在来自曲面的cluster algebras中的应用

Presentations of mapping class groups and an application to cluster algebras from surfaces

本文给出了任意亏格下稳定边界的标记曲面映射类群的展示，并以此获得了来自可行曲面$(S,M)$的cluster algebra的cluster automorphism群展示，进而刻画了不可行曲面情形下的cluster automorphism群。

自由双交换李代数的自同构，I

Automorphisms of free metabelian Lie algebras, I

本文研究了自由双交换李代数的自同构性质，证明了特定条件下自同构的 tame 性及部分自同构的几乎 tame 性，推翻了领域内长期存在的结论。

两个变量的泊松域

Poisson fields of two variables

研究两个变量有理函数泊松域的不变量与结构，分类四个特殊族，建立同构准则，描述自同构群，并探讨嵌入及Dixmier猜想的类比。

多项式代数的自同构与狄利克雷级数

Automorphisms of polynomial algebras and Dirichlet series

设GF（q）[x，y]是有限域GF（q）上q元二元多项式代数.本文给出了GF（q）[x，y]中满足max{deg（f），deg（g）}=n的自同构（f，g）的个数p（n）的一个精确公式和渐近性.我们还描述了狄利克雷级数的生成函数p（1）/1^s+p（2）/2^s+p（3）/3^s+.对于自由结合代数GF（q）<x，y>的自同构也有同样的结果。我们还得到了代数的Nielsen-Schreier簇中具有两个生成元的自由代数的类似结果。