Decompositions of algebras and post-associative algebra structures

Dietrich Burde; Vsevolod Gubarev

arxiv: 1906.09854 · v1 · pith:2OQIILBSnew · submitted 2019-06-24 · 🧮 math.RA

Decompositions of algebras and post-associative algebra structures

Dietrich Burde , Vsevolod Gubarev This is my paper

Pith reviewed 2026-05-25 16:56 UTC · model grok-4.3

classification 🧮 math.RA

keywords post-Lie algebraspost-associative algebrasRota-Baxter operatorsLie algebra decompositionsassociative algebra structuressemisimple algebrasreductive algebras

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The pith

There exists no post-Lie algebra structure on a pair of Lie algebras where one is simple and the other reductive but not isomorphic.

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

The paper defines post-associative algebra structures and connects them to post-Lie structures, Rota-Baxter operators, and algebra decompositions. It proves that no post-Lie structure can exist on pairs where one Lie algebra is simple and the other is reductive yet non-isomorphic to the first. It further shows that no post-associative structure arises from a Rota-Baxter operator when one associative algebra is semisimple and the other is not. A sympathetic reader cares because these structures are proposed as ways to extend classical algebra operations while remaining compatible with known operators and decompositions.

Core claim

We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota-Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular we prove that there exists no post-Lie algebra structure on a pair (g,n), where n is a simple Lie algebra and g is a reductive Lie algebra, which is not isomorphic to n. We also show that there is no post-associative algebra structure on a pair (A,B) arising from a Rota-Baxter operator of B, where A is a semisimple associative algebra and B is not semisimple. The proofs use results on Rota-Baxter operators and decompositions of

What carries the argument

Post-associative algebra structures on pairs of algebras, obtained from Rota-Baxter operators or decompositions, used to prove non-existence results for post-Lie structures on simple-reductive pairs.

If this is right

Post-Lie structures are impossible on non-isomorphic simple-reductive Lie algebra pairs.
Post-associative structures cannot be obtained from Rota-Baxter operators when one algebra is semisimple and the other is not.
Existence questions for these structures receive negative answers in the specified classes.

Where Pith is reading between the lines

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

The non-existence results may narrow the search for compatible structures in deformation or integrable systems contexts.
Similar proofs could be attempted for other classes such as both algebras semisimple or both non-semisimple.

Load-bearing premise

The pairs of algebras under consideration arise from Rota-Baxter operators or from decompositions of algebras.

What would settle it

An explicit construction of a post-Lie algebra structure on a pair (g,n) where n is simple and g is reductive but not isomorphic to n would disprove the non-existence claim.

read the original abstract

We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota--Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular we prove that there exists no post-Lie algebra structure on a pair $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{n}$ is a simple Lie algebra and $\mathfrak{g}$ is a reductive Lie algebra, which is not isomorphic to $\mathfrak{n}$. We also show that there is no post-associative algebra structure on a pair $(A,B)$ arising from a Rota--Baxter operator of $B$, where $A$ is a semisimple associative algebra and $B$ is not semisimple. The proofs use results on Rota--Baxter operators and decompositions of algebras.

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

The paper defines post-associative algebras and derives two non-existence results by applying existing theorems on Rota-Baxter operators and algebra decompositions to specific pairs.

read the letter

The new element is the definition of post-associative algebra structures together with the two concrete non-existence statements: no post-Lie structure on (g,n) when n is simple Lie and g is reductive but not isomorphic to n, and no post-associative structure on (A,B) coming from a Rota-Baxter operator on B when A is semisimple associative and B is not. Both results are obtained by checking that the pairs fit the hypotheses of earlier theorems on operators and decompositions. That approach is straightforward and keeps the paper short. The proofs are not self-contained, but they do not claim to be; they rest on the cited prior work. The main question is whether the modeling step is exact—whether the pairs really satisfy every condition in the invoked results on weight, directness of sums, or bracket relations. The abstract indicates the authors believe they do, so the claims stand or fall on that verification. If the reduction is clean, the non-existence statements are useful constraints inside this narrow area. If any extra freedom creeps into the definition of “arising from” or “pair,” the conclusions weaken. This is specialist material for people already working on post-Lie algebras, Rota-Baxter operators, or related decompositions. A reader outside that circle will not find much to take away. The paper shows clear engagement with the literature and produces falsifiable statements rather than vague suggestions, so it is worth sending to referees even though the results are negative and depend on prior theorems.

Referee Report

1 major / 1 minor

Summary. The paper introduces post-associative algebra structures and studies their relationship to post-Lie algebra structures, Rota-Baxter operators, and decompositions of associative algebras and Lie algebras. It proves non-existence of post-Lie algebra structures on pairs (g, n) where n is a simple Lie algebra and g is a reductive Lie algebra not isomorphic to n. It also proves non-existence of post-associative algebra structures on pairs (A, B) arising from a Rota-Baxter operator on B where A is semisimple associative and B is not semisimple. Both proofs rely on prior results about Rota-Baxter operators and algebra decompositions.

Significance. If the reductions to the hypotheses of the cited theorems on Rota-Baxter operators and decompositions are verified for the simple/reductive and semisimple/non-semisimple cases, the non-existence results would usefully constrain the possible post- structures and clarify their interplay with existing algebraic decompositions.

major comments (1)

The non-existence statements are load-bearing on the claim that the pairs (g, n) and (A, B) satisfy the exact hypotheses of the invoked prior theorems on Rota-Baxter operators and decompositions (including any conditions on operator weight or decomposition type). The manuscript must explicitly check these conditions for the simple/reductive Lie and semisimple/non-semisimple associative cases; otherwise the non-existence does not follow.

minor comments (1)

The abstract states the Lie-algebra non-existence without the qualifier 'arising from a decomposition' that appears for the associative case; this should be clarified for consistency.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and for identifying the need to strengthen the non-existence arguments. We address the single major comment below and will incorporate the requested verification in a revised version.

read point-by-point responses

Referee: The non-existence statements are load-bearing on the claim that the pairs (g, n) and (A, B) satisfy the exact hypotheses of the invoked prior theorems on Rota-Baxter operators and decompositions (including any conditions on operator weight or decomposition type). The manuscript must explicitly check these conditions for the simple/reductive Lie and semisimple/non-semisimple associative cases; otherwise the non-existence does not follow.

Authors: We agree that the non-existence claims require explicit confirmation that the pairs meet every hypothesis of the cited theorems on Rota-Baxter operators and algebra decompositions. In the revision we will insert a new subsection (or expanded paragraphs in Sections 3 and 4) that directly verifies these conditions: for the Lie case, that the decomposition induced by the post-Lie structure satisfies the precise weight and reductivity/simplicity requirements of the referenced decomposition theorem; and for the associative case, that the Rota-Baxter operator of weight zero (or the relevant weight) produces a pair (A, B) with A semisimple and B non-semisimple while satisfying all operator and decomposition hypotheses. These verifications will be stated as lemmas or propositions immediately preceding the non-existence theorems. revision: yes

Circularity Check

0 steps flagged

Non-existence claims rely on external prior results; no internal reduction by construction

full rationale

The paper states that its non-existence theorems for post-Lie structures on (g,n) and post-associative structures on (A,B) are proved using results on Rota-Baxter operators and algebra decompositions. No equations, definitions, or steps in the abstract or described claims reduce any derived quantity to a fitted parameter, self-defined input, or self-citation chain internal to this paper. The load-bearing steps are external citations whose validity is independent of the present work's constructions, satisfying the criteria for non-circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the introduction of a new algebraic structure (post-associative) and on standard background axioms of Lie and associative algebras together with the domain assumption that the pairs arise from Rota-Baxter operators. No free parameters are mentioned. The invented entity is the post-associative structure itself, with no independent evidence supplied in the abstract.

axioms (2)

standard math Standard axioms of Lie algebras and associative algebras over a field of characteristic zero.
The paper works inside the conventional framework of algebra theory.
domain assumption Pairs (A,B) arise from Rota-Baxter operators on B.
Explicitly invoked for the second non-existence result in the abstract.

invented entities (1)

post-associative algebra structure no independent evidence
purpose: New structure relating post-Lie algebras, Rota-Baxter operators, and algebra decompositions.
Introduced in the paper as the central new object of study.

pith-pipeline@v0.9.0 · 5668 in / 1428 out tokens · 41651 ms · 2026-05-25T16:56:12.797411+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 1 internal anchor

[1]

Y. A. Bahturin, O. H. Kegel: Sums of simple subalgebras . J. Math. Sci. 93, no. 6 (1999), 830–835

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[2]

C. Bai, L. Guo, X. Ni: Nonabelian generalized Lax pairs, the classical Yang-Baxt er equation and post-Lie algebras. Comm. Math. Phys. 297 (2010), no. 2, 553–596. DECOMPOSITION OF ALGEBRAS 13

work page 2010
[3]

C. Bai, O. Bellier, L. Guo, X. Ni: Splitting of operations, Manin products, and Rota–Baxter o perators. Int. Math. Res. Notices IMRN 3 (2013), 485–524

work page 2013
[4]

Burde, K

D. Burde, K. Dekimpe and K. Vercammen: Aﬃne actions on Lie groups and post-Lie algebra structures . Linear Algebra Appl. 437 (2012), no. 5, 1250–1263

work page 2012
[5]

Burde, K

D. Burde, K. Dekimpe: Post-Lie algebra structures on pairs of Lie algebras . Journal of Algebra 464 (2016), 226–245

work page 2016
[6]

Burde, V

D. Burde, V. Gubarev: Rota–Baxter operators and post-Lie algebra structures on s emisimple Lie algebras . To appear in Communications in Algebra (2019)

work page 2019
[7]

Douglas, J

A. Douglas, J. Repka: Subalgebras of the rank two semisimple Lie algebras . Linear Multilinear Algebra 66 (2018), no. 10, 2049–2075

work page 2018
[8]

Ebrahimi-Fard: Loday-type algebras and the Rota–Baxter relation

K. Ebrahimi-Fard: Loday-type algebras and the Rota–Baxter relation . Lett. Math. Phys. 61 (2002), no. 2, 139–147

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[9]

Ebrahimi-Fard, A

K. Ebrahimi-Fard, A. Lundervold, I. Mencattini, H. Z. Munthe-K aas: Post-Lie algebras and isospectral ﬂows. SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 093, 16 pp

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Gubarev: Rota–Baxter operators on a sum of ﬁelds

V. Gubarev: Rota–Baxter operators on a sum of ﬁelds . To appear in J. Algebra Appl. (2020)

work page 2020
[11]

Gubarev, P

V. Gubarev, P. Kolesnikov: Embedding of dendriform algebras into Rota–Baxter algebra s. Cent. Eur. J. Math. 11 (2013), no. 2, 226–245

work page 2013
[12]

Gubarev, P

V. Gubarev, P. Kolesnikov: Operads of decorated trees and their duals . Comment. Math. Univ. Carolin. 55 (2014), no. 4, 421–445

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[13]

Guo: An introduction to Rota–Baxter algebra

L. Guo: An introduction to Rota–Baxter algebra. Surveys of Modern Mathematics, Vol. 4 (2012), Somerville: Intern. Press; Beijing: Higher education press, 226 pp

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[14]

Hochschild: Semi-simple algebras and generalized derivations

G. Hochschild: Semi-simple algebras and generalized derivations . Amer. J. Math. 64 (1942), no. 1, 677–694

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[15]

Kegel: Zur Nilpotenz gewisser assoziativer Ringe

O. Kegel: Zur Nilpotenz gewisser assoziativer Ringe . Math. Ann. 149 (1963), 258–260

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[16]

A. W. Knapp: Advanced algebra. Birkh¨ auser Boston, Inc., Boston, MA, 2007. xxiv+730 pp

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[17]

J. L. Koszul: Variante dun th´ eor´ eme de H. Ozeki. Osaka J. Math. 15 (1978), 547–551

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[18]

Loday, M

J.-L. Loday, M. Ronco: Trialgebras and families of polytopes . Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory. Comtep. Math. 346 (2004) 369–398

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The subalgebras of G2

E. Mayanskiy: The subalgebras of G2. arXiv:1611.04070. 35 p

work page internal anchor Pith review Pith/arXiv arXiv
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A. N. Minchenko: Semisimple subalgebras of exceptional Lie algebras . Trans. Moscow Math. Soc. 67 (2006), 225–259

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A. L. Onishchik: Inclusion relations between transitive compact transformation groups. Trudy Moskov. Mat. Obs. 11 (1962) 199–242

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A. L. Onishchik: Decompositions of reductive Lie groups . Mat. Sbornik 80 (122) (1969), no. 4, 515–554

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[23]

Vallette: Homology of generalized partition posets

B. Vallette: Homology of generalized partition posets . J. Pure Appl. Algebra 208 (2007), no. 2, 699–725. F akult¨at f ¨ur Mathematik, Universit ¨at Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Aus- tria E-mail address : dietrich.burde@univie.ac.at F akult¨at f ¨ur Mathematik, Universit ¨at Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Aus- tria E-mail addre...

work page 2007

[1] [1]

Y. A. Bahturin, O. H. Kegel: Sums of simple subalgebras . J. Math. Sci. 93, no. 6 (1999), 830–835

work page 1999

[2] [2]

C. Bai, L. Guo, X. Ni: Nonabelian generalized Lax pairs, the classical Yang-Baxt er equation and post-Lie algebras. Comm. Math. Phys. 297 (2010), no. 2, 553–596. DECOMPOSITION OF ALGEBRAS 13

work page 2010

[3] [3]

C. Bai, O. Bellier, L. Guo, X. Ni: Splitting of operations, Manin products, and Rota–Baxter o perators. Int. Math. Res. Notices IMRN 3 (2013), 485–524

work page 2013

[4] [4]

Burde, K

D. Burde, K. Dekimpe and K. Vercammen: Aﬃne actions on Lie groups and post-Lie algebra structures . Linear Algebra Appl. 437 (2012), no. 5, 1250–1263

work page 2012

[5] [5]

Burde, K

D. Burde, K. Dekimpe: Post-Lie algebra structures on pairs of Lie algebras . Journal of Algebra 464 (2016), 226–245

work page 2016

[6] [6]

Burde, V

D. Burde, V. Gubarev: Rota–Baxter operators and post-Lie algebra structures on s emisimple Lie algebras . To appear in Communications in Algebra (2019)

work page 2019

[7] [7]

Douglas, J

A. Douglas, J. Repka: Subalgebras of the rank two semisimple Lie algebras . Linear Multilinear Algebra 66 (2018), no. 10, 2049–2075

work page 2018

[8] [8]

Ebrahimi-Fard: Loday-type algebras and the Rota–Baxter relation

K. Ebrahimi-Fard: Loday-type algebras and the Rota–Baxter relation . Lett. Math. Phys. 61 (2002), no. 2, 139–147

work page 2002

[9] [9]

Ebrahimi-Fard, A

K. Ebrahimi-Fard, A. Lundervold, I. Mencattini, H. Z. Munthe-K aas: Post-Lie algebras and isospectral ﬂows. SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 093, 16 pp

work page 2015

[10] [10]

Gubarev: Rota–Baxter operators on a sum of ﬁelds

V. Gubarev: Rota–Baxter operators on a sum of ﬁelds . To appear in J. Algebra Appl. (2020)

work page 2020

[11] [11]

Gubarev, P

V. Gubarev, P. Kolesnikov: Embedding of dendriform algebras into Rota–Baxter algebra s. Cent. Eur. J. Math. 11 (2013), no. 2, 226–245

work page 2013

[12] [12]

Gubarev, P

V. Gubarev, P. Kolesnikov: Operads of decorated trees and their duals . Comment. Math. Univ. Carolin. 55 (2014), no. 4, 421–445

work page 2014

[13] [13]

Guo: An introduction to Rota–Baxter algebra

L. Guo: An introduction to Rota–Baxter algebra. Surveys of Modern Mathematics, Vol. 4 (2012), Somerville: Intern. Press; Beijing: Higher education press, 226 pp

work page 2012

[14] [14]

Hochschild: Semi-simple algebras and generalized derivations

G. Hochschild: Semi-simple algebras and generalized derivations . Amer. J. Math. 64 (1942), no. 1, 677–694

work page 1942

[15] [15]

Kegel: Zur Nilpotenz gewisser assoziativer Ringe

O. Kegel: Zur Nilpotenz gewisser assoziativer Ringe . Math. Ann. 149 (1963), 258–260

work page 1963

[16] [16]

A. W. Knapp: Advanced algebra. Birkh¨ auser Boston, Inc., Boston, MA, 2007. xxiv+730 pp

work page 2007

[17] [17]

J. L. Koszul: Variante dun th´ eor´ eme de H. Ozeki. Osaka J. Math. 15 (1978), 547–551

work page 1978

[18] [18]

Loday, M

J.-L. Loday, M. Ronco: Trialgebras and families of polytopes . Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory. Comtep. Math. 346 (2004) 369–398

work page 2004

[19] [19]

The subalgebras of G2

E. Mayanskiy: The subalgebras of G2. arXiv:1611.04070. 35 p

work page internal anchor Pith review Pith/arXiv arXiv

[20] [20]

A. N. Minchenko: Semisimple subalgebras of exceptional Lie algebras . Trans. Moscow Math. Soc. 67 (2006), 225–259

work page 2006

[21] [21]

A. L. Onishchik: Inclusion relations between transitive compact transformation groups. Trudy Moskov. Mat. Obs. 11 (1962) 199–242

work page 1962

[22] [22]

A. L. Onishchik: Decompositions of reductive Lie groups . Mat. Sbornik 80 (122) (1969), no. 4, 515–554

work page 1969

[23] [23]

Vallette: Homology of generalized partition posets

B. Vallette: Homology of generalized partition posets . J. Pure Appl. Algebra 208 (2007), no. 2, 699–725. F akult¨at f ¨ur Mathematik, Universit ¨at Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Aus- tria E-mail address : dietrich.burde@univie.ac.at F akult¨at f ¨ur Mathematik, Universit ¨at Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Aus- tria E-mail addre...

work page 2007