Principal subspaces of higher-level standard $\widehat{\mathfrak{sl}(n)}$-modules

TL;DR

This paper develops a framework using algebra completions and vertex operator algebra theory to analyze principal subspaces of standard modules for affine Lie algebras, deriving exact sequences and multigraded dimensions.

Contribution

It introduces a natural setting for defining relations of principal subspaces and constructs exact sequences among them for higher-rank affine Lie algebras, extending previous results.

Findings

Derived multigraded dimensions of principal subspaces.

Constructed exact sequences among principal subspaces.

Generalized earlier work on $ ext{sl}(3)$ to higher ranks.

Abstract

Using completions of certain universal enveloping algebras, we provide a natural setting for families of defining relations for the principal subspaces of standard modules for untwisted affine Lie algebras. We also use the theory of vertex operator algebras and intertwining operators to construct exact sequences among principal subspaces of certain standard $\widehat{\mathfrak{sl}(n)}$-modules, $n \ge 3$. As a consequence, we obtain the multigraded dimensions of the principal subspaces $W(k_1\Lambda_1 + k_2 \Lambda_2)$ and $W(k_{n-2}\Lambda_{n-2} + k_{n-1} \Lambda_{n-1})$. This generalizes earlier work by Calinescu on principal subspaces of standard $\widehat{\mathfrak{sl}(3)}$-modules.