The jacobi identity for relative twisted vertex operators associated with the roots of the lie algebras A_1~((1)) and A_2~((2)), and the generating function identities for level-k standard A_1~((1)) and A_2~((2))-modules

摘要

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 A_1~((1)) and A_2~((2)) 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 A_1~((1)) and A_2~((2)) in the diagonal embedding, we construct relative twisted vertex operators equivalent to Z-operators. In the A_1~((1))-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 A_1~((1))-modules, for any positive integer k. In the A_2~((2))-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 A_2~((2))-modules), for an arbitrary integer k, the identities that, in the case k = 3, determine the Z-operator structure of level 3 standard A_2~((2))-modules.