Mirko Primc

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

Vertex algebras generated by Lie algebras

N(k\Lambda_0)

Vertex operator algebras and representations of affine Lie 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.

A BASIS OF THE BASIC

Abstract In this paper we introduce a notion of vertex Lie algebra U, in a way a “half” of vertex algebra structure sufficient to construct the corresponding local Lie algebra L(U) and a vertex algebra L(U). We show that we may consider U as a subset U ⊂ V (U) which generates V(U) and that the vertex Lie algebra structure on U is induced by the vertex algebra structure on V(U). Moreover, for any vertex algebra V a given homomorphism U → V of vertex Lie algebras extends uniquely to a homomorphism V(U) → V of vertex algebras. In the second part of paper we study under what conditions on structure constants one can construct a vertex Lie algebra U by starting with a given commutator formula for fields.

\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.

Quasi-particle fermionic formulas for ( k , 3)-admissible configurations

We construct new monomial quasi-particle bases of Feigin-Stoyanovsky type subspaces for the affine Lie algebra sl(3;ℂ)∧ from which the known fermionic-type formulas for (k, 3)-admissible configurations follow naturally. In the proof we use vertex operator algebra relations for standard modules and coefficients of intertwining operators.

Vertex Algebras and Combinatorial Identities

In the 1980s, J. Lepowsky and R. Wilson gave a Lie-theoretic interpretation of Rogers–Ramanujan identities in terms of level 3 representations of affine Lie algebra sl(2,C)~. When applied to other representations and affine Lie algebras, Lepowsky and Wilsons approach yielded a series of other combinatorial identities of the Rogers–Ramanujan type. At about the same time, R. Baxter rediscovered Rogers–Ramanujan identities within the context of statistical mechanics. The work of R. Baxter initiated another line of research which yielded numerous combinatorial and analytic generalizations of Rogers–Ramanujan identities. In this note, we describe some ideas and results related to Lepowsky and Wilsons approach and indicate the connections with some results in combinatorics and statistical physics.