Vertex algebraic intertwining operators among generalized Verma modules for ̂𝔰𝔩(2,ℂ)

Abstract

We construct vertex algebraic intertwining operators among certain generalized Verma modules for s l ( 2 , C ) ^ \widehat {\mathfrak {sl}(2,\mathbb {C})} and calculate the corresponding fusion rules. Additionally, we show that under some conditions these intertwining operators descend to intertwining operators among one generalized Verma module and two (generally non-standard) irreducible modules. Our construction relies on the irreducibility of the maximal proper submodules of generalized Verma modules appearing in the Garland-Lepowsky resolutions of standard s l ( 2 , C ) ^ \widehat {\mathfrak {sl}(2,\mathbb {C})} -modules. We prove this irreducibility using the composition factor multiplicities of irreducible modules in Verma modules for symmetrizable Kac-Moody Lie algebras of rank 2 2 , given by Rocha-Caridi and Wallach.