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Representations of Quantum Heisenberg Algebras

Published online by Cambridge University Press:  20 November 2018

Marc A. Fabbri  and
Frank Okoh
Marc A. Fabbri
Affiliation:
Centre de recherches mathématiques Université de Montréal Montréal, Quebec H3C 3JC
Frank Okoh
Affiliation:
Department of Mathematics Wayne State University Detroit, Michigan 48202 USA
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Abstract

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A Lie algebra is called a Heisenberg algebra if its centre coincides with its derived algebra and is one-dimensional. When is infinite-dimensional, Kac, Kazhdan, Lepowsky, and Wilson have proved that -modules that satisfy certain conditions are direct sums of a canonical irreducible submodule. This is an algebraic analogue of the Stone-von Neumann theorem. In this paper, we extract quantum Heisenberg algebras, q (), from the quantum affine algebras whose vertex representations were constructed by Frenkel and Jing. We introduce the canonical irreducible q ()-module Mq and a class Cq of q ()-modules that are shown to have the Stone-von Neumann property. The only restriction we place on the complex number q is that it is not a square root of 1. If q 1 and q 2 are not roots of unity, or are both primitive m-th roots of unity, we construct an explicit isomorphism between q 1() and q 2(). If q 1 is a primitive m-th root of unity, m odd, q 2 a primitive 2m-th or a primitive 4m-th root of unity, we also construct an explicit isomorphism between q 1() and q 2().

Keywords

17B3717B65

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 5 , 01 October 1994 , pp. 920 - 929
Copyright
Copyright © Canadian Mathematical Society 1994

References

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