arXiv subjects

Sihai Jin

Publications and source records attributed to Sihai Jin.

5 recordsLinked to original sources

arXiv2026

Odd-rank maximal ideals at collapsing levels of type $D$

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 $-1$.

math.QA↗

arXiv2026

Completing the Arakawa--Moreau Conjecture on Maximal Ideals of Affine Vertex Algebras

Arakawa and Moreau constructed explicit singular vectors in a family of negative-level universal affine vertex algebras of types $D$ and $E$ and conjectured that the ideals generated by these vectors are maximal. Previous work established the $n=0$ cases for $D_4$, $E_6$, $E_7$, and $E_8$, as well as the level $-2$ case for $D_\ell$ with $\ell\geq 5$. We prove all the remaining cases: the level $-1$ case for $D_\ell$ with $\ell\geq 5$, and the negative-level cases with $n>0$ for $D_4$, $E_6$, $E_7$, and $E_8$. Together with the previously known results, this completes Arakawa--Moreau Conjecture 1. The proof determines the images of the prescribed singular vectors under minimal Drinfeld--Sokolov reduction and establishes simplicity of the reduced quotients by combining a Ramond--Zhu algebra argument, a Casimir-gap argument, and Li's spectral flow. Exactness and a nonvanishing theorem for the reduction functor then lift simplicity to the corresponding affine quotients. We also formulate a general maximality principle based on minimal reduction, give an alternative reduction-theoretic proof of the known level $-2$ result for $D_\ell$, and obtain a rank-reduction proof of the maximal-ideal theorem for the collapsing family $V^{2-2r}(D_{2r})$. Consequently, every candidate quotient appearing in Arakawa--Moreau Conjecture 1 is the corresponding simple affine vertex algebra.

math.QA↗

arXiv2026

Subregular affine cells and the level $-1$ vertex algebra of type $D$

We prove the simple-object prediction of Shan--Yan--Zhao and a basis-preserving dual-cell realization for the distinguished vacuum block of the simple affine vertex algebras $L_{-1}(D_\ell)$, $\ell\ge5$. The block has exactly $\ell+1$ simple objects, indexed by the subregular affine left cell containing $s_0$. The proof combines a primitive-ideal inclusion, an independent exhaustion argument, and a finite-length step. A noncritical Sugawara lift supplies finite-dimensional weight-space detectors in the original Shan--Yan--Zhao category-$\mathcal O$ block, so dévissage applies to its ordinary Grothendieck group. We then identify this group, basis by basis, with the $q=1$ specialization of the corresponding dual affine left-cell module. An injective signed normalized-character realization identifies the resulting image with the canonical dual-cell image in the completed singular-orbit module and hence supplies the corresponding abstract $\widehat W$-module structure. We do not identify this action with a functorial action arising from affine twisting functors or Kashiwara--Tanisaki localization. The subregular inverse Kazhdan--Lusztig calculation of Bezrukavnikov--Kac--Krylov also yields uniform character formulas.

math.QA↗

arXiv2026

A Lean 4 Verification Report for Completing the Arakawa--Moreau Conjecture on Maximal Ideals of Affine Vertex Algebras

We report a Lean 4 formal audit accompanying the paper "Completing the Arakawa--Moreau Conjecture on Maximal Ideals of Affine Vertex Algebras" (arXiv:2607.25249). In a single Lean source, the development kernel-checks the paper-local deductive architecture used for the new maximal-ideal cases: affine-kernel detection and lifting, extremal PBW-line deductions, finite Ramond--Casimir arithmetic and case enumerations, reduced-simplicity contradiction schemes, and the case-level affine conclusions for the level -1 D_l family and the positive-parameter cases of D4, E6, E7, and E8. It also audits the paper's alternative D_l level -2 argument and the separate rank-reduction application of Section 8. Previously published exceptional n=0 cases are not claimed as newly formalized results of this bundle. The released project builds successfully with the pinned Lean/Mathlib environment and contains live #print axioms queries for the principal final wrappers. The development does not reconstruct affine vertex algebras, minimal W-algebras, BRST/DS reduction, Ramond Zhu theory, or Li spectral flow from foundational definitions in Mathlib. Those ingredients, together with the concrete realization of case-specific representation-theoretic objects, are exposed as theorem parameters and semantic interfaces. Accordingly, the precise verification claim is a kernel-checked deduction of the paper-local proof from explicitly stated VOA/BRST/DS boundary inputs, rather than a from-scratch formalization of the ambient representation theory.

math.QA↗

arXiv2026

A Lean 4 Verification Report for Subregular Affine Cells and the Level $-1$ Vertex Algebra of Type $D$

We report a Lean 4 formal verification accompanying the paper "Subregular Affine Cells and the Level -1 Vertex Algebra of Type D" (arXiv:2608.11997). The formalization kernel-checks substantial internal parts of the proof architecture, including the Section 4 membership/descent chain, the type-D norm-gap argument, zero-orbit energy and signed-permutation rigidity, node-weight and numerical rigidity calculations, the exhaustion logic, the passage to the candidate quotient classification, the simple-object count, the final additive Grothendieck-group comparison, and the coefficient-substitution layer of the uniform character formula. Higher representation-theoretic results whose foundational infrastructure is not presently constructed in the file are isolated as explicit semantic interfaces rather than introduced as Lean axioms. Thus the precise claim is a kernel-checked internal deduction from explicit representation-theoretic boundary inputs, not a from-scratch formalization of vertex algebras, BRST reduction, finite W-algebras, or affine Hecke theory inside Mathlib.

math.QA↗