{"id":"da7aaa9d-89ec-4c3a-9301-b34d4406af67","arxiv_id":"2412.20600","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The deformations of a Lie ideal are controlled by the cohomology of the ambient Lie algebra with coefficients in Hom(i, g/i), yielding rigidity and stability criteria.","lead":"This paper develops a deformation theory for ideals in Lie algebras, associating cohomology classes to smooth perturbations of an ideal and identifying the differential graded Lie algebra that controls them. It also provides cohomological criteria for rigidity and stability of ideals, giving new tools for classifying Lie algebras and their substructures.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6.14's proof equates surjectivity of H^0(Π) with exactness of the Aut(g)-orbit tangent sequence, but projection cocycles need not lift to derivations of g; the lifting obstruction in H^2(g;i) is unaddressed.","rationale":"I agree with the reader's identification of the load-bearing concern. The central construction of the paper—Theorem 5.3 and Proposition 4.2—is well supported: direct computation confirms that the Maurer-Cartan equation encodes exactly the Lie ideal condition for graph(φ), and the higher brackets vanish on the degree-0 elements that enter the Maurer-Cartan equation, so the controlling dgLa statement is credible. The genuine weakness is in the rigidity application. The proof of Theorem 6.14 needs to show that the image of Z^1(g;g) under ψ↦π∘ψ|_i fills Z^0(i⊳g), but its hypothesis only gives that every class in Z^0 is represented by the restriction of some 1-cocycle in g/g/i. The missing lifting of projection cocycles to derivations of g is not a cosmetic detail: the obstruction is a class in H^2(g;i) from the cochain long exact sequence associated to 0→i→g→g/i→0. Thus the advertised rigidity criterion is not established as stated, although it may be salvageable under stronger cohomological hypotheses. This does not undermine the main deformation-theoretic construction, so a conditional verdict—accepting the core results while flagging the rigidity gap—remains appropriate.","tokens_in":49011,"tokens_out":33759,"duration_ms":312052,"concrete_test":"For a concrete Lie algebra with a nontrivial abelian ideal i, compute the tangent sequence explicitly and compare Im(T) with Z^0(i⊳g). For example, take a semidirect product g = i ⋊ h where i is abelian, h acts on i, and H^2(g;i) ≠ 0; then compute H^0(Π), Z^0, and Im(T) = Π((π_{g/i})_*(Z^1(g;g))). If H^0(Π) is surjective but there exists f ∈ Z^0 not in Im(T), the proof of Theorem 6.14 fails and the hypothesis needs strengthening. This check can be done with standard Chevalley-Eilenberg cohomology computations.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The rigidity theorem's proof conflates two different images. The tangent sequence is Z^1(g;g) --T--> C^0(g;i*⊗g/i) --δ--> C^1, with T(ψ)=π_{g/i}∘ψ|_i. Exactness at C^0 requires Im(T)=Z^0. The proof factors T through Z^1(g;g/i) via (π_{g/i})_* and then Π(φ)=φ|_i, and claims that surjectivity of H^0(Π): H^1_π(g;g/i)→H^0(i⊳g) is equivalent to Im(T)=Z^0. But H^0=Z^0, and surjectivity of H^0(Π) only gives Z^0 ⊆ Π(Z^1(g;g/i)); it does not give Z^0 ⊆ Π((π_{g/i})_*(Z^1(g;g))) = Im(T). A 1-cocycle φ:g→g/i need not lift to a derivation ψ∈Z^1(g;g) with π∘ψ=φ; the obstruction lies in H^2(g;i) via the long exact sequence of 0→C•(g;i)→C•(g;g)→C•(g;g/i)→0. Without this lift, the non-degeneracy condition of Proposition 6.7 is not established, so the rigidity conclusion (and Corollary 6.15 via H^2(g/i;g/i)=0) is unjustified. The diagram in the proof does not address the lift.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a deformation theory for Lie ideals. It introduces a cochain complex C•(g; i*⊗g/i) and proves that the derivative of a smooth deformation of an ideal is a 0-cocycle (Proposition 4.2), calling H^0(i⊳g) the space of infinitesimal deformations. It compares this cohomology with several others (deformation complex of the projection g→g/i, Nijenhuis–Richardson complex, subalgebra deformation complex, and the complex for deformations of g preserving i) in Section 4.2. After choosing a complement i^c, the paper constructs a Voronov dataset and obtains an L∞[1]-algebra (claimed to be a dgL[1]a) on C•(g; i*⊗i^c); Theorem 5.3 establishes a bijection between its degree-zero Maurer–Cartan elements and small deformations of the ideal. Theorem 5.5 gives an L∞[1]-algebra controlling simultaneous deformations of the bracket and the ideal. Section 6 studies obstructions via a Kuranishi map, proves a stability theorem (Theorem 6.21) under H^1(i⊳g)=0, and claims a rigidity theorem (Theorem 6.14) under a surjectivity condition on H^0(Π). The paper is well-written, and the infinitesimal and stability parts are convincing; the rigidity theorem has a proof gap.","tokens_in":49275,"tokens_out":39688,"duration_ms":357286,"significance":"The paper addresses a real gap in the literature, since deformations of Lie ideals had not been treated systematically. The explicit identification of the infinitesimal deformation cocycles and the Maurer–Cartan description of small deformations are clean and likely useful. The systematic comparison of cohomology theories in Section 4.2 is valuable, as is the simultaneous-deformation L∞-algebra of Theorem 5.5. The stability theorem is a convincing application of standard zero-stability techniques. If the rigidity theorem is repaired, the paper will be a solid contribution to the deformation theory of Lie algebras. At present, the load-bearing proof of Theorem 6.14 is incomplete, which affects the advertised rigidity applications (Corollary 6.15 and Example 6.16).","major_comments":[{"comment":"The proof equates surjectivity of the cohomology map H^0(Π) with exactness of the tangent sequence (34) at C^0(g; i*⊗g/i). This is not justified: surjectivity of H^0(Π) gives Z^0(i⊳g) ⊆ Π(Z^1(g;g/i)), but exactness requires Z^0(i⊳g) ⊆ Π((π_{g/i})_*(Z^1(g;g))), i.e., that each 0-cocycle η be the restriction to i of a projection of an actual derivation ψ∈Z^1(g;g). The paper's diagram factors T_{Id_g}α_i through Z^1(g;g/i), but does not prove that every 1-cocycle φ∈Z^1(g;g/i) lifts to a derivation ψ with π∘ψ=φ. The obstruction to such a lift lies in H^2(g;i) via the long exact sequence of 0→C•(g;i)→C•(g;g)→C•(g;g/i)→0. Consequently, the non-degeneracy condition of Proposition 6.7 is not established, and the rigidity conclusion (and Corollary 6.15, Example 6.16) is unsupported. The theorem might be repairable with an additional hypothesis (e.g., H^2(g;i)=0) or a more refined argument; as written, this is a load-bearing gap.","section":"§6.2, Theorem 6.14 (proof, first paragraph)"},{"comment":"The assertion that 'an easy computation using I(ψ)∘I(φ)=0' implies m_k=0 for all k≥3 is not demonstrated. This vanishing is load-bearing: the paper advertises a dgL[1]a controlling deformations, and Theorem 5.3 relies on the Maurer–Cartan equation being exactly δ^Hom(φ)+½m2(φ,φ)=0. If any higher m_k is nonzero, the MC set of the L∞[1]-algebra would differ, and the bijection with small deformations would require an additional argument. Please provide the explicit computation (or a structural reason) for the vanishing of all m_k with k≥3.","section":"§5.1, after Eq. (28)"}],"minor_comments":[{"comment":"The indexing in the computation is inconsistent with the statement: the proof writes φ∈∧^k g*⊗g/i while the definition of Π uses C^{k+1}(g;g/i). This makes the proof hard to follow; please rewrite with consistent indices.","section":"§4.2.2, proof of Proposition 4.4"},{"comment":"The first sentence refers to 'Proposition 6.14', but the relevant statement is Proposition 6.10 (the Aut(g)-equivariance of σ). Please correct the reference.","section":"§6.2, proof of Theorem 6.14"},{"comment":"The symbol U_i⊥ is used for the neighborhood of i but is not defined; use a standard notation such as U_i or specify its meaning.","section":"§6.2, Definition 6.12"},{"comment":"The paper uses ⊲ in δ^Hom_{g⊲i} and ⊳ in H^•(i⊳g) to denote the ideal relation; please unify the notation.","section":"§4.1 (notational consistency)"},{"comment":"The notation 'h3(R) = : g' should be 'g := h3(R)' for readability.","section":"Example 6.4"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the rigidity theorem (Theorem 6.14), whose proof contains a genuine gap between a cohomological surjectivity condition and a tangent-space exactness condition. The other parts of the paper, in particular the infinitesimal theory, the Maurer–Cartan bijection, and the stability theorem, are solid and valuable. I recommend major revision: the authors should either repair the proof (possibly by adding a hypothesis such as H^2(g;i)=0) or adjust the statement and clearly indicate which rigidity consequences remain valid. The forward reference to Theorem 6.14 in Example 6.3 should also be revisited once the theorem is fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core of the paper is the construction of a deformation complex for Lie ideals and the proof that it controls small deformations. That part is real and, against the cited literature, new: C*(g; i*⊗g/i), the dgLa governed by m1=δ and m2 from the Voronov dataset, and the Maurer-Cartan bijection with graph(φ) in Theorem 5.3. The computations are explicit and check out, and the comparison with subalgebra and morphism deformation cohomologies is genuinely useful. Proposition 4.2, giving infinitesimal deformations as 0-cocycles, is solid. The simultaneous L∞-algebra and the Kuranishi obstruction material are standard but carefully executed. This paper fills a real gap in the deformation theory of Lie ideals.\n\nThe weak point is Theorem 6.14. The proof wants to apply Proposition 6.7 to the Aut(g)-action, and for that it needs exactness at Ti Grk(g): the image of Z^1(g;g) under ψ ↦ π∘ψ|i should equal Z^0(i⊳g). The argument reduces this to surjectivity of H^0(Π). But surjectivity only says that every 0-cochain is the restriction of a 1-cocycle φ: g → g/i. It does not say that φ lifts to a derivation ψ: g → g with π∘ψ = φ. The lift is obstructed in H^2(g;i), via the long exact sequence for 0 → C•(g;i) → C•(g;g) → C•(g;g/i) → 0. Until that lifting step is justified, or a stronger hypothesis is imposed, Corollary 6.15 is not supported by the given proof. This is a real gap in one advertised application, but it does not damage the controlling-algebra construction in Section 5.\n\nThe stability statement overlaps with Singh's result, and the authors disclose this in Remark 6.22; that is honest and fine. The paper is long and at times more detailed than necessary, but that is a minor issue.\n\nWho this is for: people working on deformation theory of Lie structures, who will get a new cohomology and a usable dgLa for ideals. It deserves a serious referee. Send it to review, with the request that the referee check the rigidity section carefully; if Theorem 6.14 is repaired or its hypothesis strengthened, this should be publishable.","headline":"Core construction (controlling dgLa for ideal deformations) is new and mostly sound; the rigidity theorem has a genuine lifting gap in its proof.","tokens_in":49859,"tokens_out":3463,"would_cite":true,"duration_ms":38143,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17B56","14D15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes a deformation theory for Lie ideals, with a controlling dgL[1]a whose Maurer-Cartan elements are exactly the small deformations of an ideal.","keywords":["deformation theory","Lie ideals","dgL[1]-algebras","L-infinity algebras","Maurer-Cartan elements","rigidity","stability","Chevalley-Eilenberg cohomology"],"falsifier":"Find a Lie algebra $\\mathfrak{g}$ and an ideal $\\mathfrak{i}$ such that $H^0(\\Pi)$ is surjective but $\\mathfrak{i}$ is not $\\mathrm{Aut}(\\mathfrak{g})$-rigid; equivalently, exhibit an element $\\eta \\in Z^0(\\mathfrak{i} \\rhd \\mathfrak{g})$ that lies in the image of the restriction map from $Z^1(\\mathfrak{g};\\mathfrak{g}/\\mathfrak{i})$ but not in the image of the map $Z^1(\\mathfrak{g};\\mathfrak{g}) \\to T_{\\mathfrak{i}}\\mathrm{Gr}_k(\\mathfrak{g})$. A concrete place to look is a nilpotent or solvable Lie algebra where the automorphism group is computable and the tangent sequence can be checked by hand.","tokens_in":48750,"feed_emoji":"","tokens_out":6993,"duration_ms":63011,"temperature":0.7,"pith_summary":"This paper develops a deformation theory for Lie ideals, treating an ideal as a structure to be deformed rather than merely as a Lie subalgebra. The central claim is that smooth one-parameter deformations of an ideal $\\mathfrak{i} \\subset \\mathfrak{g}$ differentiate to cocycles in the cohomology of $\\mathfrak{g}$ with coefficients in $\\operatorname{Hom}(\\mathfrak{i}, \\mathfrak{g}/\\mathfrak{i})$, and that, after choosing a vector-space complement, the full deformation problem is controlled by a differential graded Lie algebra whose Maurer-Cartan elements are exactly the small deformations. If correct, this gives the missing infinitesimal and formal framework for studying how many ideals a Lie algebra has and how they sit inside it, and yields sufficient cohomological conditions for rigidity and stability of ideals.","feed_headline":"Lie ideals get a controlling deformation theory","feed_subtitle":"A cohomology built from Hom(i,g/i) tracks infinitesimal deformations, rigidity, and stability of ideals in Lie algebras.","key_machinery":"The load-bearing object is the pair consisting of the ideal deformation cohomology $H^\\bullet(\\mathfrak{i} \\rhd \\mathfrak{g}) = H^\\bullet_{\\delta^{\\mathrm{Hom}}}(\\mathfrak{g}; \\mathfrak{i}^* \\otimes \\mathfrak{g}/\\mathfrak{i})$ and the dgL[1]a $(C^\\bullet(\\mathfrak{g};\\mathfrak{i}^* \\otimes \\mathfrak{i}^c), \\delta^{\\mathrm{Hom}} = m_1, m_2)$ built from a Voronov dataset $(\\mathsf L, \\mathfrak{a}, P, \\Theta)$, where $\\mathsf L = C^\\bullet(\\mathfrak{g};\\mathfrak{g})[1] \\oplus C^\\bullet(\\mathfrak{g};\\mathfrak{gl}(\\mathfrak{g}))$, $\\mathfrak{a} = C^\\bullet(\\mathfrak{g};\\mathfrak{i}^* \\otimes \\mathfrak{i}^c)$, $P$ is the projection onto the $\\mathfrak{i}^c$-components, and $\\Theta = \\mu_{\\mathfrak{g}} + \\operatorname{ad}_{\\mathfrak{g}}$ is the Maurer-Cartan element encoding the Lie bracket and its adjoint representation. The derived-bracket construction converts this datum into multibrackets whose Maurer-Cartan equation is exactly the condition that $\\operatorname{graph}(\\varphi)$ be an ideal, which is the mechanism turning the infinitesimal cohomology into a full formal deformation theory.","core_discovery":"The paper's discovery is that the deformation problem of an ideal $\\mathfrak{i} \\triangleleft \\mathfrak{g}$ is governed by the Chevalley-Eilenberg complex $C^\\bullet(\\mathfrak{g}; \\mathfrak{i}^* \\otimes \\mathfrak{g}/\\mathfrak{i})$ with the representation $\\operatorname{ad}^{\\mathrm{Hom}}$ obtained from the adjoint actions on $\\mathfrak{i}$ and on $\\mathfrak{g}/\\mathfrak{i}$. Smooth deformations differentiate to $0$-cocycles (Proposition 4.2), so $H^0(\\mathfrak{i} \\rhd \\mathfrak{g})$ consists of infinitesimal deformations. With a complement $\\mathfrak{i}^c$ chosen, the authors construct a Voronov dataset whose derived brackets give a dgL[1]a structure with $m_1 = \\delta^{\\mathrm{Hom}}_{\\mathfrak{g} \\rhd \\mathfrak{i}}$ and an explicit $m_2$; Theorem 5.3 states that degree-zero Maurer-Cartan elements of this dgL[1]a are in bijection with small deformations via $\\varphi \\mapsto \\operatorname{graph}(\\varphi)$. The same Voronov data also produces an $L_\\infty[1]$-algebra controlling simultaneous deformations of the ideal and of the ambient Lie bracket (Theorem 5.5). On the geometric side, the Kuranishi map $[\\eta] \\mapsto \\tfrac{1}{2}[m_2(\\eta,\\eta)]$ detects obstructions, surjectivity of $H^0(\\Pi) : H^1_{\\pi_{\\mathfrak{g}/\\mathfrak{i}}}(\\mathfrak{g};\\mathfrak{g}/\\mathfrak{i}) \\to H^0(\\mathfrak{i} \\rhd \\mathfrak{g})$ is claimed to imply $\\mathrm{Aut}(\\mathfrak{g})$-rigidity (Theorem 6.14), and vanishing of $H^1(\\mathfrak{i} \\rhd \\mathfrak{g})$ is shown to imply stability (Theorem 6.21).","pith_inferences":["The framework suggests a relative deformation theory in which one deforms the pair $(\\mathfrak{g},\\mathfrak{i})$ rather than fixing $\\mathfrak{g}$; the simultaneous $L_\\infty[1]$-algebra of Theorem 5.5 is the first step in that direction, and the paper leaves the corresponding Kuranishi map for the pair unexplored.","Because the deformation complex is a Chevalley-Eilenberg complex of $\\mathfrak{g}$ with coefficients in $\\mathfrak{i}^* \\otimes \\mathfrak{g}/\\mathfrak{i}$, standard cohomological vanishing theorems can likely be applied directly to produce further examples of rigid or stable ideals, beyond the semisimple cases treated in the corollaries.","The Heisenberg-center example shows that obstruction as an ideal differs from obstruction as a subalgebra; this suggests a hierarchy of deformation problems for the same geometric object, with different controlling algebras and different Kuranishi maps, whose mutual relations are only partially captured by the comparison diagrams in Section 4.2.","The rigidity theorem's missing lifting step might be repaired by adding a hypothesis on the kernel of the tangent map $T_{\\mathrm{Id}}\\alpha_{\\mathfrak{i}}$ or by requiring the projection cocycle to be a derivation; nilpotent Lie algebras, where automorphisms are computable, offer a concrete testing ground. "],"forward_implications":["Any smooth deformation of an ideal differentiates to a $0$-cocycle in $C^\\bullet(\\mathfrak{g};\\mathfrak{i}^* \\otimes \\mathfrak{g}/\\mathfrak{i})$, so $H^0(\\mathfrak{i} \\rhd \\mathfrak{g})$ is the space of infinitesimal deformations of the ideal.","Small deformations of an ideal, relative to a fixed complement, are exactly the Maurer-Cartan elements of the constructed dgL[1]a; the deformation problem is therefore controlled by a differential graded Lie algebra, not merely by a cochain complex.","The Kuranishi map $\\operatorname{Kur}_{\\mathfrak{i} \\rhd \\mathfrak{g}} : H^0(\\mathfrak{i} \\rhd \\mathfrak{g}) \\to H^1(\\mathfrak{i} \\rhd \\mathfrak{g})$ sends an infinitesimal deformation to its first obstruction; if it is nonzero, that infinitesimal deformation cannot be integrated to a smooth deformation of the ideal.","If $H^0(\\Pi)$ is surjective, the ideal is topologically rigid under the automorphism group of $\\mathfrak{g}$; in particular, rigidity follows whenever $H^2(\\mathfrak{g}/\\mathfrak{i};\\mathfrak{g}/\\mathfrak{i}) = 0$.","If $H^1(\\mathfrak{i} \\rhd \\mathfrak{g}) = 0$, the ideal is stable: nearby Lie brackets on $\\mathfrak{g}$ admit nearby ideals, and the space of ideals is locally a manifold of dimension $\\dim Z^0(\\mathfrak{i} \\rhd \\mathfrak{g})$. "],"supporting_citations":[{"why":"Supplies the deformation cohomology of Lie subalgebras, which is the baseline against which the ideal deformation cohomology is compared.","marker":"[35]"},{"why":"Provides the stability/rigidity methods, including openness of orbits and stability of zeros, that the geometric applications in Section 6 adapt to ideals.","marker":"[4]"},{"why":"Provides the Voronov dataset and derived-bracket construction used to build the controlling dgL[1]a and the simultaneous $L_\\infty[1]$-algebra.","marker":"[8]"},{"why":"Introduces higher derived brackets producing $L_\\infty[1]$-algebras from Voronov data, the foundation of Theorem 2.8 and Theorem 2.10.","marker":"[40]"},{"why":"Extends the derived-bracket construction to arbitrary derivations, used in the version of Theorem 2.10 applied to the ideal deformation problem.","marker":"[41]"},{"why":"Provides the Maurer-Cartan description of Lie algebra structures and the complex $C^\\bullet_{\\mathfrak{i}}(\\mathfrak{g};\\mathfrak{g})$ controlling deformations of the bracket that keep $\\mathfrak{i}$ an ideal.","marker":"[31]"},{"why":"Gives an alternative proof of stability under a weaker condition, cited in Remark 6.22 to improve Theorem 6.21.","marker":"[39]"},{"why":"Supplies Whitehead's lemmas, used to conclude that semisimple quotients have vanishing second cohomology and that semisimple Lie algebras yield stable ideals.","marker":"[17]"}],"fun_headline_variants":["Hom(i,g/i) cohomology governs Lie ideal deformations","Maurer-Cartan elements track deformations of Lie ideals","Rigidity and stability of Lie ideals via deformation cohomology","Simultaneous control of Lie ideals and brackets via L-infinity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The rigidity theorem rests on the assumption that every infinitesimal deformation of the ideal obtained by restricting a cocycle of the projection $\\mathfrak{g} \\to \\mathfrak{g}/\\mathfrak{i}$ is tangent to an actual automorphism of $\\mathfrak{g}$; the paper asserts this identification through a diagram without proving that projection cocycles lift to derivations.","fun_headline_variants_meta":{"raw":{"variants":["Hom(i,g/i) cohomology governs Lie ideal deformations","Maurer-Cartan elements track deformations of Lie ideals","Rigidity and stability of Lie ideals via deformation cohomology","Simultaneous control of Lie ideals and brackets via L-infinity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00096,"raw_usage":{"total_tokens":4226,"prompt_tokens":1217,"completion_tokens":3009,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":833,"completion_tokens_details":{"reasoning_tokens":2935}},"tokens_in":833,"tokens_out":3009,"duration_ms":23892,"temperature":1.0,"reasoning_tokens":2935,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:19:58.207968+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a Lie algebra $\\mathfrak{g}$ and an ideal $\\mathfrak{i}$ such that $H^0(\\Pi)$ is surjective but $\\mathfrak{i}$ is not $\\mathrm{Aut}(\\mathfrak{g})$-rigid; equivalently, exhibit an element $\\eta \\in Z^0(\\mathfrak{i} \\rhd \\mathfrak{g})$ that lies in the image of the restriction map from $Z^1(\\mathfrak{g};\\mathfrak{g}/\\mathfrak{i})$ but not in the image of the map $Z^1(\\mathfrak{g};\\mathfrak{g}) \\to T_{\\mathfrak{i}}\\mathrm{Gr}_k(\\mathfrak{g})$. A concrete place to look is a nilpotent or solvable Lie algebra where the automorphism group is computable and the tangent sequence can be checked by hand.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the deformation cohomology of Lie subalgebras, which is the baseline against which the ideal deformation cohomology is compared."},{"cited_title":"Crainic, F","cited_arxiv_id":null,"evidence_quote":"Provides the stability/rigidity methods, including openness of orbits and stability of zeros, that the geometric applications in Section 6 adapt to ideals."},{"cited_title":"Fr´ egier and M","cited_arxiv_id":null,"evidence_quote":"Provides the Voronov dataset and derived-bracket construction used to build the controlling dgL[1]a and the simultaneous $L_\\infty[1]$-algebra."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces higher derived brackets producing $L_\\infty[1]$-algebras from Voronov data, the foundation of Theorem 2.8 and Theorem 2.10."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends the derived-bracket construction to arbitrary derivations, used in the version of Theorem 2.10 applied to the ideal deformation problem."},{"cited_title":"Nijenhuis and R","cited_arxiv_id":null,"evidence_quote":"Provides the Maurer-Cartan description of Lie algebra structures and the complex $C^\\bullet_{\\mathfrak{i}}(\\mathfrak{g};\\mathfrak{g})$ controlling deformations of the bracket that keep $\\mathfrak{i}$ an ideal."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives an alternative proof of stability under a weaker condition, cited in Remark 6.22 to improve Theorem 6.21."},{"cited_title":"Jacobson","cited_arxiv_id":null,"evidence_quote":"Supplies Whitehead's lemmas, used to conclude that semisimple quotients have vanishing second cohomology and that semisimple Lie algebras yield stable ideals."}],"review_version":1}