An isomorphism between the fusion algebras of $V_L^+$ and type $D^{(1)}$ level 2

TL;DR

This paper demonstrates an explicit isomorphism between the fusion algebra of a specific vertex operator algebra associated with a rank 1 lattice and the fusion algebra of a type D affine Kac-Moody algebra at level 2, revealing deep algebraic connections.

Contribution

It establishes an explicit isomorphism between the fusion algebras of $V_L^+$ and type $D^{(1)}$ level 2, clarifying their algebraic relationship.

Findings

Fusion algebra of $V_L^+$ is isomorphic to that of type $D^{(1)}$ at level 2

The isomorphism holds in almost all cases for rank 1 lattices

Provides a new perspective on the structure of vertex operator algebras and affine Lie algebras

Abstract

The fusion algebra of the vertex operator algebra $V_L^+$ for a rank 1 even lattice $L$ is explicitly shown to be isomorphic to the fusion algebra of the Kac-Moody algebra of type $D^{(1)}$ at level 2 in almost all cases.