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Search: MSC category 17B66 ( Lie algebras of vector fields and related (super) algebras )

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1. CMB 2012 (vol 56 pp. 606)

Mazorchuk, Volodymyr; Zhao, Kaiming
Characterization of Simple Highest Weight Modules
We prove that for simple complex finite dimensional Lie algebras, affine Kac-Moody Lie algebras, the Virasoro algebra and the Heisenberg-Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro and for Heisenberg algebras.

Keywords:Lie algebra, highest weight module, triangular decomposition, locally nilpotent action
Categories:17B20, 17B65, 17B66, 17B68

2. CMB 2002 (vol 45 pp. 672)

Rao, S. Eswara; Batra, Punita
A New Class of Representations of EALA Coordinated by Quantum Tori in Two Variables
We study the representations of extended affine Lie algebras $s\ell_{\ell+1} (\mathbb{C}_q)$ where $q$ is $N$-th primitive root of unity ($\mathbb{C}_q$ is the quantum torus in two variables). We first prove that $\bigoplus s\ell_{\ell+1} (\mathbb{C})$ for a suitable number of copies is a quotient of $s\ell_{\ell+1} (\mathbb{C}_q)$. Thus any finite dimensional irreducible module for $\bigoplus s\ell_{\ell+1} (\mathbb{C})$ lifts to a representation of $s\ell_{\ell+1} (\mathbb{C}_q)$. Conversely, we prove that any finite dimensional irreducible module for $s\ell_{\ell+1} (\mathbb{C}_q)$ comes from above. We then construct modules for the extended affine Lie algebras $s\ell_{\ell+1} (\mathbb{C}_q) \oplus \mathbb{C} d_1 \oplus \mathbb{C} d_2$ which is integrable and has finite dimensional weight spaces.

Categories:17B65, 17B66, 17B68

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