The Jacobi Identity for Relative Twisted Vertex Operators Associated with the Roots of the Lie Algebras and , and the Generating Function Identities for Level-k Standard and -Modules

Abstract

The generalizations of the Jacobi identity to relative vertex operators require the introduction of “correction factors” to preserve the vertex operator structure of the identity. In the cases of relative Z ₂ and Z ₆-twisted cases associated, respectively, to the and weight lattices, these correction factors uncover the main features of the Z-operator algebras, several generalized commutator, and anticommutator relations, as residues of the suitable versions of the Jacobi identity for relative twisted vertex operators.

More specifically, using k copies of the weight lattices of the Lie algebras and in the diagonal embedding, we construct relative twisted vertex operators equivalent to Z-operators. In the -case, the residues (with respect to the untwisted vertex operator formal variable) of two versions of the Jacobi identity (differing by a rational function in the square roots of the twisted vertex operator formal variables) are the generalized commutator and anticommutator relations that determine (with suitable multi-operator extensions) the structure of level k standard -modules, for any positive integer k. In the -case, the residues (with respect to the untwisted vertex operator formal variable) of three versions of the Jacobi identity (differing by rational functions in the sixth roots of the twisted vertex operator formal variables) are the generalized commutator, anticommutator, and “partial” commutator relations that extend to level k (standard -modules), for an arbitrary integer k, the identities that, in the case k = 3, determine the Z-operator structure of level 3 standard -modules.

Key Words:

• Capparelli identities
• Generating function identities
• Infinite dimensional Lie algebra
• Jacobi identity
• Relative vertex operators
• Relative twisted vertex operators
• Rogers–Ramanujan identities
• Twisted vertex operators
• Vertex operators
• Z-operators

2000 Mathematics Subject Classification:

• 17B69
• 17B65

Notes

Communicated by K.C. Misra.