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Annihilating Fields of Standard Modules of $$\mathfrak {sl}(2, \mathbb {C})^\sim$$ and Combinatorial Identities
Arne Meurman, University of Lund, Sweden, and Mirko Primc, University of Zagreb, Croatia
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Memoirs of the American Mathematical Society
1999; 89 pp; softcover
Volume: 137
ISBN-10: 0-8218-0923-7
ISBN-13: 978-0-8218-0923-5
List Price: US$46 Individual Members: US$27.60
Institutional Members: US\$36.80
Order Code: MEMO/137/652

In this volume, the authors show that a set of local admissible fields generates a vertex algebra. For an affine Lie algebra $$\tilde{\mathfrak g}$$, they construct the corresponding level $$k$$ vertex operator algebra and show that level $$k$$ highest weight $$\tilde{\mathfrak g}$$-modules are modules for this vertex operator algebra. They determine the set of annihilating fields of level $$k$$ standard modules and study the corresponding loop $$\tilde{\mathfrak g}$$-module--the set of relations that defines standard modules. In the case when $$\tilde{\mathfrak g}$$ is of type $$A^{(1)}_1$$, they construct bases of standard modules parameterized by colored partitions, and as a consequence, obtain a series of Rogers-Ramanujan type combinatorial identities.

Graduate students and research mathematicians working in representation theory; theoretical physicists interested in conformal field theory.

• Abstract
• Introduction
• Formal Laurent series and rational functions
• Generating fields
• The vertex operator algebra $$N(k\Lambda_0)$$
• Modules over $$N(k\Lambda_0)$$
• Relations on standard modules
• Colored partitions, leading terms and the main results
• Colored partitions allowing at least two embeddings
• Relations among relations
• Relations among relations for two embeddings
• Linear independence of bases of standard modules
• Some combinatorial identities of Rogers-Ramanujan type
• Bibliography