摘要翻译：
设$K$是特征$P>0$的{em完美}域，$A_1:=K<x,\der\derx-x\der=1>$是第一Weyl代数，$Z:=K[x:=x^p,y:=\der^p]$是其中心。证明了$(i)$约束映射$\res:\aut_k(A_1)\ra\aut_k(Z),\s\mapsto\s_z$是一个单同态，其中$\im(\res)=\g:=\{\tau\in\aut_k(Z)\cj(\tau)=1\}$是$\tau$的雅可比（注意$\aut_k(Z)=k^*\ltimes\g$,如果$k$是{\em not}完美，则$\im(\res)\neq\g$)；$(ii)$双射$\res:\aut_k(A_1)\ra\g$是无穷维代数群的一个单态，它{\em not}是同构（即使$k$是代数闭的）；$(iii)$通过$Z$上的微分运算符$\cd(Z)$和Fronenius映射的负幂$F$找到了$\res^{-1}$的显式公式。本文证明了以下（非明显的）等式:$$(\frac{d}{dx}+f)^p=(\frac{d}{dx}）^p+\frac{d^{p-1}f}{dx^{p-1}}+f^p,f\in k[x]。$$
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英文标题：
《The group of automorphisms of the first Weyl algebra in prime
  characteristic and the restriction map》
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作者：
V. V. Bavula
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Rings and Algebras        环与代数
分类描述：Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups
非交换环与代数，非结合代数，泛代数与格论，线性代数，半群
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  Let $K$ be a {\em perfect} field of characteristic $p>0$, $A_1:=K< x, \der | \der x- x\der =1>$ be the first Weyl algebra and $Z:=K[X:=x^p, Y:=\der^p]$ be its centre. It is proved that $(i)$ the restriction map $\res :\Aut_K(A_1)\ra \Aut_K(Z), \s \mapsto \s|_Z$, is a monomorphism with $\im (\res) = \G :=\{\tau \in \Aut_K(Z) | \CJ (\tau) =1\}$ where $\CJ (\tau) $ is the Jacobian of $\tau$ (note that $\Aut_K(Z)=K^*\ltimes \G$ and if $K$ is {\em not} perfect then $\im (\res) \neq \G$); $(ii)$ the bijection $\res : \Aut_K(A_1) \ra \G$ is a monomorphism of infinite dimensional algebraic groups which is {\em not} an isomorphism (even if $K$ is algebraically closed); $(iii)$ an explicit formula for $\res^{-1}$ is found via differential operators $\CD (Z)$ on $Z$ and negative powers of the Fronenius map $F$. Proofs are based on the following (non-obvious) equality proved in the paper: $$ (\frac{d}{dx}+f)^p= (\frac{d}{dx})^p+\frac{d^{p-1}f}{dx^{p-1}}+f^p, f\in K[x].$$
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PDF链接：
https://arxiv.org/pdf/0708.1620