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A New Class of Representations of EALA Coordinated by Quantum Tori in Two Variables

  Published:2002-12-01
 Printed: Dec 2002
  • S. Eswara Rao
  • Punita Batra
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Abstract

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.
MSC Classifications: 17B65, 17B66, 17B68 show english descriptions Infinite-dimensional Lie (super)algebras [See also 22E65]
Lie algebras of vector fields and related (super) algebras
Virasoro and related algebras
17B65 - Infinite-dimensional Lie (super)algebras [See also 22E65]
17B66 - Lie algebras of vector fields and related (super) algebras
17B68 - Virasoro and related algebras
 

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