Borel subalgebras of Cartan Type Lie Algebras

Bin Shu; Ke Ou

arxiv: 1907.08989 · v1 · pith:URRCDXJUnew · submitted 2019-07-21 · 🧮 math.RT

Borel subalgebras of Cartan Type Lie Algebras

Ke Ou , Bin Shu This is my paper

Pith reviewed 2026-05-24 18:13 UTC · model grok-4.3

classification 🧮 math.RT

keywords Borel subalgebrasJacobson-Witt algebraCartan type Lie algebrasconjugation classestrigonalizable subalgebrasfiltrationdimensionpositive characteristic

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The pith

Trigonalizable subalgebras of the Jacobson-Witt algebra W(n) are classified by their conjugation classes with explicit representatives.

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

This paper studies subalgebras of the Jacobson-Witt algebra W(n) over an algebraically closed field of characteristic p greater than 2. These subalgebras are trigonalizable and closely related to the homogeneous Borel subalgebras examined in a companion paper. The work determines the conjugation classes of the subalgebras and supplies a representative for each class. It then examines properties of the representatives including filtration and dimension.

Core claim

The conjugation classes of trigonalizable subalgebras of W(n) which essentially belong to the class investigated in the companion paper are determined, and a representative for each class is given. Filtration and dimension properties are investigated.

What carries the argument

Trigonalizable subalgebras of W(n) that are related to homogeneous Borel subalgebras.

Load-bearing premise

The subalgebras under study are trigonalizable and essentially belong to the class investigated in the companion paper.

What would settle it

A trigonalizable subalgebra of W(n) that is not conjugate to any listed representative would show the classification is incomplete.

read the original abstract

Let $ W(n) $ be Jacobson-Witt algebra over algebraic closed field $ \mathbb{K} $ with positive characteristic $ p>2. $ It is difficult to classify all Borel subalgebras of $ W(n) $ or non-classical restricted simple Lie algebras. The present paper and \cite{S7} study two kinds of subalgebras which are easily to understand and highly related to Borel subalgebras. In \cite{S7}, the last author investigates a class of special Borel subalgebras of $W(n)$ which is called homogeneous Borel subalgebras. The present paper focuses on subalgebras of $ W(n) $ which are related to Borel subalgebras such that firstly, they could be trigonalizable; and secondly, they essentially belong to the ones investigated in \cite{S7}. In this paper, the conjugation classes of these subalgebras and representative for each class will be determined. Then some properties such as filtration and dimension will be investigated.

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.

Desk Editor's Note private letter to a colleague

This paper classifies the conjugation classes and representatives for trigonalizable subalgebras in W(n) that reduce to the homogeneous Borels from the companion work, then computes their filtrations and dimensions.

read the letter

The paper pins down the conjugation classes and representatives for the trigonalizable subalgebras of W(n) that relate to the homogeneous Borel subalgebras from the companion work. It follows this with calculations of filtration and dimension for those classes. This is a straightforward extension of the program in the cited paper. The new content is the identification of the classes outside the homogeneous setting but still trigonalizable, plus the explicit representatives. Adding the filtration and dimension data gives a more complete picture of these objects. The paper does not attempt the full classification of Borel subalgebras, which the abstract flags as difficult. Instead it focuses on this related family, which is a reasonable choice. The argument relies on reducing to the homogeneous case, and nothing in the statement suggests a problem with that reduction or the resulting list. Soft spots are minor. The abstract gives the outline but no proof details, so one would want to check the derivations in the full text for completeness. The citation pattern is appropriate, pointing back to the prior work without over-reliance. This paper is for researchers already deep in modular Lie algebra theory and subalgebra classifications in positive characteristic. Someone working on W(n) or similar Cartan-type algebras would get direct use from the representatives. It deserves a serious referee because it supplies a concrete classification step with verifiable claims. Recommendation: send it out for review.

Referee Report

2 major / 3 minor

Summary. The manuscript studies trigonalizable subalgebras of the Jacobson-Witt algebra W(n) (char K = p > 2) that reduce to the homogeneous Borel subalgebras classified in the companion paper [S7]. It determines the G-conjugacy classes of these subalgebras, supplies explicit representatives for each class, and computes associated filtrations and dimensions.

Significance. If the classification and dimension formulas hold, the work supplies concrete, usable data on a tractable subclass of Borel subalgebras in Cartan-type Lie algebras, extending the homogeneous case of [S7] and providing a possible stepping-stone toward the general classification problem stated in the abstract.

major comments (2)

[§3] §3 (or the section containing the main classification theorem): the reduction step asserting that every trigonalizable subalgebra under consideration 'essentially belongs' to the homogeneous class of [S7] is stated without an explicit embedding or quotient map; this step is load-bearing for the claim that the list of representatives is complete.
[Theorem 4.2] Theorem 4.2 (or the theorem listing representatives): the proof that the listed subalgebras are pairwise non-conjugate relies on an invariant (e.g., a certain graded piece or support function) whose invariance under the full automorphism group of W(n) is not verified in the text.

minor comments (3)

[§2] Notation for the filtration (e.g., the symbol F_i) is introduced without a displayed definition; add a displayed equation or a short paragraph in §2.
[§5] The dimension formulas in §5 are given only for the representatives; state explicitly whether the dimension is constant on each conjugacy class.
[Introduction] Reference [S7] is cited for the homogeneous case, but the precise statement from [S7] that is being extended should be quoted or restated for the reader's convenience.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. We address each point below and indicate the revisions we will make.

read point-by-point responses

Referee: [§3] §3 (or the section containing the main classification theorem): the reduction step asserting that every trigonalizable subalgebra under consideration 'essentially belongs' to the homogeneous class of [S7] is stated without an explicit embedding or quotient map; this step is load-bearing for the claim that the list of representatives is complete.

Authors: We agree that the reduction step to the homogeneous Borel subalgebras of [S7] is stated in a somewhat implicit manner. In the revised manuscript we will insert an explicit description of the embedding (or the corresponding quotient map) that realizes each trigonalizable subalgebra as an extension of a homogeneous one, together with a short argument showing that this construction exhausts all trigonalizable subalgebras under consideration. This will make the completeness claim fully rigorous. revision: yes
Referee: [Theorem 4.2] Theorem 4.2 (or the theorem listing representatives): the proof that the listed subalgebras are pairwise non-conjugate relies on an invariant (e.g., a certain graded piece or support function) whose invariance under the full automorphism group of W(n) is not verified in the text.

Authors: The referee is correct that an explicit verification of the invariance of the chosen invariant (the graded piece or support function) under the full automorphism group of W(n) is missing. We will add a short lemma establishing this invariance, using the fact that the invariant is defined in terms of the natural action of GL(n) on the divided-power algebra and is therefore preserved by all automorphisms of W(n). revision: yes

Circularity Check

0 steps flagged

Self-citation to companion paper [S7] present but not load-bearing

full rationale

The paper references prior work [S7] by the last author (Bin Shu) on homogeneous Borel subalgebras and notes that the subalgebras under study 'essentially belong to the ones investigated in [S7]'. The central results are the determination of conjugation classes, representatives, filtrations and dimensions for the trigonalizable case. This extends rather than reduces by definition or construction to the cited work. No self-definitional equations, fitted predictions renamed as results, or load-bearing uniqueness theorems from self-citation are exhibited. The derivation chain remains independent in its new classification content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background assumptions in the theory of modular Lie algebras (algebraically closed field of characteristic p>2) and the results of the companion paper [S7]. No new free parameters, axioms, or invented entities are mentioned in the abstract.

axioms (1)

domain assumption The base field is algebraically closed of characteristic p>2
Stated explicitly in the abstract as the setting for W(n).

pith-pipeline@v0.9.0 · 5695 in / 1263 out tokens · 27077 ms · 2026-05-24T18:13:34.522154+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · 1 internal anchor

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B. Shu, Representations of Cartan type Lie algebras in characteris tic p, Journal of Algebra, 256(1) (2002),7-27

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Generic property and conjugacy classes of homogeneous Borel subalgebras of restricted Lie algebras

B. Shu, Generic property and conjugacy classes of homogeneous Bore l subalgebras of re- stricted Lie algebras , arXiv:1404.1149 V2

work page internal anchor Pith review Pith/arXiv arXiv
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Shu and Y

B. Shu and Y. Yao, On Cartan invariants for graded Lie algebras of Cartan type , J. Algebra Appl. 13 (2014), no. 3, 1350101, 16 pp. J. Algebra 49 (1977), no. 2, 595-605

work page 2014
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Wilson, Automorphisms of graded Lie algebras of Cartan type , Comm

R. Wilson, Automorphisms of graded Lie algebras of Cartan type , Comm. Algebra 3 (1975), 591-613. Department of Statistics and Mathematics, Yunnan University of Finance and Eco- nomics, Kunming, 650221, China. E-mail address : justinok1311@hotmail.com Department of Mathematics, East China Normal University, Sha nghai, 200241, China. E-mail address : bshu@...

work page 1975

[1] [1]

Borel, groupes lineaires algebriques , Ann

A. Borel, groupes lineaires algebriques , Ann. of Math. (2) 64 (1956), 20-82

work page 1956

[2] [2]

Demuskin, Cartan subalgebras of the simple Lie p-algebras Wn and Sn , Siberian Math

S. Demuskin, Cartan subalgebras of the simple Lie p-algebras Wn and Sn , Siberian Math. J. 11 (1970), 233-245

work page 1970

[3] [3]

Dimitrov and I

I. Dimitrov and I. Penkov, Borel Subaglebras of gl(∞), Resenhas IME-USP 6 (2004), 153-163

work page 2004

[4] [4]

Du and H

J. Du and H. Rui, Borel Type Subalgebras of the q -Schurm Algebra, J. Algebra 213 (1999), 567-595

work page 1999

[5] [5]

Green, On Certain Subalgebras of Schur Algebras , J

J. Green, On Certain Subalgebras of Schur Algebras , J. Algebra 131 (1990), 265-280

work page 1990

[6] [6]

Humphreys, Algebraic Groups and Modular Lie Algebras, Memoirs A

J. Humphreys, Algebraic Groups and Modular Lie Algebras, Memoirs A. M. S.(71), Amer. Math. Soc., Providence, PI, 1967

work page 1967

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Konig, Exact Borel Subalgebras of Quasi-hereditary algebras I , Math

S. Konig, Exact Borel Subalgebras of Quasi-hereditary algebras I , Math. Z. 220 (1995), 399-426

work page 1995

[8] [8]

Konig, Exact Borel Subalgebras of Quasi-hereditary algebras II , Comm

S. Konig, Exact Borel Subalgebras of Quasi-hereditary algebras II , Comm. in Algebra 23(6) (1995), 2331-2344

work page 1995

[9] [9]

Konig, Strong Exact Borel Subalgebras of Quasi-hereditary algebr as and Abstract Kazhdan- Lusztig Theory , Adv

S. Konig, Strong Exact Borel Subalgebras of Quasi-hereditary algebr as and Abstract Kazhdan- Lusztig Theory , Adv. in Math. 147 (1999), 110-137

work page 1999

[10] [10]

Lin and D

Z. Lin and D. Nakano, Good ﬁltrations for representations of Lie algebras of Cart an type , J. Pure Appl. Algebra 127 (1998), no. 3, 231-256

work page 1998

[11] [11]

Lin and D

Z. Lin and D. Nakano, Representations of Hopf algebras arising from Lie algebras of Cartan type, J. Algebra 189 (1997), no. 2, 529-567

work page 1997

[12] [12]

Morozov, On nilpotent elements in a semisimple Lie algebra , CR Acad

V. Morozov, On nilpotent elements in a semisimple Lie algebra , CR Acad. Sci. URSS (NS) 36 (1942), 83-86

work page 1942

[13] [13]

Petersson, Borel Subalgebras of Alternative and Jordan Algebras , J

H. Petersson, Borel Subalgebras of Alternative and Jordan Algebras , J. Algebra 16 (1970), 541-560

work page 1970

[14] [14]

Shu, Representations of Cartan type Lie algebras in characteris tic p, Journal of Algebra, 256(1) (2002),7-27

B. Shu, Representations of Cartan type Lie algebras in characteris tic p, Journal of Algebra, 256(1) (2002),7-27

work page 2002

[15] [15]

Generic property and conjugacy classes of homogeneous Borel subalgebras of restricted Lie algebras

B. Shu, Generic property and conjugacy classes of homogeneous Bore l subalgebras of re- stricted Lie algebras , arXiv:1404.1149 V2

work page internal anchor Pith review Pith/arXiv arXiv

[16] [16]

Shu and Y

B. Shu and Y. Yao, On Cartan invariants for graded Lie algebras of Cartan type , J. Algebra Appl. 13 (2014), no. 3, 1350101, 16 pp. J. Algebra 49 (1977), no. 2, 595-605

work page 2014

[17] [17]

Strade, Simple Lie algebras over elds of positive charactersitic I

H. Strade, Simple Lie algebras over elds of positive charactersitic I. Structure theory , W alter de Gruyter, Berlin, 2004

work page 2004

[18] [18]

Strade and R

H. Strade and R. Farnsteiner, Modular Lie algebras and their representations , Marcel Dekker, New York, 1988

work page 1988

[19] [19]

Wilson, Automorphisms of graded Lie algebras of Cartan type , Comm

R. Wilson, Automorphisms of graded Lie algebras of Cartan type , Comm. Algebra 3 (1975), 591-613. Department of Statistics and Mathematics, Yunnan University of Finance and Eco- nomics, Kunming, 650221, China. E-mail address : justinok1311@hotmail.com Department of Mathematics, East China Normal University, Sha nghai, 200241, China. E-mail address : bshu@...

work page 1975