Abstract

Let R be the set of isomorphism classes of right ideals in the Weyl algebra A = A1(ℂ), and let C be the set of isomorphism classes of triples (V;X,Y), where V is a finite-dimensional (complex) vector space, and X,Y are endomorphisms of V such that [X,Y]+I has rank 1. Following a suggestion of L. Le Bruyn, we define a map θRC by appropriately extending an ideal of A to a sheaf over a quantum projective plane, and then using standard methods of homological algebra. We prove that θ is inverse to a bijection ω:CR, constructed, by the authors in 2000, by a completely different method. The main step in the proof is to show that θ is equivariant with respect to natural actions of the group G = Aut(A) on R and C: for that we have to study also the extensions of an ideal to certain weighted quantum projective planes. Along the way, we find an elementary description of θ.

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