Memoirs of the American Mathematical Society 1999; 89 pp; softcover Volume: 137 ISBN10: 0821809237 ISBN13: 9780821809235 List Price: US$49 Individual Members: US$29.40 Institutional Members: US$39.20 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 \(\tilde{\mathfrak g}\)modulethe 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 RogersRamanujan 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 RogersRamanujan type
 Bibliography
