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Averaging operators on groups and Hopf algebras

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Pith's one-line read The paper constructs the free averaging group on any set as an explicit set of bracketed words with a recursively defined multiplication and operator, and proves the universal property.

desk verdict Good new definitions and bridges, but the free averaging group is not well-defined as printed: the multiplication in Definition 4.2 depends on a non-canonical choice of bracketing exponents. read the letter →

arxiv 2412.11600 v3 pith:7ABUDN3Y submitted 2024-12-16 math.RA math.GR

Huhu Zhang , Xing Gao This is my paper · ORCID
classification math.RAmath.GR MSC 22E6008B2017B4016W99
keywords operatedgroupsaveragingfreegroupHopfalgebrasLiedisemigroupsracksbracketedwords
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces averaging groups: a group G together with a map A satisfying A(g)A(h)=A(A(g)h)=A(gA(h)) for all g,h, the group-level version of the familiar averaging-operator identity. Its central result is that the category of averaging groups has an explicit free object: on any set X, the averaging group words A(X), with a recursively defined multiplication ⋄ and bracketing operator A_X, form the free averaging group on X, so every averaging group generated by X is a quotient of A(X). The same framework proves structural bridges: an averaging group induces a disemigroup, and when A(e)=e it induces a rack; a smooth averaging operator on a Lie group differentiates to an averaging Lie algebra; and the linear extension to the group algebra turns averaging groups into averaging Hopf algebras. The upshot is that averaging operators on groups belong to the same network of structures already developed for Rota-Baxter operators and averaging Lie algebras, with a concrete universal object as the base.

What carries the argument

The central object is the set A(X) of averaging group words: elements of the free operated group G(X) whose standard factorizations contain no subword of the forms ⌊u⌋⌊v⌋, ⌊⌊u⌋v⌋, or ⌊u⌊v⌋^{(2)}⌋. These are the normal forms that survive the relations forced by the averaging identity. The load-bearing mechanism is the recursive definition of the multiplication ⋄ in equations (10)-(11) and of the operator A_X in equation (12), together with the lexicographic inductions (on depth and breadth for ⋄, on degree and breadth for A_X) that prove (A(X),⋄) is a group, that A_X satisfies the averaging identities, and finally that the universal factorization exists and is unique.

What would settle it

Directly compute both sides of (u ⋄ v) ⋄ w = u ⋄ (v ⋄ w) for the example words u = ⌊x⌊y⌋⌋^{(2)}, v = ⌊z⌋^{-1}, and w = ⌊z⌋^{(3)} from Example 4.3, and check the two averaging identities for the same words; a mismatch, or a word on which equations (10)-(12) do not apply, would falsify Theorem 4.7. More systematically, enumerate all words in A(X) of operator degree at most 4 over a two-element set X and verify the defining identities in that finite set.

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Extended reading notes

Core claim

On its own terms, the paper establishes that averaging operators on groups admit a free object and a set of structural transports. The free averaging group on X is realized inside the free operated group G(X): A(X) consists of bracketed words containing no subword of the form ⌊u⌋⌊v⌋, ⌊⌊u⌋v⌋, or ⌊u⌊v⌋^{(2)}⌋, the patterns whose collapse is forced by the averaging identity. A multiplication ⋄ on A(X) is defined by lexicographic induction on depth and breadth, and an operator A_X by induction on operator degree; Theorem 4.6 verifies the group axioms and the averaging identity for (A(X),⋄,A_X), and Theorem 4.7 verifies the universal property that any map from X into an averaging group extends uniquely as an averaging-group homomorphism. The same framework yields the accompanying results that averaging groups induce disemigroups and, under A(e)=e, racks, that differentiation at the identity sends averaging Lie groups to averaging Lie algebras, and that the linear extension over the group algebra makes (G,A) and (k[G],A) equivalent as averaging structures.

Load-bearing premise

The load-bearing premise is that the recursively defined multiplication ⋄ and operator A_X on the set of averaging group words are well-defined for every word and that the lexicographic inductions used to prove associativity and the averaging identities cover all cases.

Editorial extensions

If this is right

  • Every averaging group generated by a set X is a quotient of the bracketed-word model A(X), so the free object gives a universal normal form for identities among averaging operators on groups.
  • An averaging group with A(e)=e carries a rack operation g ⊲ h = A(g)hA(g)^{-1}, connecting group-level averaging operators to rack and Leibniz-algebra machinery.
  • Smooth averaging Lie groups with A(e)=e differentiate to averaging Lie algebras at the tangent space of the identity, giving a Lie-theoretic integration direction for averaging operators.
  • The linear extension of A to the group algebra k[G] is an averaging Hopf algebra exactly when (G,A) is an averaging group, so the two notions coincide for group algebras.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • My inference: because A(X) is defined by forbidding subwords, the bracketed reductions should form a terminating rewriting system, which would give a decision procedure for the word problem of free averaging groups; the paper does not prove termination or confluence.
  • My inference: the rack structure induced by an averaging group with A(e)=e may yield set-theoretic solutions of the Yang-Baxter equation or rack-theoretic invariants whenever the underlying group is finite; the paper records the rack structure but does not pursue these consequences.
  • My inference: the same bracketed-word construction should adapt to free averaging algebras and free averaging Hopf algebras by taking the linear span of A(X) and adjoining the relevant operations; the paper stops at groups and group algebras.
  • My inference: the paper explicitly leaves open, in Remark 3.14, whether an averaging operator on a Lie algebra lifts to its universal enveloping algebra; a natural test is whether the averaging Lie group construction of Theorem 3.10 integrates through U(g).
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. This paper introduces averaging operators on groups, defined by the identities A(g)A(h)=A(A(g)h)=A(gA(h)), and studies their relationships with averaging Lie algebras, averaging Hopf algebras, disemigroups, and racks. The main contribution is an explicit construction of the free averaging group on a set: inside the free operated group on X, the authors define a set A(X) of 'averaging group words', equip it with a multiplication ⋄ and an operator A_X, and claim in Theorem 4.7 that (A(X),⋄,A_X) is the free averaging group on X. The paper also proves that smooth averaging operators on Lie groups differentiate to averaging operators on Lie algebras and that averaging operators on groups correspond exactly to averaging Hopf algebra structures on the associated group Hopf algebra.

Significance. The motivation via Koszul duality is attractive, and an explicit bracketed-word model for the free averaging group would give a concrete universal object with a clean universal property. The ancillary results on disemigroups, racks, and group Hopf algebras are mostly straightforward but provide useful context, and the authors are commendably explicit about the complexity of their induction arguments. However, the central construction is not well-defined as printed: the representation of iterated brackets in Definition 4.2 and in equations (10)-(12) is ambiguous, and the multiplication ⋄ can send averaging words to words outside A(X). As a result, Theorem 4.6 and Theorem 4.7 are not established by the manuscript as written.

major comments (4)
  1. [Definition 4.2 and Eq. (10)] The multiplication ⋄ is not well-defined as printed. The representation u=⌊u′⌋(s) is not unique when u′ itself begins with a bracket. For example, take X={a,b}, u=⌊a⌋, and v=⌊⌊b⌋⌋. The word v can be written as ⌊v′⌋(1) with v′=⌊b⌋ or as ⌊v′⌋(2) with v′=b. Both choices satisfy the first-branch condition of (10), since u′=a∉⌊A(X)⌋. With v′=⌊b⌋, (10) yields ⌊a⋄⌊⌊b⌋⌋⌋ = ⌊a⌊b⌋(2)⌋, which is explicitly of the forbidden form in Definition 4.2 and hence is not in A(X). With v′=b, the same formula yields ⌊a⌊b⌋⌋(2), a different element of G(X). Thus (10) does not define a binary map A(X)×A(X)→A(X) as printed. Since Theorem 4.6(i) and Theorem 4.7 both use this multiplication, a canonical representation of iterated brackets and additional conditions in both branches of (10) are needed, together with a proof that every output lies in A(X).
  2. [Definition 4.2, forbidden subwords] The definition of A(X) is internally ambiguous about iterated brackets. Taken literally, the forbidden form ⌊⌊u⌋v⌋ with v=1 excludes the word ⌊b⌋(2)=⌊⌊b⌋⌋, and words of exactly this shape are used as elements of A(X) throughout Section 4, including Example 4.3 and the branches of (10). If 'subword' is intended to mean proper subword, or if the word v in the forbidden forms is required to be nonempty, this must be stated explicitly; otherwise A(X) is not the set that the rest of the paper works with. This ambiguity affects the very definition of the carrier set for the claimed free object.
  3. [Eq. (12), definition of A_X] The operator A_X suffers from the same non-uniqueness as ⋄. In (12), the cases are split according to whether the first factor w1 equals ⌊w′1⌋(t) or the last factor equals ⌊w′k⌋(s), with side conditions w′1,w′k∉⌊A(X)⌋. Since the leading or trailing bracketed factor can be peeled at different depths, these cases are not mutually exclusive and the outputs need not agree. The display also asserts that auxiliary words such as ⌊w2⋯wk⌋ or ⌊w2⋯⌊w′k⌋⌋ belong to A(X) without proof, and this can fail when the middle part begins with a bracket. Therefore A_X is not yet shown to be a well-defined map A(X)→A(X), which is required for Theorem 4.6(ii) and for the universal property in Theorem 4.7.
  4. [Theorem 3.10] The proof of Theorem 3.10 uses equation (6) to replace A(exp tu) by exp(tA(u)) inside a second mixed derivative at t=s=0. Equation (6) only records equality of the first derivatives at t=0; it does not by itself justify substituting the two curves in the argument of ∂t∂s of a product. A short additional argument is needed, for example showing that the difference of the two curves is o(t) and that this error does not contribute to the mixed derivative at zero. As written, the step marked 'by (6)' is a gap in the proof that A is an averaging operator on the Lie algebra.
minor comments (4)
  1. [Eq. (2)] In the displayed chain after Definition 3.1(ii), the equality [A(a),A(b)] = A([A(a),b]) = A([A(a),b]) should end with A([a,A(b)]); the final expression appears to be a typographical repetition.
  2. [Theorem 4.6(i)] In the proof of associativity, the phrase 'bre(u)+bre(v)+bre(w))' contains an unmatched parenthesis, and the proof does not explicitly justify that w^{-1} belongs to A(X) for every w∈A(X), although this is needed for the claim that (A(X),⋄) is a group.
  3. [Section 2 and Definition 4.2] The notation ⌊w⌋(n) for iterates of the bracket operator is introduced in Section 2, but near Definition 4.2 it is used with n=1 and n≥2 in ways that are easy to confuse with powers or with the single bracket ⌊w⌋; a short restatement of the convention at the beginning of Section 4 would improve readability.
  4. [Figure 2] The arrow labels in Figure 2 appear partially garbled in the typeset version; please check the alignment of the labels so that the claimed relationships are legible.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the free averaging group is constructed directly from the free operated group and its universal property is proved, not assumed.

full rationale

The central claim, Theorem 4.7, asserts that the explicitly defined triple (A(X), *, A_X) together with j_X is the free averaging group on X. The construction is transparently non-circular: A(X) is defined as a concrete subset of the free operated group G(X), the multiplication * and operator A_X are defined recursively by equations (10)-(12), and Theorem 4.6 verifies directly that (A(X), *) is a group and that (A(X), *, A_X) satisfies the averaging identities. Theorem 4.7 then constructs the unique extension of an arbitrary map phi: X -> G by induction on depth and breadth, and verifies that it is a group homomorphism and intertwines A_X with A. Nothing is fitted and no parameter is renamed as a prediction. The use of the free operated group theorem from [23] involves a coauthor but is a published, externally checkable result used only to provide the ambient bracketed-word environment and the standard factorization; it does not assume the existence of free averaging groups. The quotient presentation at the start of Section 4 is the standard universal-algebra construction by generators and relations and is explicitly followed by the direct construction, so it is not a hidden circular input. The motivating identities around equation (9) are used only to select the word set, and the resulting algebra is then re-proved from the recursive definitions. Two caveats are present but they are not circularity: Definition 4.2's recursive cases may require a more explicit canonical bracketing to be fully well-defined, and the assertion in the proof of Theorem 4.7 that 'By (18), we have phi ∘ A_X = A ∘ phi' is compressed and would need an induction over all of A(X). These are correctness or proof-order concerns, not reductions of the theorem to its own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters or empirical constants appear in this paper. The construction relies on standard prior results: the free operated group from [23], universal-algebra quotient reasoning, and standard Lie group and Hopf algebra facts. The new definitions, averaging groups and averaging Hopf algebras, are introduced as algebraic structures and are proven to exist via the free construction, so no unexplained postulated entity is needed.

assumptions (4)
  • standard math The free operated group construction on a set via bracketed words is valid and yields the ambient group G(X).
    Invoked in Section 4 to define averaging group words and the operator P_X; the explicit free averaging group is built inside this object, using Lemma 2.4 from [23].
  • standard math A quotient of a free operated group by the normal operated subgroup generated by the averaging relations is the free averaging group.
    Used in the paragraph after Definition 4.1; standard universal algebra gives existence of the free object before the explicit construction.
  • domain assumption For a smooth map A on a Lie group with A(e) = e, the tangent map A_*e exponentiates to first order in the sense needed for the bracket computation in Theorem 3.10.
    Used in Theorem 3.10; equation (6) only proves equality of first derivatives at t = 0, and the proof substitutes these curves inside a second mixed derivative, so an additional smoothness and exponential-map argument is required.
  • standard math The group algebra k[G] with diagonal coproduct is a cocommutative Hopf algebra, and any set map A extends linearly to a coalgebra map.
    Used in Theorem 3.13 to reduce averaging Hopf algebras on k[G] to averaging groups; this is a standard Hopf algebra fact.

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Pith. "Pith review of Averaging operators on groups and Hopf algebras." pith.science (2026). https://pith.science/paper/7ABUDN3Y

@misc{pith2026241211600,
  author       = {Pith},
  title        = {Pith review of: Averaging operators on groups and Hopf algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ABUDN3Y}},
  note         = {Machine review of arXiv:2412.11600}
}
read the original abstract

Rota-Baxter operators on groups were studied quite recently. Motivated mainly by the fact that weight zero Rota-Baxter operators and averaging operators are Koszul dual to each other, we propose the concepts of averaging group and averaging Hopf algebra, and study relationships among them and the existing averaging Lie algebras. We also show that an averaging group induces a disemigroup and a rack, respectively. As the free object is one of the most significant objects in a category, we also construct explicitly the free averaging group on a set.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Induced structures of averaging commutative and cocommutative infinitesimal bialgebras via a new splitting of perm algebras

    math.RA 2025-09 accept novelty 6.0 of 10

    Averaging commutative and cocommutative infinitesimal bialgebras induce special apre-perm bialgebras via a new splitting of perm algebras.

Reference graph

Works this paper leans on

64 extracted references · 63 canonical work pages · cited by 1 Pith paper

  1. [23]

    X. Gao, L. Guo, Y . Liu and Z. Zhu, Operated groups, di fferential groups and Rota-Baxter groups with an emphasis on the free objects, Communications in Algebra 51 (2023), 4481-4500. 3, 5, 6, 12

    work page 2023

  2. [1]

    Aguiar, Pre-Poisson algebras, Lett

    M. Aguiar, Pre-Poisson algebras, Lett. Math. Phys. 54 (2000), 263-277. 2, 7

    work page 2000

  3. [2]

    C. Bai, O. Bellier, L. Guo and X. Ni, Splitting of operatio ns, Manin products, and Rota-Baxter operators, International Mathematics Research Notices, 2013(3) (2013), 485-524. 3, 4

    work page 2013

  4. [3]

    C. Bai, L. Guo and X. Ni, Nonabelian generalized Lax pairs , the classical Y ang-Baxter equation and PostLie algebras, Comm. Math. Phys. 297 (2010), 553-596. 2

    work page 2010

  5. [4]

    C. Bai, L. Guo, Y . Sheng and R. Tang, Post-groups, (Lie)-B utcher groups and the Y ang-Baxter equation, Math. Ann. 388 (2024), 3127-3167. 3

    work page 2024

  6. [5]

    V . G. Bardakov and V . Gubarev, Rota-Baxter groups, skew left braces, and the Y ang-Baxter equation,J. Algebra 596 (2022), 328-351. 3

    work page 2022

  7. [6]

    V . G. Bardakov and V . Gubarev, Rota-Baxter operators on groups, Proc. Math. Sci. 133 (2023), 4. 3

    work page 2023

  8. [7]

    V . G. Bardakov, M. V . Neshchadim and M. K. Y adav, Symmetric skew braces and brace systems, F orum Math- ematicum 35(3) (2023), 713-738. 3

    work page 2023

Show all 64 references
  1. [8]

    Baxter, An analytic problem whose solution follows fr om a simple algebraic identity, Pacific J

    G. Baxter, An analytic problem whose solution follows fr om a simple algebraic identity, Pacific J. Math. 10 (1960), 731-742. 1

    1960

  2. [9]

    E. A. Bergshoe ff, M. Deroo, O. Hohm, Multiple M2-branes and the embedding tensor, Classical Quantum Gravity 25(14) (2008), 142001, 10 pp. 2

    2008

  3. [10]

    Bourbaki, Elements of Mathematics: Algebra I, Chapt er 1-3, Hermann, 1974

    N. Bourbaki, Elements of Mathematics: Algebra I, Chapt er 1-3, Hermann, 1974. 2

    1974

  4. [11]

    Brainerd, On the structure of averaging operators, J

    B. Brainerd, On the structure of averaging operators, J. Math. Anal. 5 (1962), 347-377. 2

    1962

  5. [12]

    Carin˜ ena, J

    J. Carin˜ ena, J. Grabowski and G. Marmo, Quantum bi-ham iltonian systems, Internat. J. Modern Phys. A. 15 (2000), 4797-4810. 1

    2000

  6. [13]

    Caranti and L

    A. Caranti and L. Stefanello, Skew braces from Rota-Bax ter operators: A cohomological characterisation and an example, Annali di Matematica 202 (2023), 1-13. 3

    2023

  7. [14]

    Catino, M

    F. Catino, M. Mazzotta and P . Stefanelli, Rota-Baxter o perators on Cli fford semigroups and the Y ang-Baxter equation, J. Algebra 622 (2023), 587-613. 3

    2023

  8. [15]

    Ced´ o, E

    F. Ced´ o, E. Jespers and J. Okni´ nski, Braces and the Y an g-Baxter equation. Comm. Math. Phys. 327 (2014), 101-116. 3

    2014

  9. [16]

    Connes and D

    A. Connes and D. Kreimer, Renormalization in quantum fie ld theory and the Riemann-Hilbert problem. I, Comm. Math. Phys. 210 (2000), 249-273. 2

    2000

  10. [17]

    Das, Deformations of associative Rota-Baxter opera tors, J

    A. Das, Deformations of associative Rota-Baxter opera tors, J. Algebra 560 (2020), 144-180. 1

    2020

  11. [18]

    Das, Averaging operators on groups, racks and Leibni z algebras, arXiv:2403.06250v1

    A. Das, Averaging operators on groups, racks and Leibni z algebras, arXiv:2403.06250v1. 2, 7

    arXiv

  12. [19]

    Etingof, T

    P . Etingof, T. Schedler and A. Soloviev, Set-theoretic al solutions to the quantum Y ang-Baxter equation. Duke Math. J. 100 (1999), 169-209. 3

    1999

  13. [20]

    Gao and L

    X. Gao and L. Guo, Rota’s Classification Problem, rewrit ing systems and Gr¨ obner-Shirshov bases,J. Algebra 470 (2017), 219-253. 2 A VERAGING OPERA TORS ON GROUPS AND HOPF ALGEBRAS 23

    2017

  14. [21]

    X. Gao, L. Guo and Z. Han, Rota-Baxter groups with weight zero and integration on topological groups, arXiv:2405.11288 . 3

    arXiv

  15. [22]

    X. Gao, L. Guo, Z. Han and Y . Zhang, Rota-Baxter operators, differential operators, pre- and Novikov structures on groups and Lie algebras, arXiv:2408.06096v2. 3, 4

    arXiv

  16. [24]

    Gao and L

    X. Gao and L. Guo and H. Zhang, Rota’s program on algebrai c operators, rewriting systems and Gr¨ obner- Shirshov bases, Adv. Math. (China) 51 (2022), 1-31. 2

    2022

  17. [25]

    Goncharov, Rota-Baxter operators on cocommutative Hopf algebras, J

    M. Goncharov, Rota-Baxter operators on cocommutative Hopf algebras, J. Algebra 582 (2021), 39-56. 3, 12

    2021

  18. [26]

    Gubarev, Universal enveloping associative Rota-Ba xter algebras of preassociative and postassociative alge- bra, J

    V . Gubarev, Universal enveloping associative Rota-Ba xter algebras of preassociative and postassociative alge- bra, J. Algebra 516 (2018), 298-328. 1

    2018

  19. [27]

    V . Y u. Gubarev and P . S. Kolesnikov, On embedding of dend riform algebras into Rota-Baxter algebras, Cent. Eur. Jour. Math 11 (2013), 226-245. 2

    2013

  20. [28]

    Guo, Operated semigroup, Motzkin paths and rooted tr ee, J

    L. Guo, Operated semigroup, Motzkin paths and rooted tr ee, J. Algebraic Combinatorics 29 (2009), 35-62. 2

    2009

  21. [29]

    Guo, An introduction to Rota-Baxter Algebras, Surve ys of Modern Mathematics 4

    L. Guo, An introduction to Rota-Baxter Algebras, Surve ys of Modern Mathematics 4. Somerville: International Press; Beijing: Higher Education Press, 2012. 1

    2012

  22. [30]

    L. Guo, R. Gustavson and Y . Li, An algebraic study of V olt erra integral equations and their operator linearity, J. Algebra 595 (2022), 398-433. 2

    2022

  23. [31]

    Guo and W

    L. Guo and W . Keigher, On di fferential Rota-Baxter algebras, J. Pure Appl. Algebra 212 (2008), 522-540. 1

    2008

  24. [32]

    L. Guo, H. Lang and Y . Sheng, Integration and geometrization of Rota-Baxter operators, Adv. Math. 387 (2021), 107834. 3, 5

    2021

  25. [33]

    Guo and J

    L. Guo and J. Pei, Averaging algebras, Schr¨ oder numbers and rooted trees, Journal of Algebraic Combinatorics, 42 (2015), 73-109. 2

    2015

  26. [34]

    L. Guo, W . Sit and R. Zhang, Di fferential type operators and Gr¨ obner-Shirshov bases,J. Symbolic Comput. 52 (2013), 97-123. 2

    2013

  27. [35]

    Jiang, Y

    J. Jiang, Y . Sheng and C. Zhu, Lie theory and cohomology o f relative Rota-Baxter operators, J. Lond. Math. Soc. 109 (2024), e12863. 3

    2024

  28. [36]

    Kamp´ ede F´ eriet, L’etat actuel du probl´ eme de la turbulaence (I and II), La Sci

    J. Kamp´ ede F´ eriet, L’etat actuel du probl´ eme de la turbulaence (I and II), La Sci. A´ erienne3 (1934), 9-34, 4 (1935), 12-52. 2

    1934

  29. [37]

    M. K. Kinyon, Leibniz algebras, Lie racks, and digroups , J. Lie Theory 17 (2007), 99-114. 4, 5, 9

    2007

  30. [38]

    E. R. Kolchin, Di fferential Algebra and Algebraic Group, Academic Press, New Y ork, 1973. 1

    1973

  31. [39]

    Kolesnikov and V .Y u

    P .S. Kolesnikov and V .Y u. V oronin, On special identities for dialgebras, Linear and Multilinear Algebra 61(3), 377-391. 2

  32. [40]

    Kotov and T

    A. Kotov and T. Strobl, The embedding tensor, Leibniz-L oday algebras, and their higher Gauge theories, Comm. Math. Phys. 376 (2020), 235-258. 2

    2020

  33. [41]

    B. A. Kupershmidt, What a classical r-matrix really is, J. Nonlinear Math. Phys. 6 (1999), 448-488. 2

    1999

  34. [42]

    Lang and Y

    H. Lang and Y . Sheng, Factorizable Lie bialgebras, quad ratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebras, Commun. Math. Phys. 397 (2023), 763-791. 3

    2023

  35. [43]

    Loday, Dialgebras, in Dialgebras and related ope rads, Lecture Notes in Math

    J.-L. Loday, Dialgebras, in Dialgebras and related ope rads, Lecture Notes in Math. 1763 (2002), 7-66. 4, 9

    2002

  36. [44]

    J. Lu, M. Y an and Y . Zhu, On the set-theoretical Y ang-Baxter equation. Duke Math. J. 104 (2000), 1-18. 3

    2000

  37. [45]

    Mackenzie, General Theories of Lie Groupoids and Lie Algebroids, Cambridge University Press, 2005

    K. Mackenzie, General Theories of Lie Groupoids and Lie Algebroids, Cambridge University Press, 2005. 3

    2005

  38. [46]

    H. Z. Munthe-Kaas and A. Lundervold, On post-Lie algebr as, Lie-Butcher series and moving frames. F ound. Comput. Math. 13 (2013), 583-613. 3

    2013

  39. [47]

    H. Z. Munthe-Kaas and W . M. Wright, On the Hopf algebraic structure of Lie group integrators. F ound. Comput. Math. 8 (2008), 227-257. 3

    2008

  40. [48]

    Nicolai and H

    H. Nicolai and H. Samtleben, Maximal gauged supergravi ty in three dimensions, Phys. Rev. Lett. 86(9) (2001), 1686-1689. 2

    2001

  41. [49]

    Nijenhuis, Xn−1-forming sets of eigenvectors

    A. Nijenhuis, Xn−1-forming sets of eigenvectors. Indag. Math. 13 (1951), 200-212. 2

    1951

  42. [50]

    J. Pei, C. Bai, L. Guo and X. Ni, Replicators, Manin white product of binary operads and average operators, New Trends in Algebras and Combinatorics (2020), 317-353. 2, 3, 4, 5

    2020

  43. [51]

    J. Pei, C. Bai, L. Guo and X. Ni, Disuccessors and duplica tors of operads, Manin products and operators, In ”Symmetries and Groups in Contemporary Physics”, Nankai Se ries in Pure, Applied Mathematics and Theo- retical Physics 11 (2013), 191-196. 2, 3

    2013

  44. [52]

    van der Put and M

    M. van der Put and M. Singer, Galois Theory of Di fference Equations, Springer, 1997. 1 24 HUHU ZHANG AND XING GAO ∗

    1997

  45. [53]

    A. G. Reyman and M. A. Semenov-Tian-Shansky, Reduction of Hamilton systems, a ffine Lie algebras and Lax equations, Invent. Math. 54 (1979), 81-100. 3

    1979

  46. [54]

    A. G. Reyman and M. A. Semenov-Tian-Shansky, Reduction of Hamilton systems, a ffine Lie algebras and Lax equations II, Invent. Math. 63 (1981), 423-432. 3

    1981

  47. [55]

    Reynolds, On the dynamic theory of incompressible vi scous fluids, Phil

    O. Reynolds, On the dynamic theory of incompressible vi scous fluids, Phil. Trans. Roy. Soc. 136 (1895), 123-

  48. [56]

    Rota, Baxter algebras and combinatorial identit ies, I, II, Bull

    G.-C. Rota, Baxter algebras and combinatorial identit ies, I, II, Bull. Amer . Math. Soc.75 (1969), 325-329. 1

    1969

  49. [57]

    Rota, Baxter operators, an introduction, in: J.P .S

    G.-C. Rota, Baxter operators, an introduction, in: J.P .S. Kung (Ed.), Gian-Carlo Rota on Combinatorics, Intro- ductory Papers and Commentaries, Birkh¨ auser, Boston, 1995. 2

    1995

  50. [58]

    M. A. Semenov-Tian-Shansky, What is a classical r-matrix? Funct. Anal. Appl. 17(4) (1983), 259-272. 2, 3

    1983

  51. [59]

    Serre, Galois Cohomology, Springer, 1997

    J.-P . Serre, Galois Cohomology, Springer, 1997. 5

    1997

  52. [60]

    Zhang, X

    H. Zhang, X. Gao and L. Guo, Operator identities on Lie al gebras, rewriting systems and Gr¨ obner-Shirshov bases, J. Algebra 620 (2023), 585-629. 2

    2023

  53. [61]

    Zhang, X

    T. Zhang, X. Gao and L. Guo, Reynolds algebras and their f ree objects from bracketed words and rooted trees, J. Pure. Appl. Algebra 225 (2021), 106766. 2

    2021

  54. [62]

    Zhang, X

    Y . Zhang, X. Guo and L. Guo, Matching Rota-Baxter algebr as, matching dendriform algebras and matching pre-Lie algebras, J. Algebra 552 (2020), 134-170. 2

    2020

  55. [63]

    Zhang, X

    Y . Zhang, X. Gao and L. Guo, Hopf algebra of multidecorat ed rooted forests, free matching Rota-Baxter alge- bras and Gr¨ obner-Shirshov bases,Pacific J. Math. 317 (2022), 441-475. 2

    2022

  56. [64]

    Zhang, X

    Y . Zhang, X. Gao and D. Manchon, Free (tri)dendriform fa mily algebras, J. Algebra 547 (2020), 456-493. 2 School of Mathematics and StatisticsYulin University, Yulin, Shaanxi 719000, China Email address: huhuzhang@yulinu.edu.cn School of Mathematics and Statistics, L anzhou U...

    2020

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