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

TL;DR

This paper constructs and analyzes vertex algebraic intertwining operators among generalized Verma modules for f7a0sl(2,C)f7, computing fusion rules and exploring their descent to irreducible modules, based on irreducibility proofs.

Contribution

It introduces a new construction of intertwining operators among generalized Verma modules for f7a0sl(2,C)f7 and establishes their properties and fusion rules.

Findings

Constructed vertex algebraic intertwining operators for generalized Verma modules.

Calculated fusion rules for these operators.

Proved irreducibility of certain submodules using composition factor multiplicities.

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.