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

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 $W(Λ)$.

💡 Research Summary

The paper investigates the structure of standard modules for the affine Lie algebra of type D₄^{(1)} by constructing explicit generating sets for both the Feigin‑Stoyanovsky type subspaces and the whole modules. Starting from the finite‑dimensional simple Lie algebra 𝔤 of type D₄, the authors fix a minuscule weight ω (which equals the first fundamental weight ω₁) and use it to induce a ℤ‑gradation 𝔤 = 𝔤_{‑1} ⊕ 𝔤₀ ⊕ 𝔤₁. The subspaces 𝔤₁ and 𝔤_{‑1} are commutative and are spanned by root vectors corresponding to the set of “colors” Γ = {γ₂,…,γ₄,γ₄,…,γ₂}, where each γ_i is a root of the form ε₁ ± ε_i. The affine algebra 𝔤̃ = 𝔤⊗ℂ