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On Differential Operators $D(X)$'s automorphisms group in $Aut_{k}(A_{1}(k))$

Kouakou M. K.
UFR Mathématiques et Informatique
Université de Cocody, 22 BP 582 Abidjan 22, Côte d'Ivoire

Tchoudjem A.
Université Claude Bernard Lyon I, Institut Camille-Jordan, France


Let MATH be the first algebra over a field $k$ of characteristic zero and $I$ a right ideal of $A_{1}(k)$. The subgroup of Stafford associated to $I$ denoted $H(I)$ is :
By J.T. Stafford in [5], it is known that subgroups $H(I)$ are isomorphic to automorphisms groups MATH, where $\QTR{cal}{D}(X)$ is the $k$-algebra of differentials operators over an algebraic affine curve $X$. Due to Stafford, it is known at this step that the $H(I_{0})$ isomorphic to MATH where $X_{0}$ is the well-known algebraic affine curve defined by the equation: $x^{2} = y^{3}$, is equal to its own normalizer in MATH.
We will show in this paper that for any ideal $I$, $H(I)$ is equal to its own normalizer, precisely we show that for any right ideal $I$ and $J$ non principal, one has :
from which it follows that $H(I)$ is equal to its own normalizer in $G$.

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