Arne Meurman

Annihilating ideals of standard modules of

Abstract Introduction Formal Laurent series and rational functions Generating fields The vertex operator algebra

{rm sl}(2, Bbb C)sp sim

N(k\Lambda_0)

and combinatorial identities

Modules over

A moonshine module for the Monster

N(k\Lambda_0)

Highest weight representations of the Neveu-Schwarz and Ramond algebras

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.

Vertex operator algebras and representations of affine Lie algebras

The theory of finite simple groups was for a long time a rather isolated and unusual branch of mathematics. It achieved its goal in 1981 when a proof of the classification theorem was completed. This unique proof, comprising several thousand pages of published articles and preprints, leads to the following list of finite simple groups (see [12] for the details): the groups of Lie type, the alternating groups and the 26 sporadic groups. While each of the first two classes has a uniform description, the groups in the third class still have quite different constructions. The largest sporadic group, called the Monster and denoted F1, was predicted independently by B. Fischer and R. Griess in 1973. It contains 20 or 21 of the sporadic groups and has order >8 – 1053. This group gave rise to many mysteries even before its actual appearance, promising deep connections with different areas of mathematics.

A BASIS OF THE BASIC

We construct a family of representationsKξ,w of the Neveu-Schwarz and Ramond algebras, which generalize the Fock representations of the Virasoro algebra. We show that the representationsKξ,w are intertwined by a vertex operator.The above results are used to give the proof of the conjectured formulas for the determinant of the contravariant form on the highest weight representations of the Neveu-Schwarz and Ramond algebras. Further results on the representation theory of the latter are derived from the determinant formulas.

\mathfrak{sl} ({\bf 3}, {\mathbb C})^~

We announce the construction of an explicit basis for all integrable highest weight modules over the Lie algebra A1(1). The construction uses representations of vertex operator algebras and leads to combinatorial identities of Rogers-Ramanujan-type.

-MODULE

J. Lepowsky and R. L. Wilson initiated the approach to combinatorial Rogers–Ramanujan type identities via the vertex operator constructions of representations of affine Lie algebras. In this approach the first new combinatorial identities were discovered by S. Capparelli through the construction of the level 3 standard -modules. We obtained several infinite series of new combinatorial identities through the construction of all standard -modules; the identities associated to the fundamental modules coincide with the two Capparelli identities. In this paper we extend our construction to the basic -module and, by using the principal specialization of the Weyl–Kac character formula, we obtain a Rogers–Ramanujan type combinatorial identity for colored partitions. The new combinatorial identity indicates the next level of complexity which one should expect in Lepowsky–Wilsons approach for affine Lie algebras of higher ranks, say for , n ≥ 2, in a way parallel to the next level of complexity seen when passing from the Rogers–Ramanujan identities (for modulus 5) to the Gordon identities for odd moduli ≥7.