### Abstract

Let W_{n} (R) be the Weyl algebra of index n. We have shown that by using extension and localization, it is possible to construct homomorphisms of W_{n} (R) onto its image in a localization, or a quotient thereof, of U(so(2, q)), the universal enveloping algebra of so(2, q), forn depending upon q [1]. Here we treat the so(2, 1) case in complete detail. We establish an isomorphism of skew fields, specifically, D(so(2, 1)) ~ D_{(1,1)}(R) where D_{1,1}(R) is the fraction field of W_{1.1}(R) ~ W_{1}(R) ⊗ R(y) with R(y) being the ring of polynomials in the indeterminate y and D(so(2, 1)) is a certain extension of the skew field of fractions of U(so(2, 1)), which is described below. We give applications of this result to representations. In particular we are able to construct representations of W_{1} (R) out of representations of so (2, 1). Thus, we are able, for this lowest dimensional case, to obtain the canonical commutation relations and representations of them out of so(2, 1) symmetry. Using similar results in higher dimensions [1] we are able to construct representations of W_{n}(R) out of representations of so (2, q).

Original language | English (US) |
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Title of host publication | Lie Theory and Its Applications in Physics |

Editors | Vladimir Dobrev |

Publisher | Springer New York LLC |

Pages | 423-430 |

Number of pages | 8 |

ISBN (Print) | 9789811026355 |

DOIs | |

State | Published - Jan 1 2016 |

Event | Proceedings of the 11th International Workshop on Lie Theory and Its Applications in Physics, 2015 - Varna, Bulgaria Duration: Jun 15 2015 → Jun 21 2015 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 191 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Other

Other | Proceedings of the 11th International Workshop on Lie Theory and Its Applications in Physics, 2015 |
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Country | Bulgaria |

City | Varna |

Period | 6/15/15 → 6/21/15 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Lie Theory and Its Applications in Physics*(pp. 423-430). (Springer Proceedings in Mathematics and Statistics; Vol. 191). Springer New York LLC. https://doi.org/10.1007/978-981-10-2636-2_30