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IRREDUCIBLE HARISH CHANDRA MODULES OVER THE DERIVATION ALGEBRAS OF RATIONAL QUANTUM TORI

Published online by Cambridge University Press:  25 February 2013

GENQIANG LIU  and
KAIMING ZHAO
GENQIANG LIU
Affiliation:
College of Mathematics and Information Science, Henan University, Kaifeng 475004, China e-mail: liugenqiang@amss.ac.cn
KAIMING ZHAO
Affiliation:
Department of Mathematics, Wilfrid Laurier University, Waterloo, ON N2L 3C5, Canada, and College of Mathematics and Information Science, Hebei Normal (Teachers) University, Shijiazhuang, Hebei 050016, China e-mail: kzhao@wlu.ca
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Abstract

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Let d be a positive integer, q=(qij)d×d be a d×d matrix, ℂq be the quantum torus algebra associated with q. We have the semidirect product Lie algebra $\mathfrak{g}$=Der(ℂq)⋉Z(ℂq), where Z(ℂq) is the centre of the rational quantum torus algebra ℂq. In this paper, we construct a class of irreducible weight $\mathfrak{g}$-modules $\mathcal{V}$α (V,W) with three parameters: a vector α∈ℂd, an irreducible $\mathfrak{gl}$d-module V and a graded-irreducible $\mathfrak{gl}$N-module W. Then, we show that an irreducible Harish Chandra (uniformaly bounded) $\mathfrak{g}$-module M is isomorphic to $\mathcal{V}$α(V,W) for suitable α, V, W, if the action of Z(ℂq) on M is associative (respectively nonzero).

Keywords

17B1017B2017B6517B6617B68
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 3 , September 2013 , pp. 677 - 693
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

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