Localization and the Weyl algebras

Research output: Contribution to journalArticle

Abstract

Let Wn(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 Wn(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 Wn(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 Wn(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 Wq(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.

Original languageEnglish (US)
Pages (from-to)590-597
Number of pages8
JournalPhysics of Atomic Nuclei
Volume80
Issue number3
DOIs
Publication statusPublished - May 1 2017

    Fingerprint

All Science Journal Classification (ASJC) codes

  • Atomic and Molecular Physics, and Optics
  • Nuclear and High Energy Physics

Cite this