AI 中文总结
本文确定了奇秩类型 $D$ 仿射顶点代数在坍缩层级 $k=2-\ell$ 的极大理想，由单个二次奇异向量生成，并通过秩约化证明其简单性。
AI 中文摘要
我们确定了每个奇数 $\ell\ge5$ 的简单仿射顶点代数 $L_{2-\ell}(\mathfrak{so}_{2\ell})$ 的定义理想。在此层级上，Perše 的二次奇异向量单独生成泛仿射顶点代数的极大理想。结合已建立的偶秩表示，这给出了该类型 $D$ 坍缩族在 $k=2-\ell$ 处的完整奇偶依赖描述：奇秩中有一个二次生成元，偶秩中有一个二次生成元以及两个 Pfaffian 生成元。证明建立了在极小 Drinfeld--Sokolov 约化下二次商环的秩约化，该约化在两种奇偶性下均有效。约化在非零分次子商上的非零性将简单性沿奇秩链从已知基例 $D_3\cong A_3$ 在层级 $-1$ 处提升。
英文摘要
We determine the defining ideal of the simple affine vertex algebra $L_{2-\ell}(\mathfrak{so}_{2\ell})$ for every odd $\ell\ge5$. Perše's quadratic singular vector alone generates the maximal ideal of the universal affine vertex algebra at this level. Together with the established even-rank presentation, this gives a complete parity-dependent description of this type-$D$ collapsing family at $k=2-\ell$: one quadratic generator in odd rank, and a quadratic generator together with two Pfaffian generators in even rank. The proof establishes a rank reduction for the quadratic quotients under minimal Drinfeld--Sokolov reduction, valid in both parities. Nonvanishing of reduction on nonzero graded subquotients then lifts simplicity along the odd-rank chain from the known base case $D_3\cong A_3$ at level $-1$.
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