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arxiv: 2604.10388 · v1 · submitted 2026-04-12 · 🧮 math.RT

Koszulity in the category mathcal{O} of the periplectic Lie Superalgebra mathfrak{pe}(2)

Finn Kinley This is my paper

Pith reviewed 2026-05-10 16:40 UTC · model grok-4.3

classification 🧮 math.RT
keywords KoszulityCategory Operiplectic Lie superalgebrape(2)integral blocksendomorphism algebraExt groups
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The pith

The remaining integral block in the category O of the periplectic Lie superalgebra pe(2) is Koszul.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The category O for the periplectic Lie superalgebra pe(2) has three blocks up to equivalence. The generic block is Koszul and one integral block is not. This paper proves that the other integral block is also Koszul. The proof proceeds by explicitly determining the endomorphism algebra of the projective modules in the block with the help of a computer algebra system and then showing inductively that this algebra is Koszul. As a byproduct, all Ext groups between simple modules in the block are computed.

Core claim

We prove that the remaining integral block of O(pe(2)) is Koszul. This is achieved by explicitly computing the endomorphism algebra of the projective modules in this block and proving that it is Koszul by induction on the structure of the algebra.

What carries the argument

The endomorphism algebra of the projective modules in the non-principal integral block, computed explicitly and verified to be Koszul inductively.

If this is right

  • The Ext groups between all pairs of simple modules in the block are now known explicitly.
  • The classification of Koszul blocks in O(pe(2)) is complete.
  • The inductive verification technique applies to the specific relations found in this endomorphism algebra.
  • Projective modules in this block have a graded structure compatible with Koszul duality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method of explicit computation followed by induction may extend to blocks of category O for other small-rank Lie superalgebras.
  • Similar computer-assisted approaches could classify Koszulity in related categories for larger superalgebras.
  • The computed algebra might admit a presentation in terms of generators and relations that reveals a pattern across different blocks.

Load-bearing premise

The explicit computation of the endomorphism algebra, assisted by the Mathematica tool, accurately captures all relations without omissions or errors, and the inductive argument applies without gaps to every module in the block.

What would settle it

Finding a specific pair of modules or a relation in the endomorphism algebra that violates the Koszul condition, such as a non-vanishing higher Ext group that contradicts the inductive proof.

Figures

Figures reproduced from arXiv: 2604.10388 by Finn Kinley.

Figure 1
Figure 1. Figure 1: Roots in Φ+ and Φ− for basis elements of g. We write b = h ⊕ n + and call it the Borel sub-algebra of g. Denote by U(g) the universal enveloping algebra of g (as defined in [Mus12], for example). Define the Verma module of weight λ as ∆(λ) := Indg bCλ = U(g) ⊗U(b) Cλ, where Cλ is the b-module spanned by w with action n + · w = 0 and hw = λ(h)w ∀h ∈ h. Let b0 = h0⊕n + 0 be the Borel sub-algebra of g0, then … view at source ↗
Figure 2
Figure 2. Figure 2: Weight spaces represented by a • are one dimensional, by a × are two dimensional and by a ∗ are three dimensional [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The vectors shown above in P0(λ) for a ≤ −3 are those that, upon tensoring with specific elements in U(g−1), lie in weight spaces of P(λ) known to contain target vectors. Note that we omit the component of each weight corresponding to the action of ξ∂ξ for simplicity. Proposition 4.13. Let λ = aε + bδ ∈ Oodd with a ≤ −5. There are twelve target vectors in P(λ) which we list below. (A1) 1 ⊗ v0 (A2) 1 ⊗ u0 (… view at source ↗
Figure 4
Figure 4. Figure 4: Local pictures of A Notice the labeled edges and note that f, g, p, q are all unambiguous (up to scalar) as their corresponding homomorphism spaces are one dimensional. However, there is ambiguity for g ′ and f ′ , as their corresponding homomorphism spaces are two dimensional, we will define g ′ and f ′ after we state the theorem. In what follows, the order in which these labeled edges appear (when compos… view at source ↗
Figure 5
Figure 5. Figure 5: b b + 1 b + 2 a + 4 • a + 2 ◦ a • • a − 2 ◦ a − 4 • . . . . . . −a • −a − 2 ◦ −a − 4 • [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: n = 3, f ′ , g′ relation b b + 1 b + 2 b + 3 −a + 2 • −a ◦ −a − 2 • −a − 4 ◦ −a − 6 • . . . . . . a + 4 ◦ a + 2 a ◦ a − 2 a − 4 ◦ [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: n > 3, f ′ , g′ relation b + n − 3 b + n − 2 b + n − 1 b + n −a + 2 + m2 −a + m2 ◦ −a − 2 + m2 • −a − 4 + m2 ◦ −a − 6 + m2 . . . . . . a + 4 − m2 a + 2 − m2 a − m2 ◦ a − 2 − m2 a − 4 − m2 g f q [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: 2n = 1 − a b + n − 3 b + n − 2 b + n − 1 b + n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 ◦ 5 • 3 ◦ 1 ? −1 • −3 ◦ −5 • −7 ◦ −9 . . . . . . . . . . . . . . [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
read the original abstract

The main result of this paper is to establish precisely which blocks in the Category $\mathcal{O}$ of the periplectic Lie superalgebra $\mathfrak{pe}(2)$ are Koszul. It is known that $\mathcal{O}(\mathfrak{pe}(2))$ has three blocks up to equivalence; one generic block and two integral blocks. The generic block is known to be Koszul, and the principle integral block is verifiably not Koszul. In this paper, we prove that the remaining of the three blocks of $\mathcal{O}(\mathfrak{pe}(2))$ is Koszul. This is done by explicitly computing the endomorphism algebra of projective modules in this block and then proving that it is Koszul inductively. Along the way, we compute all $\mathrm{Ext}$ groups between simples in this block. To compute the endomorphism algebra we are aided by a computer algebra tool developed in Mathematica, inspired by a post on Stack Exchange.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript proves that the remaining integral block of category O for the periplectic Lie superalgebra pe(2) is Koszul. It does so by using a custom Mathematica implementation to compute the endomorphism algebra of the projective modules in this block, followed by an inductive argument establishing Koszulity; all Ext groups between simple modules in the block are computed along the way. The generic block is already known to be Koszul and the principal integral block is known not to be.

Significance. If the explicit computation and induction are accurate and complete, the result finishes the classification of Koszul blocks in O(pe(2)). The concrete presentation of the endomorphism algebra and the full Ext data supply explicit algebraic information that may be useful for testing broader conjectures on Koszulity in category O for Lie superalgebras.

major comments (2)
  1. [Section describing the endomorphism algebra computation] The central claim rests on the accuracy of the Mathematica-computed multiplication table and relations for the endomorphism algebra of the projective modules. The manuscript must either include the full notebook (or equivalent verifiable output) or provide a hand-checkable derivation of the key relations, because any undetected error in the algebra presentation would invalidate the subsequent inductive Koszulity proof.
  2. [Section containing the inductive argument] The inductive proof of Koszulity must be shown to cover every module in the block without omitted cases. The base cases, the precise inductive step (including how it uses the computed Ext groups), and the verification that all possible extensions are handled should be stated explicitly; gaps here would leave the Koszulity conclusion incomplete.
minor comments (3)
  1. The Stack Exchange post that inspired the Mathematica tool should be cited with a precise reference.
  2. The introduction should state more explicitly how the two integral blocks are distinguished (e.g., by their highest weights or central characters) before referring to “the remaining block.”
  3. Notation for the simple modules and their projective covers should be introduced once and used consistently throughout the computations and the induction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help us strengthen the presentation of our results on the Koszulity of the non-principal integral block in category O for pe(2). We address each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Section describing the endomorphism algebra computation] The central claim rests on the accuracy of the Mathematica-computed multiplication table and relations for the endomorphism algebra of the projective modules. The manuscript must either include the full notebook (or equivalent verifiable output) or provide a hand-checkable derivation of the key relations, because any undetected error in the algebra presentation would invalidate the subsequent inductive Koszulity proof.

    Authors: We agree that the reliability of the computer-assisted computation is central to the argument. In the revised version we will include the full Mathematica notebook as supplementary material, together with a self-contained description of the input data (weights, action matrices, and basis choices), the verification steps against independently computed Ext dimensions, and the extracted multiplication table and relations. This will allow readers to reproduce and check the algebra presentation directly. revision: yes

  2. Referee: [Section containing the inductive argument] The inductive proof of Koszulity must be shown to cover every module in the block without omitted cases. The base cases, the precise inductive step (including how it uses the computed Ext groups), and the verification that all possible extensions are handled should be stated explicitly; gaps here would leave the Koszulity conclusion incomplete.

    Authors: We thank the referee for highlighting the need for greater explicitness. In the revision we will add an enumerated list of all simple modules appearing in the block, state the base cases for the lowest weights explicitly, describe the inductive step in terms of the already-computed Ext groups between simples, and verify that the induction applies to every possible extension by using the explicit presentation of the endomorphism algebra. This will make the coverage of the entire block transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit computation and induction are self-contained

full rationale

The paper proves Koszulity of the remaining integral block in O(pe(2)) by directly computing the endomorphism algebra of its projective modules (aided by a Mathematica tool) and then establishing Koszulity of that algebra via induction on the block structure, while also computing all Ext groups between simples. This chain relies on explicit calculation and standard inductive reasoning rather than any self-definitional steps, fitted parameters renamed as predictions, load-bearing self-citations, or imported uniqueness theorems. Prior statements about the generic and principal integral blocks are treated as external known facts and do not enter the derivation for the target block. The argument is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result relies on the known decomposition of O(pe(2)) into three blocks and standard background facts about Koszul algebras and category O; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption O(pe(2)) decomposes into exactly three blocks up to equivalence (one generic, two integral).
    Stated as known background in the abstract.
  • domain assumption Standard definitions and properties of Koszul algebras and Ext groups in category O for Lie superalgebras hold.
    Used as the foundation for the inductive proof and computations.

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

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    Representation Theory of a Semisimple Extension of the Takiff Superalgebra.International Mathematics Research Notices, 2022(18):14454–14495, 06

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    On semisimplicity of jantzen middles for the periplectic lie superal- gebra.International Mathematics Research Notices, 2023(7):5660–5684, 02

    [Che22] Chih-Whi Chen. On semisimplicity of jantzen middles for the periplectic lie superal- gebra.International Mathematics Research Notices, 2023(7):5660–5684, 02

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    Khovanov algebras for the periplectic lie superalgebras.International Mathematics Research Notices, 2024(22):14008–14060, 10

    [Neh24] Jonas Nehme. Khovanov algebras for the periplectic lie superalgebras.International Mathematics Research Notices, 2024(22):14008–14060, 10

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