Automorphisms and Ideals of the Weyl Algebra

2001/02/25 by Yuri Berest, George Wilson · 1 citation
Mathematics · #math.QA #math.AG

paper · pdf

published as Math. Ann. 318, 127-147 (2000) · 17 pages

arxiv created 2001/02/25 · arxiv updated 2009/11/30

Abstract

Let A₁ be the (first) Weyl algebra, and let G be its automorphism group. We study the natural action of G on the space of isomorphism classes of right ideals of A₁ (equivalently, of finitely generated rank 1 torsion-free right A₁-modules). We show that this space breaks up into a countable number of orbits each of which is a finite dimensional algebraic variety. Our results are strikingly similar to those for the commutative algebra of polynomials in two variables; however, we do not know of any general principle that would allow us to predict this in advance. As a key step in the proof, we obtain a new description of the bispectral involution of \citeW1. We also make some comments on the group G from the viewpoint of Shafaravich's theory of infinite dimensional algebraic groups.

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