### 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 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### 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

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*Lie Theory and Its Applications in Physics.*Springer Proceedings in Mathematics and Statistics, vol. 191, Springer New York LLC, pp. 423-430, Proceedings of the 11th International Workshop on Lie Theory and Its Applications in Physics, 2015, Varna, Bulgaria, 6/15/15. https://doi.org/10.1007/978-981-10-2636-2_30

**Localization and the canonical commutation relations.** / Moylan, Patrick J.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Localization and the canonical commutation relations

AU - Moylan, Patrick J.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - Let Wn (R) be the Weyl algebra of index n. We have shown that by using extension and localization, it is possible to construct homomorphisms of Wn (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 D1,1(R) is the fraction field of W1.1(R) ~ W1(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 W1 (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 Wn(R) out of representations of so (2, q).

AB - Let Wn (R) be the Weyl algebra of index n. We have shown that by using extension and localization, it is possible to construct homomorphisms of Wn (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 D1,1(R) is the fraction field of W1.1(R) ~ W1(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 W1 (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 Wn(R) out of representations of so (2, q).

UR - http://www.scopus.com/inward/record.url?scp=85009730035&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85009730035&partnerID=8YFLogxK

U2 - 10.1007/978-981-10-2636-2_30

DO - 10.1007/978-981-10-2636-2_30

M3 - Conference contribution

SN - 9789811026355

T3 - Springer Proceedings in Mathematics and Statistics

SP - 423

EP - 430

BT - Lie Theory and Its Applications in Physics

A2 - Dobrev, Vladimir

PB - Springer New York LLC

ER -