December 7
Hans Schoutens,
CUNY
Towards a theory of Weyl Algebras
The Weyl algebra (in $n$ variables) is a well-studied object, playing a key role in non-commutative algebra, differential algebra, theory of $D$-modules, mathematical physics, etc. It reflects the 'canonical commutation relations' from quantum physics, also known as Heisenberg's uncertainty principle: $p_nq_n-q_np_n=1$ (where $q_n$ and $p_n$ are the respective position and moment operators on the $n$-th particle). I’ll give a brief introduction to this object, and tell a little about its 'pathologies' in positive characteristic. The main motivation for this project is a paper by Kontsevich and Belov-Kanel, where they prove the equivalence of the Jacobian Conjecture (about the surjectivity of endomorphisms of polynomial rings) with the Dixmier Conjecture (about the surjectivity of endomorphisms of the Weyl algebra) using reduction modulo positive characteristic. They remark that a proof using 'non-standard analysis'--by which they presumably mean via model-theory/ultraproducts--would perhaps be more natural.
In reply to this request, I will propose a first-order theory of Weyl algebras, based on my study of non-standard polynomial rings (which, therefore, I will review as well). The key tool is an embedded model of PA, which in case of ordinary polynomials/Weyl algebra is just the natural numbers (viewed as the set of exponents), together with a Darboux action (corresponding to the differential action on a polynomial ring). At the moment, however, I can only do this when the (internal) characteristic is zero; this, however, is not the case if one takes ultraproducts of Weyl algebras in positive characteristic, and so, at present, the right framework for doing the Konsevich/Belov-Kanel argument is still wanting.