The graph of a Weyl algebra endomorphism

Niels Lauritzen*, Jesper Funch Thomsen*

*Corresponding author for this work

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal articleResearchpeer-review


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.

Original languageEnglish
JournalBulletin of the London Mathematical Society
Pages (from-to)161-176
Number of pages16
Publication statusPublished - Feb 2021


  • 16D20
  • 16S32 (primary)
  • 16W20 (secondary)


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