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

摘要

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