The group of automorphisms of the first Weyl algebra in prime characteristic and the restriction map
| dc.creator | Bavula, V. V. | |
| dc.date | 2007-08-12 | |
| dc.date.accessioned | 2026-07-07T08:23:21Z | |
| dc.date.available | 2026-07-07T08:23:21Z | |
| dc.description | 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 $\im (\res) = \G :=\{τ\in \Aut_K(Z) | \CJ (τ) =1\}$ where $\CJ (τ) $ is the Jacobian of $τ$ (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: $$ (\frac{d}{dx}+f)^p= (\frac{d}{dx})^p+\frac{d^{p-1}f}{dx^{p-1}}+f^p, f\in K[x].$$ | |
| dc.description | 12 pages | |
| dc.identifier | https://arxiv.org/abs/0708.1620 | |
| dc.identifier | http://arxiv.org/abs/0708.1620 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/135964 | |
| dc.subject | Rings and Algebras | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14J50, 16W20, 14L17, 14R10, 14R15, 14M20 | |
| dc.title | The group of automorphisms of the first Weyl algebra in prime characteristic and the restriction map | |
| dc.type | text |