Generating sets of standard modules for $D_4^{(1)}$

Abstract

Let $\tilde{\mathfrak{g}}$ be an affine Lie algebra of type $D_4^{(1)}$ and $L(\hat{I}›)$ its standard module of level $k$ with highest weight vector $v_{\hat{I}}›$. We define Feigin--Stoyanovsky's type subspace as $W(\hat{I}›)=U({\tilde{\mathfrak{g}}}_1)v_{\hat{I}}›$, where $\tilde{\mathfrak{g}}={\tilde{\mathfrak{g}}}_{-1}\oplus {\tilde{\mathfrak{g}}}_0\oplus {\tilde{\mathfrak{g}}}_1$ is a $\mathbb{Z}$-gradation of $\tilde{\mathfrak{g}}$ associated with a $\mathbb{Z}$-gradation $\mathfrak{g}={\mathfrak{g}}_{-1}\oplus {\mathfrak{g}}_0\oplus {\mathfrak{g}}_1$. Using vertex operator relations, we reduce the Poincaré--Birkhoff--Witt spanning set of $W(\hat{I}›)$, and describe it in terms of difference and initial conditions. The spanning set of the whole standard module $L(\hat{I}›)$ can be obtained as a limit of the spanning set for $W(\hat{I}›)$.

13 pages, 1 figure