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# Verma Modules over Quantum Torus Lie Algebras

Published:2009-12-04
Printed: Apr 2010
• Rencai Lü
• Kaiming Zhao
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## Abstract

Representations of various one-dimensional central extensions of quantum tori (called quantum torus Lie algebras) were studied by several authors. Now we define a central extension of quantum tori so that all known representations can be regarded as representations of the new quantum torus Lie algebras $\mathfrak{L}_q$. The center of $\mathfrak{L}_q$ now is generally infinite dimensional. In this paper, $\mathbb{Z}$-graded Verma modules $\widetilde{V}(\varphi)$ over $\mathfrak{L}_q$ and their corresponding irreducible highest weight modules $V(\varphi)$ are defined for some linear functions $\varphi$. Necessary and sufficient conditions for $V(\varphi)$ to have all finite dimensional weight spaces are given. Also necessary and sufficient conditions for Verma modules $\widetilde{V}(\varphi)$ to be irreducible are obtained.
 MSC Classifications: 17B10 - Representations, algebraic theory (weights) 17B65 - Infinite-dimensional Lie (super)algebras [See also 22E65] 17B68 - Virasoro and related algebras

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