Verma Modules over Quantum Torus Lie Algebras

Lü, Rencai ; Zhao, Kaiming

Canadian journal of mathematics, Tome 62 (2010) no. 2, pp. 382-399

Résumé

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.

Détail

DOI : 10.4153/CJM-2010-022-1

Mots-clés : 17B10, 17B65, 17B68

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