Vertex-algebraic structure of the principal subspaces of level one modules for the untwisted affine Lie algebras of types A,D,E

TL;DR

This paper provides explicit presentations and recursive formulas for the graded dimensions of principal subspaces of level one modules for untwisted affine Lie algebras of types A, D, and E, using vertex operator algebra techniques.

Contribution

It introduces natural presentations and recursions for these principal subspaces, extending previous work and employing intertwining operators in vertex algebra theory.

Findings

Derived explicit presentations of principal subspaces.

Established recursive q-difference equations for graded dimensions.

Computed graded dimensions explicitly.

Abstract

Generalizing some of our earlier work, we prove natural presentations of the principal subspaces of the level one standard modules for the untwisted affine Lie algebras of types A, D and E, and also of certain related spaces. As a consequence, we obtain a canonical complete set of recursions (q-difference equations) for the (multi-)graded dimensions of these spaces, and we derive their graded dimensions. Our methods are based on intertwining operators in vertex operator algebra theory.