Automorphisms and Ideals of the Weyl Algebra

Details Automorphisms and Ideals of the Weyl Algebra Automorphisms and Ideals of the Weyl Algebra

2001-02-25

arxiv.org/abs/math/0102190v1

Math. Ann. 318, 127-147 (2000)

Mathematics

Quantum Algebra

17 pages

Scientific paper

Let $A_1$ 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 $A_1$ (equivalently, of finitely generated rank 1 torsion-free right $A_1$-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 \cite{W1}. 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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