Localization and the Weyl algebras

Let W_n(ℝ) be the Weyl algebra of index n. It is well known that so(p, q) Lie algebras can be viewed as quadratic polynomial (Lie) algebras in W_n(ℝ) for p + q = n with the Lie algebra multiplication being given by the bracket [a, b] = ab − ba for a, b quadratic polynomials in W_n(ℝ). What does not seem to be so well known is that the converse statement is, in a certain sense, also true, namely, that, by using extension and localization, it is possible, at least in some cases, to construct homomorphisms of W_n(ℝ) onto its image in a localization of U(so(p + 2, q)), the universal enveloping algebra of so(p + 2, q), and m = p + q. Since Weyl algebras are simple, these homomorphisms must either be trivial or isomorphisms onto their images. We illustrate this remark for the so(2, q) case and construct a mappping from W_q(ℝ) onto its image in a localization of U(so(2, q)). We prove that this mapping is a homomorphism when q = 1 or q = 2. Some specific results about representations for the lowest dimensional case of W₁(ℝ) and U(so(2, 1)) are given.