I just read about Weyl algebras, and they sound like neat little toys that are similar in a number of ways to polynomials. However, it's curious to me that they are non-commutative, and I was wondering if there are unique factorizations for Weyl algebras the way there are for Hurwitz quaternions.
Further, there's a polynomial time factorization algorithm for polynomials, and at first glance Weyl algebras are extensions of polynomials. Do they have polynomial time factorization algorithms?