Leonid Makar-Limanov, Umut Turusbekova, Ualbai Umirbaev
We describe the automorphism groups of elliptic Poisson algebras on polynomial algebras in three variables and give an explicit set of generators and defining relations for this group.
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Let P be a free Poisson algebra in two variables over a field of characteristic zero. We prove that the automorphisms of P are tame and that the locally nilpotent derivations of P are triangulable.
We prove Pursell-Shanks type results for the Lie algebra D(M) of all linear differential operators of a smooth manifold M, for its Lie subalgebra D^1(M) of all linear first-order differential operators of M, and for the Poisson algebra S(M)=Pol(T*M) of all polynomial functions on the cotangent bundle T*M, the symbols of the operators in D(M). Chiefly however we provide explicit formulas describing completely the automorphisms of the Lie algebras D^1(M), S(M), and D(M).
An Anick type wild automorphism of free Poisson algebras
本文在 3 生成自由 Poisson 代数中构造 Anick 型野生自同构 $δ$，证明其在多项式代数上诱导驯顺自同构且本身稳定驯顺。
A skew Laurent polynomial ring R[x^{\pm 1};α] is reversible if it has a reversing automorphism, that is, an automorphism of period two that transposes x and x^{-1} and restricts to an automorphism of R. We study invariants for reversing automorphisms and apply our methods to determine the rings of invariants of reversing automorphisms of two simple skew Laurent polynomial rings, namely a localization of the enveloping algebra of the two-dimensional non-abelian solvable Lie algebra and the coordinate ring of the quantum torus. These two skew Laurent polynomial rings are deformations of Poisson …
Let $k$ be an arbitrary field of characteristic 0. We prove that the group of automorphisms of a free Poisson field $P(x,y)$ in two variables $x,y$ over $k$ is isomorphic to the Cremona group $\mathrm{Cr}_2(k)$. We also prove that the universal enveloping algebra $P(x_1,...,x_n)^e$ of a free Poisson field $P(x_1,...,x_n)$ is a free ideal ring and give a characterization of the Poisson dependence of two elements of $P(x_1,...,x_n)$ via universal derivatives.
Kanel-Belov and Kontsevich's conjecture in \cite[Conjecture 1]{BeKo} is proved: The automorphism group of the $n$-th Weyl algebra is isomorphic to the Poisson automorphism group of the $n$-th Poisson Weyl algebra.
We prove a rigidity theorem for the Poisson automorphisms of the function fields of tori with quadratic Poisson structures over fields of characteristic 0. It gives an effective method for classifying the full Poisson automorphism groups of ${\mathbb{N}}$-graded connected cluster algebras equipped with Gekhtman-Shapiro-Vainshtein Poisson structures. Based on this, we classify the groups of algebraic Poisson automorphisms of the open Schubert cells of the full flag varieties of semisimple algebraic groups over fields of characteristic 0, equipped with the standard Poisson structures. Their coor…
Transposed Poisson structure on the Witt-type algebra $\mathcal{W}(a,-1)$: Derivations, Automorphisms, and Rota--Baxter operators
分类了Witt型代数W(a,-1)上的导子、自同构和Rota-Baxter算子，证明了非平凡δ-导子仅存在于δ=1和1/2，并建立了刚性结果。
Let $P(\mathrm{sl}_2(K))$ be the Poisson enveloping algebra of the Lie algebra $\mathrm{sl}_2(K)$ over an algebraically closed field $K$ of characteristic zero. The quotient algebras $ $ $P(\mathrm{sl}_2(K))/(C_P-λ)$, where $C_P$ is the standard Casimir element of $\mathrm{sl}_2(K)$ in $P(\mathrm{sl}_2(K))$ and $0\neq λ\in K$, are proven to be simple in \cite{UZh}. Using a result by L. Makar-Limanov \cite{ML90}, we describe generators of the automorphism group of $P(\mathrm{sl}_2(K))/(C_P-λ)$ and represent this group as an amalgamated product of its subgroups. Moreover, using similar results b…
Hessian matrices, automorphisms of $p$-groups, and torsion points of elliptic curves
本文研究由椭圆曲线Hessian行列式表示产生的有限p-群的自同构群，对j-不变量为1728的曲线给出自同构群阶的显式公式，并用有限域上3-挠点数目解释这些阶，推广并概念化了du Sautoy和Vaughan-Lee的先前工作。
Valuation method for Nambu-Poisson algebras
本文通过Poisson赋值方法研究Nambu-Poisson代数的自同构、同构与嵌入问题，为这类代数结构提供了新的赋值分析工具。
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。
We prove the Freiheitssatz for Poisson algebras in characteristic zero. We also give a proof of the tameness of automorphisms for two generated free Poisson algebras and prove that an analogue of the commutator test theorem is equivalent to the two dimensional classical Jacobian conjecture using the Freiheitssatz and Jung's Theorem.
Automorphisms of Veronese subalgebras of polynomial algebras and free Poisson algebras
本文研究多项式代数与自由 Poisson 代数的 d 次 Veronese 子代数，证明其导子、局部幂零导子及自同构在相应域条件下均可由原代数的对应映射诱导。
We discuss a conjecture which says that the automorphism group of the Weyl algebra in characteristic zero is canonically isomorphic to the automorphism group of the corresponding Poisson algebra of classical polynomial symbols. Several arguments in favor of this conjecture are presented, all based on the consideration of the reduction of the Weyl algebra to positive characteristic.
We give the 3-dimensional Sklyanin algebras $S$ that are module-finite over their center $Z$ the structure of a Poisson $Z$-order (in the sense of Brown-Gordon). We show that the induced Poisson bracket on $Z$ is non-vanishing and is induced by an explicit potential. The ${\mathbb Z}_3 \times \Bbbk^\times$-orbits of symplectic cores of the Poisson structure are determined (where the group acts on $S$ by algebra automorphisms). In turn, this is used to analyze the finite-dimensional quotients of $S$ by central annihilators: there are 3 distinct isomorphism classes of such quotients in the case …
Poisson valuations
本文研究 Poisson 赋值，并将其应用于解决 Poisson 代数和域的刚性、自同构、Dixmier 性质、同构及嵌入等问题。
We prove a result that can be applied to determine the finite-dimensional simple Poisson modules over a Poisson algebra and apply it to numerous examples. In the discussion of the examples, the emphasis is on the correspondence with the finite-dimensional simple modules over deformations and on the behaviour of finite-dimensional simple Poisson modules on the passage from a Poisson algebra to the Poisson subalgebra of invariants for the action of a finite group of Poisson automorphisms.
We give a description of the Hochschild cohomology for noncommutative planes (resp. quadrics) using the automorphism groups of the elliptic triples (resp. quadruples) that classify the Artin-Schelter regular $\mathbb{Z}$-algebras used to define noncommutative planes and quadrics. For elliptic triples the description of the automorphism groups is due to Bondal-Polishchuk, for elliptic quadruples it is new.
Applications of Poisson cohomology to the inducibility problems and study of deformation maps
本文利用泊松上同调研究泊松代数阿贝尔扩张中自同构与导子的不可诱导性并构造 Wells 型正合序列，同时引入形变映射统一多种算子并研究其诱导结构与形式形变。
Invariants of Unimodular Quadratic Polynomial Poisson Algebras of Dimension 3
本文研究三维幺模二次 Poisson 代数及其 Poisson 包络代数在有限分次 Poisson 自同构群作用下的不变量，推广了 Shephard-Todd-Chevalley 与 Watanabe 定理。
Reflection Groups and Rigidity of Quadratic Poisson Algebras
本文研究二次Poisson代数的不变量理论，证明了Shephard-Todd-Chevalley定理的Poisson版本，并揭示了多种代数族的Poisson刚性，即固定子环A^G与A同构当且仅当群G平凡。
Results on derivations and automorphisms of some quantum and classical Poisson algebras, as well as characterizations of manifolds by the Lie structure of such algebras, are revisited and extended. We prove in particular somehow unexpected fact that the algebras of linear differential operators acting on smooth sections of two real vector bundles of rank 1 are isomorphic as Lie algebras if and only if the base manifolds are diffeomorphic, independently whether the line bundles themselves are isomorphic or not.
Log-ozone groups and centers of polynomial Poisson algebras
本文引入正特征域上泊松代数的对数臭氧群，用以刻画具有斜对称结构的多项式泊松代数，并证明此类幺模代数的中心为Gorenstein，同时给出三维分次情形的相关结论。
We study Jacobi pairs in details and obtained some properties. We also study the natural Poisson algebra structure $(\PP,[...,...],...)$ on the space $\PP:=\C[y]((x^{-\frac1N}))$ for some sufficient large $N$, and introduce some automorphisms of $(\PP,[...,...],...)$ which are (possibly infinite but well-defined) products of the automorphisms of forms $e^{\ad_H}$ for $H\in x^{1-\frac1N}\C[y][[x^{-\frac1N}]]$ and $τ_c:(x,y)\mapsto(x,y-cx^{-1})$ for some $c\in\C$. These automorphisms are used as tools to study Jacobi pairs in $\PP$. In particular, starting from a Jacobi pair $(F,G)$ in $\C[x,y]$…
An Elliptic Generalization of $A_1$ Spherical DAHA at $K=2$
本文构造 $K=2$ 时 $A_1$ 球面 DAHA 的椭圆推广，证明 $PSL(2,\mathbb Z)$ 的自同构作用并给出矩阵表示，还猜想其与仿射 Laumon 空间 K 理论的联系。
Classification of Transposed Poisson 3-Lie algebras of dimension 3
本文通过确定唯一非平凡三维复3-李代数的所有1/3-导子与自同构，并利用1/3-导子与转置泊松3-李代数的一一对应，在L_{e_1}平凡条件下完成三维转置泊松3-李代数的同构分类。
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