On transposed Poisson conformal superalgebras

Hao Fang; Lamei Yuan

arxiv: 2605.17747 · v1 · pith:V4QKDFHXnew · submitted 2026-05-18 · 🧮 math.RA · math-ph· math.MP

On transposed Poisson conformal superalgebras

Hao Fang , Lamei Yuan This is my paper

Pith reviewed 2026-05-20 01:12 UTC · model grok-4.3

classification 🧮 math.RA math-phmath.MP

keywords transposed Poisson conformal superalgebrasLie conformal superalgebrastransposed conformal super-Leibniz ruletensor product constructionHom-Lie conformal superalgebrascompatibility conditionsNovikov-Poisson structures

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The pith

Transposed Poisson conformal superalgebras on rank (1+1) Lie conformal superalgebras are fully determined

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

This paper introduces transposed Poisson conformal superalgebras as the conformal and Z2-graded versions of transposed Poisson algebras. It derives the identities imposed by the transposed conformal super-Leibniz rule and shows that the tensor product over the ring of differential operators preserves the structure. The work also establishes a direct link to Hom-Lie conformal superalgebras and provides explicit compatibility conditions when both Poisson and transposed Poisson structures coexist on the same object. Using the complete classification of Lie conformal superalgebras of rank (1+1), it then enumerates every possible transposed Poisson structure that can be added to these superalgebras.

Core claim

All compatible transposed Poisson conformal superalgebra structures on the classified Lie conformal superalgebras of rank (1+1) are determined by solving the conditions coming from the transposed conformal super-Leibniz rule.

What carries the argument

The family of identities forced by the transposed conformal super-Leibniz rule, which must hold for the multiplication to be compatible with the Lie conformal bracket.

If this is right

The tensor product of any two transposed Poisson conformal superalgebras again carries a natural transposed Poisson conformal superalgebra structure.
Explicit constructions are obtained by modifying Lie conformal brackets or by starting from Novikov-Poisson and pre-Lie Poisson conformal superalgebras.
Compatibility conditions are given that allow a superalgebra to carry both a Poisson conformal structure and a transposed Poisson conformal structure simultaneously.
Results about Hom-Lie conformal superalgebras can be used to produce new transposed Poisson examples.

Where Pith is reading between the lines

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

These low-rank classifications may provide a foundation for inductive constructions of structures on higher-rank or deformed versions of conformal superalgebras.
The tensor product operation suggests a method for building infinite families of examples that could be studied for their representation theory or cohomology.
Connections to vertex algebras or conformal field theory could be explored if the differential operator actions satisfy additional locality conditions.

Load-bearing premise

The prior classification of Lie conformal superalgebras of rank (1+1) is complete and accurate.

What would settle it

Discovery of a transposed Poisson conformal superalgebra structure on one of the rank-(1+1) Lie conformal superalgebras that does not satisfy the listed identities or is missing from the enumeration would disprove the determination.

read the original abstract

We introduce and study transposed Poisson conformal superalgebras, the $\mathbb Z_2$-graded conformal analogues of transposed Poisson algebras, as well as their noncommutative variants. We derive a family of identities forced by the transposed conformal super-Leibniz rule and prove that the tensor product over $\mathbb C[\partial]$ of two such superalgebras again carries a natural transposed Poisson conformal superalgebra structure. Moreover, we display a close relationship between transposed Poisson conformal superalgebras and Hom-Lie conformal superalgebras, and give the compatibility conditions between a Poisson conformal superalgebra and a transposed Poisson conformal superalgebra. In addition, several constructions are obtained from modified Lie conformal brackets and from Novikov-Poisson, pre-Lie commutative, differential Novikov-Poisson, and pre-Lie Poisson conformal superalgebras. Finally, using the known classification of Lie conformal superalgebras of rank (1+1), we determine all compatible transposed Poisson conformal superalgebra structures on such superalgebras.

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 defines transposed Poisson conformal superalgebras and classifies compatible structures on rank-(1+1) Lie conformal superalgebras using an external classification.

read the letter

This paper introduces transposed Poisson conformal superalgebras as the Z2-graded conformal version of transposed Poisson algebras. It also looks at their noncommutative versions and works out classifications in the rank-(1+1) setting. What stands out is the explicit classification of all compatible transposed Poisson structures on the known Lie conformal superalgebras of that rank. The authors derive a set of identities that follow from the transposed conformal super-Leibniz rule. They prove that the tensor product over the polynomial ring in partial preserves the transposed Poisson structure. They also map out the relationship to Hom-Lie conformal superalgebras and give conditions for compatibility with ordinary Poisson conformal superalgebras. Several constructions come from modified Lie brackets and from Novikov-Poisson, pre-Lie, and differential versions. The work builds directly on standard definitions and the prior classification of the underlying Lie conformal superalgebras. That prior list carries the weight for the all claim in the classification. The derivations of the compatibility conditions appear to follow from the rules without obvious circularity, but the completeness of the base objects is taken as given. The soft spots are minor here. The paper does not re-verify the earlier classification or provide an independent enumeration, so any gaps there would carry over. The abstract outlines the program clearly, and the results seem internally consistent based on the description. This is aimed at researchers in conformal algebras, superalgebras, and related algebraic structures in mathematical physics. A reader who follows work on Poisson-type algebras in graded or conformal settings will pick up new definitions and concrete low-rank examples. It is worth a serious referee report because the new structure is motivated and the classification is concrete, even though it depends on previous results.

Referee Report

1 major / 2 minor

Summary. The paper introduces transposed Poisson conformal superalgebras (and noncommutative variants) as Z_2-graded conformal analogues of transposed Poisson algebras. It derives identities forced by the transposed conformal super-Leibniz rule, proves that the tensor product over C[∂] again carries such a structure, establishes a relationship to Hom-Lie conformal superalgebras, gives compatibility conditions between Poisson and transposed Poisson conformal superalgebras, provides constructions from modified Lie conformal brackets and from Novikov-Poisson, pre-Lie commutative, differential Novikov-Poisson, and pre-Lie Poisson conformal superalgebras, and finally uses the known classification of Lie conformal superalgebras of rank (1+1) to determine all compatible transposed Poisson structures on them.

Significance. If the results hold, the work adds a new family of structures to the theory of conformal superalgebras, supplies explicit constructions and a tensor-product closure property, and gives a classification in the rank-(1+1) case that rests on an external but standard classification. These contributions could facilitate further study of nonassociative conformal structures and their applications in related algebraic and physical contexts. The internal derivations appear consistent with the given definitions.

major comments (1)

Final classification result (using known classification of Lie conformal superalgebras of rank (1+1)): the determination of 'all' compatible transposed Poisson structures is load-bearing on the completeness of the cited external classification. While the compatibility conditions are derived directly from the transposed conformal super-Leibniz rule and appear internally consistent, the paper does not re-list or independently verify the base Lie conformal superalgebras of rank (1+1), so any gaps in the prior classification would render the enumeration incomplete.

minor comments (2)

Introduction and definitions section: the precise statement of the transposed conformal super-Leibniz rule and its relation to the ordinary conformal super-Leibniz rule could be displayed side-by-side for immediate comparison.
Notation: the symbol for the conformal product and the action of ∂ on the superalgebra should be fixed consistently throughout the constructions in the later sections.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive overall assessment. We address the major comment below.

read point-by-point responses

Referee: Final classification result (using known classification of Lie conformal superalgebras of rank (1+1)): the determination of 'all' compatible transposed Poisson structures is load-bearing on the completeness of the cited external classification. While the compatibility conditions are derived directly from the transposed conformal super-Leibniz rule and appear internally consistent, the paper does not re-list or independently verify the base Lie conformal superalgebras of rank (1+1), so any gaps in the prior classification would render the enumeration incomplete.

Authors: We appreciate the referee drawing attention to this dependence. The classification of Lie conformal superalgebras of rank (1+1) is a standard, previously established result in the literature (cited as [reference to the known classification paper]), and our work applies the newly derived compatibility conditions to determine which of these admit transposed Poisson structures. Re-deriving or independently verifying the base classification lies outside the scope of the present paper. To address the concern explicitly, we will revise the manuscript to include a short remark stating that the enumeration is conditional upon the completeness of the cited classification, together with a brief recall of the base structures for the reader's convenience. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central results derive from external classification and internal identities

full rationale

The paper derives identities from the transposed conformal super-Leibniz rule, proves tensor product and Hom-Lie relationships, gives compatibility conditions, and constructs examples from modified brackets and other structures entirely internally. The final enumeration applies a cited external classification of Lie conformal superalgebras of rank (1+1) to list compatible transposed Poisson structures on those objects. This classification is treated as a known, independent input rather than derived or fitted within the paper, and no self-citation chain, self-definitional loop, or fitted-input-renamed-as-prediction appears in the load-bearing steps. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper rests primarily on the new definition of the transposed conformal super-Leibniz rule and on the external classification of low-rank Lie conformal superalgebras; no numerical parameters are fitted.

axioms (2)

domain assumption The transposed conformal super-Leibniz rule holds for the product and bracket operations.
This is the central defining relation introduced for the new structures.
domain assumption The known classification of Lie conformal superalgebras of rank (1+1) is complete.
Invoked to determine all compatible structures in the final section.

invented entities (1)

transposed Poisson conformal superalgebra no independent evidence
purpose: To provide Z2-graded conformal analogues of transposed Poisson algebras.
Newly defined object whose properties are studied throughout the paper.

pith-pipeline@v0.9.0 · 5698 in / 1500 out tokens · 64445 ms · 2026-05-20T01:12:29.400522+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/ArithmeticFromLogic.lean LogicNat ≃ Nat recovery; no transposed Leibniz or conformal product unclear

?

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

using the known classification of Lie conformal superalgebras of rank (1+1), we determine all compatible transposed Poisson conformal superalgebra structures on such superalgebras (Theorem 5.2, cases R1–R5)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J(x) = ½(x + x⁻¹) − 1 unique) unclear

?

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

transposed conformal super-Leibniz rule: 2(a◦λ[bμc]) = [(a◦λb)λ+μc] + (−1)|a||b|[bμ(a◦λc)]

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

22 extracted references · 22 canonical work pages

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[1] [1]

Transposed Poisson algebras, Novikov-Poisson algebras and 3-Lie algebras.Journal of Algebra, 632:535–566, 2023.doi:10.1016/j.jalgebra.2023.06.006

Chengming Bai, Ruipu Bai, Li Guo, and Yong Wu. Transposed Poisson algebras, Novikov-Poisson algebras and 3-Lie algebras.Journal of Algebra, 632:535–566, 2023.doi:10.1016/j.jalgebra.2023.06.006. 1

work page doi:10.1016/j.jalgebra.2023.06.006 2023

[2] [2]

Kac, and Alexander A

Bojko Bakalov, Victor G. Kac, and Alexander A. Voronov. Cohomology of conformal algebras.Comm. Math. Phys., 200(3):561–598, 1999.doi:10.1007/s002200050541. 1

work page doi:10.1007/s002200050541 1999

[3] [3]

Kac, and Jos´ e I

Carina Boyallian, Victor G. Kac, and Jos´ e I. Liberati. Irreducible modules over finite simple Lie conformal superalgebras of typeK.J. Math. Phys., 51(6):063507, 37, 2010.doi:10.1063/1.3397419. 1

work page doi:10.1063/1.3397419 2010

[4] [4]

Kac, Jose I

Carina Boyallian, Victor G. Kac, Jose I. Liberati, and Alexei Rudakov. Representations of simple finite Lie conformal superalgebras of typeWandS.J. Math. Phys., 47(4):043513, 25, 2006.doi:10.1063/1.2191788. 1

work page doi:10.1063/1.2191788 2006

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A new construction of Lie conformal algebras from formal distribution Lie algebras.J

Fulin Chen, Xiaoling Liao, Shaobin Tan, and Qing Wang. A new construction of Lie conformal algebras from formal distribution Lie algebras.J. Algebra, 680:96–133, 2025.doi:10.1016/j.jalgebra.2025.04.0

work page doi:10.1016/j.jalgebra.2025.04.0 2025

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Hom-Gel’fand-Dorfman conformal superbialgebras.Hacet

Taoufik Chtioui. Hom-Gel’fand-Dorfman conformal superbialgebras.Hacet. J. Math. Stat., 53(3):577–585, 2024.doi:10.15672/hujms.1196147. 4

work page doi:10.15672/hujms.1196147 2024

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Alessandro D’Andrea and Victor G. Kac. Structure theory of finite conformal algebras.Selecta Math. (N.S.), 4(3):377–418, 1998.doi:10.1007/s000290050036. 3

work page doi:10.1007/s000290050036 1998

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Alberto De Sole and Victor G. Kac. The variational Poisson cohomology.Jpn. J. Math., 8(1):1–145, 2013. doi:10.1007/s11537-013-1124-3. 1, 3

work page doi:10.1007/s11537-013-1124-3 2013

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work page doi:10.1016/s0021-8 2002

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Kac, and Alexander Retakh

Davide Fattori, Victor G. Kac, and Alexander Retakh. Structure theory of finite Lie conformal superalge- bras. InLie theory and its applications in physics V, pages 27–63. World Sci. Publ., River Edge, NJ, 2004. doi:10.1142/9789812702562\_0002. 1, 3, 4

work page doi:10.1142/9789812702562 2004

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1 2-derivations of Lie algebras and transposed Poisson algebras.Rev

Bruno Leonardo Macedo Ferreira, Ivan Kaygorodov, and Viktor Lopatkin. 1 2-derivations of Lie algebras and transposed Poisson algebras.Rev. R. Acad. Cienc. Exactas F´ ıs. Nat. Ser. A Mat. RACSAM, 115(3):Paper No. 142, 19, 2021.doi:10.1007/s13398-021-01088-2. 1, 5

work page doi:10.1007/s13398-021-01088-2 2021

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A class of Lie conformal superalgebras in higher dimensions.J

Yanyong Hong. A class of Lie conformal superalgebras in higher dimensions.J. Lie Theory, 26(4):1145– 1162, 2016. 4

work page 2016

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Left-symmetric conformal algebras and vertex algebras.J

Yanyong Hong and Fang Li. Left-symmetric conformal algebras and vertex algebras.J. Pure Appl. Algebra, 219(8):3543–3567, 2015.doi:10.1016/j.jpaa.2014.12.012. 4

work page doi:10.1016/j.jpaa.2014.12.012 2015

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Victor G. Kac. The idea of locality. InPhysical Applications and Mathematical Aspects of Geometry, Groups and Algebras, pages 16–32. World Scientific, Singapore, 1997. 1

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Kac.Vertex algebras for beginners, volume 10 ofUniversity Lecture Series

Victor G. Kac.Vertex algebras for beginners, volume 10 ofUniversity Lecture Series. American Mathe- matical Society, Providence, RI, second edition, 1998.doi:10.1090/ulect/010. 1, 3

work page doi:10.1090/ulect/010 1998

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Kolesnikov, Roman A

Pavel S. Kolesnikov, Roman A. Kozlov, and Aleksander S. Panasenko. Quadratic Lie conformal su- peralgebras related to Novikov superalgebras.J. Noncommut. Geom., 15(4):1485–1500, 2021.doi: 10.4171/jncg/445. 1, 3, 5

work page doi:10.4171/jncg/445 2021

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Local systems of vertex operators, vertex superalgebras and modules.J

Hai-Sheng Li. Local systems of vertex operators, vertex superalgebras and modules.J. Pure Appl. Algebra, 109(2):143–195, 1996.doi:10.1016/0022-4049(95)00079-8. 1

work page doi:10.1016/0022-4049(95)00079-8 1996

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Schr¨ odinger-Virasoro Lie conformal algebra.J

Yucai Su and Lamei Yuan. Schr¨ odinger-Virasoro Lie conformal algebra.J. Math. Phys., 54(5):053503, 16, 2013.doi:10.1063/1.4803029. 1

work page doi:10.1063/1.4803029 2013

[20] [20]

Hom-Gel’fand-Dorfman super-bialgebras and Hom-Lie conformal superalgebras.Acta Math

Lamei Yuan, Sheng Chen, and Caixia He. Hom-Gel’fand-Dorfman super-bialgebras and Hom-Lie conformal superalgebras.Acta Math. Sin. (Engl. Ser.), 33(1):96–116, 2017.doi:10.1007/s10114-016-6074-2. 4

work page doi:10.1007/s10114-016-6074-2 2017

[21] [21]

On transposed Poisson conformal algebras

Lamei Yuan and Hao Fang. On transposed Poisson conformal algebras. Preprint, 21 pp., 2026. URL: https://arxiv.org/abs/2603.14735. 1, 2, 5, 13

work page arXiv 2026

[22] [22]

Deformations and generalized derivations of Lie conformal superalgebras.J

Jun Zhao, Liangyun Chen, and Lamei Yuan. Deformations and generalized derivations of Lie conformal superalgebras.J. Math. Phys., 58(11):111702, 17, 2017.doi:10.1063/1.5012886. 2, 19 Hao F ang, School of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China Email address:23s012028@stu.hit.edu.cn Lamei Yuan (corresponding au...

work page doi:10.1063/1.5012886 2017