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The graph of a Weyl algebra endomorphism

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Endomorphisms of Weyl algebras are studied using bimodules. Initially, for a Weyl algebra over a field of characteristic zero, Bernstein's inequality implies that holonomic bimodules finitely generated from the right (respectively, left) form a monoidal category. The most important bimodule in this paper is the graph of an endomorphism. We prove that the graph of an endomorphism of a Weyl algebra over a field of characteristic zero is a simple bimodule. The simplicity of the tensor product of the dual graph and the graph is equivalent to the Dixmier conjecture. It is also shown how the graph construction leads to a non-commutative Gröbner basis algorithm for detecting invertibility of an endomorphism for Weyl algebras and computing the inverse over arbitrary fields.

OriginalsprogEngelsk
TidsskriftBulletin of the London Mathematical Society
Antal sider16
ISSN0024-6093
DOI
StatusAccepteret/In press - 11 aug. 2020

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