{"id":"72d7a1be-3852-4ca0-a35c-5c77ad28aaf4","arxiv_id":"2509.03648","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Associative-Yamaguti algebras are introduced and shown to admit an enveloping associative algebra, induce Lie-Yamaguti algebras, and support a (2,3)-cohomology theory classifying deformations and extensions.","lead":"The paper introduces associative-Yamaguti algebras, a new algebraic structure combining a multiplication with two triple operations, and develops their representation, cohomology, and deformation theory. The result connects several known algebraic frameworks, but some key constructions have unverified details.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.10's multiplication on M(A) lacks a well-definedness proof; the gap is load-bearing but likely fillable via the composition identities.","rationale":"The reader correctly identifies that Theorem 3.10's multiplication is defined on representatives and that well-definedness is not proved. This is indeed the most load-bearing concern because the enveloping algebra theorem depends on M(A) being an associative algebra. However, the concern is not a demonstrated error: the axioms appear to imply the product is just composition in the first coordinate and reverse composition in the second, which is manifestly well-defined. Thus the central claim likely survives, but the paper should supply the missing proof. The numerous typos and other omitted 'straightforward' verifications further support the reader's CONDITIONAL verdict rather than unconditional acceptance.","tokens_in":36107,"tokens_out":19459,"duration_ms":182540,"concrete_test":"Derive from the rephrased identities after Definition 3.1 that for all a,b,c,d in A, σ_{σ_{a,b}(c), d} = σ_{a,b}σ_{c,d} and τ_{σ_{a,b}(c), d} = τ_{c,d}τ_{a,b}. If these equalities hold, the product on M(A) is well-defined and associative, closing the gap. If the derivation fails, find a finite-dimensional associative-Yamaguti algebra (e.g., from Example 3.5) where two different pairs give the same (σ,τ) but the formula yields different results, which would refute Theorem 3.10.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that every associative-Yamaguti algebra embeds in an enveloping associative algebra—rests on Theorem 3.10, where M(A) is spanned by pairs (σ_{a,b}, τ_{a,b}) and a product is declared on these representatives. The paper does not prove that this product is independent of the representing pair, nor that it extends bilinearly to all of M(A). This is a genuine gap because the proof is omitted ('straightforward') and the definition of the enveloping algebra in Proposition 3.9 requires a genuine associative algebra structure on M(A). If the product were ill-defined, the entire theorem would fail. However, the gap is likely resolvable: the identities listed immediately after Definition 3.1 include σ_{a,b}σ_{c,d} = σ_{σ_{a,b}(c), d} and τ_{c,d}τ_{a,b} = τ_{σ_{a,b}(c), d}. Together they imply (σ_{a,b}, τ_{a,b}) ∗ (σ_{c,d}, τ_{c,d}) = (σ_{a,b}∘σ_{c,d}, τ_{c,d}∘τ_{a,b}), an expression depending only on the endomorphisms, not on the chosen representatives. This also settles associativity. Since this derivation is not shown, the theorem is at least under-proved.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces associative-Yamaguti algebras as the associative analogue of Lie-Yamaguti algebras: a vector space with a binary operation and two ternary operations satisfying eleven identities. It develops the basic theory: examples (associative algebras, reductive associative algebras, associative triple systems, diassociative algebras), an enveloping associative algebra theorem, a skew-symmetrization functor to Lie-Yamaguti algebras, representations and a (2,3)-cohomology theory, and applications to formal deformations and abelian extensions. It then defines Yamaguti multiplications on nonsymmetric operads, introduces dendriform-Yamaguti algebras, and relates them to relative Rota-Baxter operators.","tokens_in":36371,"tokens_out":14263,"duration_ms":133578,"significance":"If the main results hold, the paper provides a broad new framework that connects associative algebras, triple systems, diassociative algebras, and Lie-Yamaguti algebras. The most striking claim—Theorem 3.10, that every associative-Yamaguti algebra arises from a reductive associative algebra—would give a strong bridge between the new axioms and classical associative algebra theory. The cohomology and deformation results are standard in structure but are useful additions for a new class. However, the significance is currently conditional: the central enveloping theorem is not actually proved in the manuscript, and one theorem statement contains a typo in a key definition. The paper also contains many 'straightforward' proofs, some of which are not immediate.","major_comments":[{"comment":"The enveloping-algebra theorem, highlighted in the abstract, is not proved. M(A) is defined as the span of the pairs (σ_{a,b}, τ_{a,b}), and the product is declared on representatives as (σ_{a,b}, τ_{a,b}) * (σ_{c,d}, τ_{c,d}) = (σ_{ {a,b,c}, d}, τ_{ {a,b,c}, d}). Because the same endomorphism pair may arise from different (a,b), the product must be shown independent of the chosen representatives and bilinear. The paper only says 'The proof of this result is straightforward.' This is load-bearing: without a well-defined associative product on M(A), the reductive algebra (M(A)⊕A, ⊛) and hence the existence of an enveloping algebra are not established. The gap is likely repairable: the identities immediately after Definition 3.1 imply σ_{ {a,b,c}, d} = σ_{a,b} σ_{c,d} and a similar derivation gives τ_{ {a,b,c}, d} = τ_{c,d} τ_{a,b}, so the product is the componentwise product in End(A) ⊕ E","section":"Theorem 3.10"},{"comment":"The displayed definitions of the induced ternary operations contain a typo: {u,v,w}[1] is defined as {u, R(u), R(v)} (and similarly for { {u,v,w} }[1]), which uses u twice and omits w. The proof immediately uses {u, R(v), R(w)}, which is evidently the intended definition. As printed, the theorem does not define a ternary operation on M and must be corrected. Please also check the corresponding line in the statement of Theorem 6.14 and all downstream references to it.","section":"Theorem 6.14"}],"minor_comments":[{"comment":"There is a typo: '{ {a, b, G(c, d, e)} }+G(a, b, { {c, , d, e} })' should read '{ {a, b, G(c, d, e)} }+G(a, b, { {c, d, e} })'.","section":"Definition 4.10, Eq. (26)"},{"comment":"The proof of part (i) is compressed to 'direct calculations' and a statement that the 58 dendriform-Yamaguti identities correspond to the 58 representation conditions. Since this theorem is the converse of Theorem 6.14 and the operations are defined using the total products, a reader cannot easily verify the claim. Please include at least one representative verification or an explicit correspondence table.","section":"Theorem 6.15"},{"comment":"The proof is very long and contains several typographical slips (e.g., an unmatched brace in the verification of (AY7)). The result itself is convincing, but the exposition would benefit from a cleaned-up proof or a more systematic presentation.","section":"Section 3, Theorem 3.7"},{"comment":"Several statements are justified by 'straightforward' or 'easy to see' even when they are not immediate (e.g., Proposition 4.2, Theorem 6.6, the last part of Proposition 3.9). For a journal submission, please expand the most important of these, especially those on which later results depend.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper introduces a reasonable new definition and develops it in several directions. The main mathematical claims appear plausible, and the defects I found are local: the missing well-definedness proof in Theorem 3.10 and a typo in Theorem 6.14. I recommend major revision rather than rejection. The editor may wish to ask the author to add a complete proof of Theorem 3.10 and to fix the typo before further review. No concerns about citation patterns or scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper defines associative-Yamaguti algebras, the associative analogue of Lie-Yamaguti algebras. That is genuinely new, and the paper does a lot with it: examples (associative algebras, reductive algebras, triple systems, diassociative algebras), an enveloping algebra theorem, a skew-symmetrization functor to Lie-Yamaguti algebras, a (2,3)-cohomology with deformations and extensions, and an operadic/dendriform/Rota-Baxter package. The author knows the surrounding literature, the citations look appropriate, and the paper is self-contained.\n\nThe main advertised result, Theorem 3.10, states that every associative-Yamaguti algebra embeds in a reductive associative algebra. The proof is omitted as 'straightforward' and the product on M(A) is defined on representatives (sigma_{a,b}, tau_{a,b}) without proving it is well-defined. That is a genuine gap, and it is load-bearing because the whole theorem rests on M(A) being an associative algebra. The good news is that the identities listed right after Definition 3.1 likely make the product depend only on the pair of endomorphisms, so the gap is probably fillable. But as written it is under-proved.\n\nThere is also a typo in Theorem 6.14: the definition of {u,v,w}[1] appears to use R(u) and R(v) instead of R(v) and R(w), contradicting the proof. The paper has other typos and several 'straightforward' verifications omitted. None of these strike me as fatal; the framework is plausible and the cohomology/deformation theorems follow standard patterns.\n\nIf you work on non-associative algebras or Loday-type structures, this is worth your time. It is not a breakthrough that changes practice elsewhere, but it is a serious foundational contribution. I would send it to a good referee, asking them to check Theorem 3.10 carefully and to require the author to fill in the well-definedness argument and fix the typos before publication.","headline":"A genuinely new associative analogue of Lie-Yamaguti algebras with a solid framework, but the key enveloping algebra theorem needs a real well-definedness proof.","tokens_in":36876,"tokens_out":2268,"would_cite":false,"duration_ms":21470,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17A30","17A40","17B60","16E99","17A36"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper introduces associative-Yamaguti algebras, an associative counterpart of Lie-Yamaguti algebras, and proves that every such algebra admits an enveloping associative algebra, so the entire structure is realized inside a reductive as","keywords":["associative-Yamaguti algebras","Lie-Yamaguti algebras","enveloping associative algebra","reductive associative algebra","(2,3)-cohomology","dendriform-Yamaguti algebras","relative Rota-Baxter operators","diassociative algebras"],"falsifier":"Take a small associative-Yamaguti algebra, preferably one induced from a diassociative algebra or an associative triple system, compute the pairs (σ,τ), and search for two distinct pairs (a,b) and (a',b') with equal (σ,τ) but unequal (σ_{ {a,b,c}, d }, τ_{ {a,b,c}, d }) for some c,d. One such example would make Theorem 3.10 false; verifying the equality in general would confirm the missing well-definedness step.","tokens_in":35958,"feed_emoji":"🔗","tokens_out":8533,"duration_ms":73328,"temperature":0.7,"pith_summary":"The paper introduces a new algebraic structure, an associative-Yamaguti algebra, which packages a binary multiplication together with two ternary products subject to eleven axioms. Its central claim is that this structure is not exotic: every associative-Yamaguti algebra embeds into an ordinary associative algebra with a reductive decomposition, in such a way that the binary and ternary operations are the ones induced by that ambient algebra. If the claim holds, the new axioms are a controlled slice of classical associative algebra theory, and many questions about them reduce to questions about associative algebras and their bimodules. The paper also shows that a suitable skew-symmetrization turns an associative-Yamaguti algebra into a Lie-Yamaguti algebra, and it builds a partial cohomology theory that controls formal deformations and abelian extensions.","feed_headline":"Every associative-Yamaguti algebra embeds in an associative algebra","feed_subtitle":"The new binary-plus-two-ternary algebra always arises from a reductive associative algebra.","key_machinery":"The load-bearing object is the pair of endomorphism-valued maps σ and τ attached to the two ternary operations: σ_{a,b}(c) = {a,b,c} and τ_{a,b}(c) = {{c,a,b}}. The space M(A) spanned by the pairs (σ_{a,b}, τ_{a,b}) is given a product that mimics the composition of the ternary operations, and the paper uses the reductive direct sum M(A) ⊕ A with the action (σ_{a,b}, τ_{a,b}) ▷ c = {a,b,c}, c ◁ (σ_{a,b}, τ_{a,b}) = {{c,a,b}} to build a genuine associative algebra whose induced ternary operations reproduce the original ones. This construction is the mechanism that turns the eleven axioms of an associative-Yamaguti algebra into ordinary associativity of an enveloping algebra.","core_discovery":"The paper's central claim is Theorem 3.10: given an associative-Yamaguti algebra (A, ·, {,,}, {{,,}}), let M(A) be the subspace of End(A) ⊕ End(A) spanned by pairs (σ_{a,b}, τ_{a,b}), where σ_{a,b}(c) = {a,b,c} and τ_{a,b}(c) = {{c,a,b}}. With the product (σ_{a,b}, τ_{a,b}) ∗ (σ_{c,d}, τ_{c,d}) = (σ_{ {a,b,c}, d }, τ_{ {a,b,c}, d }), M(A) is asserted to be an associative algebra, and A is a bimodule over it. The direct sum M(A) ⊕ A then carries a reductive associative algebra structure whose induced associative-Yamaguti operations on the A summand coincide with the original ones. In other words, every associative-Yamaguti algebra is realized as the 'odd part' of a reductive associative algeb","pith_inferences":["Editorial inference: if Theorem 3.10 survives the omitted well-definedness check, the enveloping algebra construction gives a concrete route to define a full cochain complex for associative-Yamaguti algebras by pulling back Hochschild cohomology of M(A) ⊕ A; the paper itself constructs only the partial (2,3) complex.","Editorial inference: the same reductive-realization idea may apply to the weak associative triple systems proposed in the concluding remarks, whose two ternary operations satisfy only (AY7) and (AY9); a corresponding enveloping construction would tie diassociative algebras more tightly to associative algebras.","Editorial inference: the compatibility between the diassociative-to-associative-Yamaguti and Leibniz-to-Lie-Yamaguti constructions suggests that future Lie-Yamaguti groups, when defined, may have their infinitesimal identities readable from the eleven AY identities."],"forward_implications":["Every associative-Yamaguti algebra is isomorphic to the induced structure on the A1 component of a reductive associative algebra A0 ⊕ A1, so the entire class is a subclass of structures coming from classical associative algebras.","The functor from diassociative algebras to associative-Yamaguti algebras, followed by skew-symmetrization, agrees with the standard passage from Leibniz algebras to Lie-Yamaguti algebras; the two routes to Lie-Yamaguti algebras commute.","The (2,3)-cohomology group is the right invariant for deformation theory: first-order infinitesimals of formal deformations are (2,3)-cocycles, equivalent deformations give the same cohomology class, and abelian extensions are classified by H^(2,3)(A,M).","Relative Rota-Baxter operators produce dendriform-Yamaguti algebras, and every dendriform-Yamaguti algebra arises this way, making the Rota-Baxter operator the splitting device for the total structure."],"supporting_citations":[{"why":"Supplies the enveloping Lie algebra construction for Lie-Yamaguti algebras that the paper's enveloping associative algebra generalizes.","marker":"[13]"},{"why":"Provides diassociative algebras, which canonically yield associative-Yamaguti algebras in Theorem 3.7, and dendriform algebras used later.","marker":"[15]"},{"why":"Introduced the general Lie triple systems underlying Lie-Yamaguti algebras, the object being mimicked associatively.","marker":"[23]"},{"why":"Defined representations and cohomology for Lie-Yamaguti algebras, the template for the (2,3)-cohomology.","marker":"[24]"},{"why":"Supplies the deformation and abelian-extension classification for Lie-Yamaguti algebras that Theorem 5.3 adapts to the associative setting.","marker":"[25]"},{"why":"Gives the cohomology of associative triple systems, the closest existing cohomology theory that the new partial complex extends.","marker":"[5]"},{"why":"A source of associative triple systems, included as examples of the new algebras.","marker":"[14]"},{"why":"Defines multiplications on nonsymmetric operads, which the paper generalizes to Yamaguti multiplications.","marker":"[8]"},{"why":"Constructs the dendriform operad Dend_A whose multiplications are dendriform algebras, the basis for dendriform-Yamaguti algebras.","marker":"[6]"},{"why":"Supplies the classical result that an associative triple system becomes a Lie triple system by skew-symmetrization, the prototype for Theorem 3.11.","marker":"[16]"}],"fun_headline_variants":["Associative-Yamaguti algebras always fit inside an associative algebra","New theorem: every associative-Yamaguti algebra has an enveloping algebra","Theorem: every associative-Yamaguti algebra embeds in an associative algebra","Associative-Yamaguti algebras are always embedded in associative algebras"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that the multiplication on M(A) in Theorem 3.10 is independent of the chosen representative pair: whenever two pairs (a,b) and (a',b') induce the same endomorphisms σ and τ, their products with any (c,d) must still agree, a verification the paper omits by calling the proof 'straightforward'.","fun_headline_variants_meta":{"raw":{"variants":["Associative-Yamaguti algebras always fit inside an associative algebra","New theorem: every associative-Yamaguti algebra has an enveloping algebra","Theorem: every associative-Yamaguti algebra embeds in an associative algebra","Associative-Yamaguti algebras are always embedded in associative algebras"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000871,"raw_usage":{"total_tokens":3645,"prompt_tokens":820,"completion_tokens":2825,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":564,"completion_tokens_details":{"reasoning_tokens":2746}},"tokens_in":564,"tokens_out":2825,"duration_ms":19489,"temperature":1.0,"reasoning_tokens":2746,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T10:46:52.219015+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small associative-Yamaguti algebra, preferably one induced from a diassociative algebra or an associative triple system, compute the pairs (σ,τ), and search for two distinct pairs (a,b) and (a',b') with equal (σ,τ) but unequal (σ_{ {a,b,c}, d }, τ_{ {a,b,c}, d }) for some c,d. One such example would make Theorem 3.10 false; verifying the equality in general would confirm the missing well-definedness step.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the enveloping Lie algebra construction for Lie-Yamaguti algebras that the paper's enveloping associative algebra generalizes."},{"cited_title":"Dialgebras and related operads","cited_arxiv_id":null,"evidence_quote":"Provides diassociative algebras, which canonically yield associative-Yamaguti algebras in Theorem 3.7, and dendriform algebras used later."},{"cited_title":"Yamaguti, On the Lie triple system and its generalization, J","cited_arxiv_id":null,"evidence_quote":"Introduced the general Lie triple systems underlying Lie-Yamaguti algebras, the object being mimicked associatively."},{"cited_title":"Yamaguti, On cohomology groups of general Lie triple systems, Kumamoto J","cited_arxiv_id":null,"evidence_quote":"Defined representations and cohomology for Lie-Yamaguti algebras, the template for the (2,3)-cohomology."},{"cited_title":"Zhang and J","cited_arxiv_id":null,"evidence_quote":"Supplies the deformation and abelian-extension classification for Lie-Yamaguti algebras that Theorem 5.3 adapts to the associative setting."},{"cited_title":"Carlsson, Cohomology of associative triple systems, Proc","cited_arxiv_id":null,"evidence_quote":"Gives the cohomology of associative triple systems, the closest existing cohomology theory that the new partial complex extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"A source of associative triple systems, included as examples of the new algebras."},{"cited_title":"Gerstenhaber and A","cited_arxiv_id":null,"evidence_quote":"Defines multiplications on nonsymmetric operads, which the paper generalizes to Yamaguti multiplications."},{"cited_title":"Das, Cohomology and deformations of dendriform algebras, and Dend∞-algebras, Comm","cited_arxiv_id":null,"evidence_quote":"Constructs the dendriform operad Dend_A whose multiplications are dendriform algebras, the basis for dendriform-Yamaguti algebras."},{"cited_title":"Meyberg, Lectures on algebras and triple systems, Lecture nores, Charlottesville V A, University of Virgina (1972)","cited_arxiv_id":null,"evidence_quote":"Supplies the classical result that an associative triple system becomes a Lie triple system by skew-symmetrization, the prototype for Theorem 3.11."}],"review_version":1}