Verma Modules over Quantum Torus Lie Algebras
Verma Modules over Quantum Torus Lie Algebras
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DOI：
10.4153/cjm-2010-022-1
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发表时间：
2010-04
期刊：
影响因子：
--
通讯作者：
Rencai Lü;K. Zhao
中科院分区：
文献类型：
--
作者：
Rencai Lü;K. Zhao
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 $\tilde{V}\left( \varphi \right)$ over ${{\mathfrak{L}}_{q}}$ and their corresponding irreducible highest weight modules $V\left( \varphi \right)$ are defined for some linear functions $\varphi $ . Necessary and sufficient conditions for $V\left( \varphi \right)$ to have all finite dimensional weight spaces are given. Also necessary and sufficient conditions for Verma modules $\tilde{V}\left( \varphi \right)$ to be irreducible are obtained.
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 $\tilde{V}\left( \varphi \right)$ over ${{\mathfrak{L}}_{q}}$ and their corresponding irreducible highest weight modules $V\left( \varphi \right)$ are defined for some linear functions $\varphi $ . Necessary and sufficient conditions for $V\left( \varphi \right)$ to have all finite dimensional weight spaces are given. Also necessary and sufficient conditions for Verma modules $\tilde{V}\left( \varphi \right)$ to be irreducible are obtained.