### Abstract

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 W_{1}(ℝ) and U(so(2, 1)) are given.

Original language | English (US) |
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Pages (from-to) | 590-597 |

Number of pages | 8 |

Journal | Physics of Atomic Nuclei |

Volume | 80 |

Issue number | 3 |

DOIs | |

Publication status | Published - May 1 2017 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Physics of Atomic Nuclei*,

*80*(3), 590-597. https://doi.org/10.1134/S106377881703022X