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

Rao, S. Eswara ; Batra, Punita

Canadian mathematical bulletin, Tome 45 (2002) no. 4, pp. 672-685

Résumé

We study the representations of extended affine Lie algebras $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$ where $q$ is $N$ -th primitive root of unity ( $({{\mathbb{C}}_{q}}$ is the quantum torus in two variables). We first prove that $\oplus \,s{{\ell }_{\ell +1}}\left( \mathbb{C} \right)$ for a suitable number of copies is a quotient of $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$ . Thus any finite dimensional irreducible module for $\oplus \,s{{\ell }_{\ell +1}}\left( \mathbb{C} \right)$ lifts to a representation of $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$ . Conversely, we prove that any finite dimensional irreducible module for $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$ comes from above. We then construct modules for the extended affine Lie algebras $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)\oplus \mathbb{C}{{d}_{1}}\oplus \mathbb{C}{{d}_{2}}$ which is integrable and has finite dimensional weight spaces.

Détail

DOI : 10.4153/CMB-2002-060-9

Mots-clés : 17B65, 17B66, 17B68