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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 q1 and q2 are not roots of unity, or are both primitive m-th roots of unity, we construct an explicit isomorphism between q1() and q2(). If q1 is a primitive m-th root of unity, m odd, q2 a primitive 2m-th or a primitive 4m-th root of unity, we also construct an explicit isomorphism between q1() and q2().

Keywords

17B3717B65
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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