Generating sets of standard modules for $D_4^{(1)}$
Ivana Baranović, Miroslav Jerkovic, Goran Trupčević
TL;DR
This work extends the Feigin–Stoyanovsky framework to the affine Lie algebra of type $D_4^{(1)}$ by constructing a PBW-type spanning set for Feigin–Stoyanovsky's type subspaces $W(\Lambda)$ and reducing it to a combinatorial description via difference and initial conditions. It then establishes a basis-like validity for these spanning sets by counting leading terms and comparing with Weyl dimensions, and finally obtains a semi-infinite, limit-based description of the standard module $L(\Lambda)$ as a union of shifted subspaces $W_{-2m}$ under the simple current operator $[\omega]$. The approach blends the vertex-operator realization, simple-current symmetry, and a careful combinatorial analysis of monomial conditions to produce explicit generating sets for $L(\Lambda)$, with fundamental weights yielding genuine bases. Collectively, the results provide explicit, constructive bases (and generating sets) for $D_4^{(1)}$ standard modules, linking vertex-operator relations to semi-infinite monomial bases and enabling potential character recurrences.
Abstract
Let $\widetilde{\mathfrak g}$ be an affine Lie algebra of type $D_4^{(1)}$ and $L(Λ)$ its standard module of level $k$ with highest weight vector $v_Λ$. We define Feigin--Stoyanovsky's type subspace as $W(Λ)=U(\widetilde{\mathfrak g}_{1})\,v_Λ$, where $\widetilde{\mathfrak g}=\widetilde{\mathfrak g}_{-1}\oplus\widetilde{\mathfrak g}_{0}\oplus\widetilde{\mathfrak g}_{1}$ is a $\mathbb{Z}$-gradation of $\widetilde{\mathfrak g}$ 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 $W(Λ)$, and describe it in terms of difference and initial conditions. The spanning set of the whole standard module $L(Λ)$ can be obtained as a limit of the spanning set for $W(Λ)$.