An analogue of the conjecture of Dixmier is true for the algebra of polynomial integro-differential operators. (English) Zbl 1278.16024
The associative algebra \(\mathbb I_1\) over a field of characteristic zero is the algebra of polynomial integro-differential operators generated by elements \(\partial,x,\int\) with defining relations
\[
\partial\int =1,\quad\left[H,\int\right]=\int,\quad H\left(1-\int\partial\right)=\left(1-\int\partial\right)H=\left(1-\int\partial\right),
\]
where \(H=\partial x\). It is proved that any algebra endomorphism of \(\mathbb I_1\) is an automorphism.
Reviewer: Vyacheslav A. Artamonov (Moskva)
MSC:
| 16S32 | Rings of differential operators (associative algebraic aspects) |
| 16W20 | Automorphisms and endomorphisms |
| 14R15 | Jacobian problem |
Keywords:
rings of differential operators; Jacobian conjecture; Weyl algebras; endomorphisms; automorphisms; Dixmier conjectureReferences:
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