Vertex algebraic intertwining operators among generalized Verma modules for $\widehat{\mathfrak{sl}(2,\mathbb{C})}$

Abstract

We construct vertex algebraic intertwining operators among certain generalized Verma modules for $\widehat{\mathfrak{sl}(2,\mathbb{C})}$ 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 $\widehat{\mathfrak{sl}(2,\mathbb{C})}$-modules. We prove this irreducibility using the composition factor multiplicities of irreducible modules in Verma modules for symmetrizable Kac-Moody Lie algebras of rank $2$, given by Rocha-Caridi and Wallach.

39 pages, updated version incorporates a comment of Antun Milas, who informed us that Theorem 3.8 can be proved using a result of Rocha-Caridi and Wallach

References in corpus (1)

• Tensor categories and the mathematics of rational and logarithmic conformal field theory

Cited by in corpus (5)

• Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$
• A general mirror equivalence theorem for coset vertex operator algebras
• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge
• Vertex algebraic intertwining operators among generalized Verma modules for affine Lie algebras
• The non-semisimple Kazhdan-Lusztig category for affine ${\mathfrak{sl}}_2$ at admissible levels