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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
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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.

Readership

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

Table of Contents

  • 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
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