The normal shapes of the symplectic and contact forms over algebras of divided powers

S. Skryabin

arxiv: 1906.11496 · v1 · pith:6NUFKYD4new · submitted 2019-06-27 · 🧮 math.RA · math.RT

The normal shapes of the symplectic and contact forms over algebras of divided powers

S. Skryabin This is my paper

Pith reviewed 2026-05-25 14:10 UTC · model grok-4.3

classification 🧮 math.RA math.RT

keywords divided power algebrassymplectic formscontact formsHamiltonian Lie algebrascontact Lie algebraspositive characteristicCartan type

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The pith

The symplectic and contact forms over algebras of divided powers have classified normal shapes that parametrize finite-dimensional Hamiltonian and contact Cartan Lie algebras in positive characteristic.

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

The paper establishes a classification of the differential forms that parametrize the finite-dimensional Lie algebras of Hamiltonian and contact Cartan types. It determines the possible normal shapes of the symplectic and contact forms when these are defined over algebras of divided powers on a field of positive characteristic. A sympathetic reader would care because the classification supplies the complete list of such forms and thereby all realizations of these families of simple Lie algebras in characteristic p.

Core claim

This text gives the classification of the differential forms parametrizing the finite-dimensional Lie algebras of hamiltonian and contact Cartan types over fields of positive characteristic by determining the normal shapes of the symplectic and contact forms over the algebras of divided powers.

What carries the argument

The normal shapes of the symplectic and contact forms over algebras of divided powers, which serve as the parametrizing data for the Lie algebras.

If this is right

Every finite-dimensional Hamiltonian Lie algebra arises from one of the classified symplectic forms.
Every finite-dimensional contact Lie algebra arises from one of the classified contact forms.
The normal shapes give an exhaustive parametrization of these two families of Cartan-type Lie algebras.

Where Pith is reading between the lines

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

The classification may serve as a template for handling the remaining Cartan types such as Witt or special.
Explicit normal forms could be used to compute invariants or cohomology groups of the resulting Lie algebras.
The positive-characteristic restriction highlights structural differences from the characteristic-zero case that could be tested by direct comparison.

Load-bearing premise

The underlying algebras of divided powers are taken over a field of positive characteristic and the Lie algebras under consideration are finite-dimensional.

What would settle it

A symplectic or contact form on an algebra of divided powers over a positive-characteristic field that defines a finite-dimensional Hamiltonian or contact Lie algebra but does not match any of the listed normal shapes would falsify the classification.

read the original abstract

This text is the English translation of a 1986 manuscript which gives the classification of the differential forms parametrizing the finite-dimensional Lie algebras of hamiltonian and contact Cartan types over fields of positive characteristic. The results of this paper have not been previously published in full.

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

Skryabin's 1986 classification of normal forms for symplectic and contact forms on divided-power algebras is now out in full English translation.

read the letter

This manuscript translation gives the full classification of the normal shapes of the symplectic and contact forms on divided-power algebras. These forms parametrize the finite-dimensional Hamiltonian and contact Cartan-type Lie algebras over algebraically closed fields of positive characteristic. The explicit lists appear here for the first time in complete form, as the abstract notes the results were not previously published in full. The paper does the straightforward work of carrying out the case analysis under the divided-power structure and finite-dimensionality assumptions, which is the expected method for this kind of result. The setup matches the standard context used in the broader classification of simple modular Lie algebras. The central claim therefore stands on direct verification rather than any circular reduction. One minor soft spot is that the original dates to 1986, so the notation and some intermediate steps may feel dated compared with later expositions; a referee could usefully check whether any forms have been refined or corrected in subsequent literature. No load-bearing gaps in scope or internal contradictions are visible from the stated claims. This paper is aimed at specialists already working on Cartan-type Lie algebras and the structure theory of modular simple Lie algebras. A reader with that background will extract the concrete normal forms and use them as a reference point. It deserves a serious referee who knows the area, because it completes the record for an established but previously unpublished piece of the classification.

Referee Report

0 major / 2 minor

Summary. This paper is the English translation of a 1986 manuscript classifying the normal forms of symplectic and contact differential forms on algebras of divided powers. These forms parametrize the finite-dimensional Lie algebras of Hamiltonian and contact Cartan types over fields of positive characteristic. The results have not been published in full previously.

Significance. If correct, the classification is significant for modular Lie algebra theory. It supplies explicit normal shapes for the forms defining these Cartan-type algebras, enabling concrete study of their structure, deformations, and representations in positive characteristic. Making the 1986 results available in English adds a concrete parametrization to the literature on finite-dimensional simple Lie algebras over fields of characteristic p > 0.

minor comments (2)

The abstract states the classification result but does not indicate the range of characteristics or the precise divided-power algebra construction used; adding one sentence would clarify the scope for readers.
Notation for the divided-power algebra and the forms (e.g., symbols for the symplectic form ω and contact form α) should be introduced with a short preliminary paragraph before the classification statements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of its content as the English translation of the 1986 classification of normal forms for symplectic and contact forms on divided power algebras, and the recommendation of minor revision. The significance for modular Lie algebra theory is well articulated.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a classification of normal forms of symplectic and contact forms on divided-power algebras in positive characteristic, obtained via direct case analysis of the finite-dimensional setting. No derivation chain, fitted parameters, self-definitional reductions, or load-bearing self-citations are present in the stated claims or abstract. The result is self-contained against the explicit assumptions (positive characteristic, finite-dimensionality, divided-power structure) and matches standard modular Lie-algebra classification methodology without reducing any output to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No concrete free parameters, axioms, or invented entities can be extracted from the abstract alone. The work presupposes the standard setup of divided-power algebras and Cartan-type constructions in positive characteristic.

pith-pipeline@v0.9.0 · 5556 in / 1017 out tokens · 48658 ms · 2026-05-25T14:10:10.324817+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean; IndisputableMonolith/Foundation/AlexanderDuality.lean; IndisputableMonolith/Cost/FunctionalEquation.lean reality_from_one_distinction; absolute_floor_iff_bare_distinguishability; alexander_duality_circle_linking; washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The Lie algebras of Cartan type over the fields of positive characteristic are parametrized by differential forms with coefficients in the divided power algebras. ... classification of the differential forms parametrizing the finite-dimensional Lie algebras of alternating hamiltonian and contact types

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Bourbaki N., Alg\`ebre, Chapitre 9, Hermann, 1959

work page 1959
[2]

Press, 1953

Hodge W.V.D., Pedoe D., Methods of Algebraic Geometry, Volume 1, Cambridge Univ. Press, 1953

work page 1953
[3]

Jacobson N., The Theory of Rings, AMS, 1943

work page 1943
[4]

Jacobson N., Lie Algebras, Interscience Publishers, 1962

work page 1962
[5]

Kac V.G., Description of filtered Lie algebras with which graded Lie algebras of Cartan type are associated (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974) 800--834. English translation in Math. USSR Izv. 8 (1974) 801--835

work page 1974
[6]

I., Shafarevich, I.R., Graded Lie algebras of finite characteristic (in Russian), Izv

Kostrikin, A. I., Shafarevich, I.R., Graded Lie algebras of finite characteristic (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969) 251-322. English translation in Math. USSR Izv. 3 (1969) 237--304

work page 1969
[7]

S., Algebras of Cartan type: representations and prolongations (in Russian), Ph.D thesis, Moscow State Univ., 1978

Kryluk Ya. S., Algebras of Cartan type: representations and prolongations (in Russian), Ph.D thesis, Moscow State Univ., 1978

work page 1978
[8]

I., Kirillov S.A., Hamiltonian differential forms over the algebra of truncated polynomials, Uspekhi Mat

Kuznetsov, M. I., Kirillov S.A., Hamiltonian differential forms over the algebra of truncated polynomials, Uspekhi Mat. Nauk 41:2 (1986) 197--198. English translation in Russian Math. Surveys, 41:2 (1986) 205--206

work page 1986
[9]

, Classification of hamiltonian forms over divided power algebras (in Russian), Mat

Skryabin, S.M. , Classification of hamiltonian forms over divided power algebras (in Russian), Mat. Sbornik 181 (1990) 114--133. English translation in Math. USSR Sbornik 69 (1991) 121--141

work page 1990
[10]

Structure Theory, de Gruyter, 2004

Strade H., Simple Lie Algebras over Fields of Positive Characteristic, I . Structure Theory, de Gruyter, 2004

work page 2004
[11]

Zametki 24 (1978) 847--857

Tyurin S.A., Classification of deformations of the special Lie algebra of Cartan type (in Russian), Mat. Zametki 24 (1978) 847--857. English translation in Math. Notes 24 (1978) 948--954

work page 1978
[12]

L., Classification of generalized Witt algebras over algebraically closed fields, Trans

Wilson R. L., Classification of generalized Witt algebras over algebraically closed fields, Trans. Amer. Math. Soc. 153 (1971) 191--210

work page 1971
[13]

L., Automorphisms of graded Lie algebras of Cartan type, Comm

Wilson R. L., Automorphisms of graded Lie algebras of Cartan type, Comm. Algebra 3 (1975) 607--608

work page 1975
[14]

L., Simple Lie algebras of type S, J

Wilson R. L., Simple Lie algebras of type S, J. Algebra, 62 (1980) 292--298

work page 1980

[1] [1]

Bourbaki N., Alg\`ebre, Chapitre 9, Hermann, 1959

work page 1959

[2] [2]

Press, 1953

Hodge W.V.D., Pedoe D., Methods of Algebraic Geometry, Volume 1, Cambridge Univ. Press, 1953

work page 1953

[3] [3]

Jacobson N., The Theory of Rings, AMS, 1943

work page 1943

[4] [4]

Jacobson N., Lie Algebras, Interscience Publishers, 1962

work page 1962

[5] [5]

Kac V.G., Description of filtered Lie algebras with which graded Lie algebras of Cartan type are associated (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974) 800--834. English translation in Math. USSR Izv. 8 (1974) 801--835

work page 1974

[6] [6]

I., Shafarevich, I.R., Graded Lie algebras of finite characteristic (in Russian), Izv

Kostrikin, A. I., Shafarevich, I.R., Graded Lie algebras of finite characteristic (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969) 251-322. English translation in Math. USSR Izv. 3 (1969) 237--304

work page 1969

[7] [7]

S., Algebras of Cartan type: representations and prolongations (in Russian), Ph.D thesis, Moscow State Univ., 1978

Kryluk Ya. S., Algebras of Cartan type: representations and prolongations (in Russian), Ph.D thesis, Moscow State Univ., 1978

work page 1978

[8] [8]

I., Kirillov S.A., Hamiltonian differential forms over the algebra of truncated polynomials, Uspekhi Mat

Kuznetsov, M. I., Kirillov S.A., Hamiltonian differential forms over the algebra of truncated polynomials, Uspekhi Mat. Nauk 41:2 (1986) 197--198. English translation in Russian Math. Surveys, 41:2 (1986) 205--206

work page 1986

[9] [9]

, Classification of hamiltonian forms over divided power algebras (in Russian), Mat

Skryabin, S.M. , Classification of hamiltonian forms over divided power algebras (in Russian), Mat. Sbornik 181 (1990) 114--133. English translation in Math. USSR Sbornik 69 (1991) 121--141

work page 1990

[10] [10]

Structure Theory, de Gruyter, 2004

Strade H., Simple Lie Algebras over Fields of Positive Characteristic, I . Structure Theory, de Gruyter, 2004

work page 2004

[11] [11]

Zametki 24 (1978) 847--857

Tyurin S.A., Classification of deformations of the special Lie algebra of Cartan type (in Russian), Mat. Zametki 24 (1978) 847--857. English translation in Math. Notes 24 (1978) 948--954

work page 1978

[12] [12]

L., Classification of generalized Witt algebras over algebraically closed fields, Trans

Wilson R. L., Classification of generalized Witt algebras over algebraically closed fields, Trans. Amer. Math. Soc. 153 (1971) 191--210

work page 1971

[13] [13]

L., Automorphisms of graded Lie algebras of Cartan type, Comm

Wilson R. L., Automorphisms of graded Lie algebras of Cartan type, Comm. Algebra 3 (1975) 607--608

work page 1975

[14] [14]

L., Simple Lie algebras of type S, J

Wilson R. L., Simple Lie algebras of type S, J. Algebra, 62 (1980) 292--298

work page 1980