{"id":"745c0918-78c1-44be-994d-51f5f95eefb9","arxiv_id":"2411.17093","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Schur-Weyl duality yields a surjective map from tensor-algebra invariants to the center of U(g) for gl_{m|n}, q_n, osp_{2m+1|2n}, and p_n, and a new proof that Z(p_n)=0.","lead":"This paper shows that, for four families of classical Lie superalgebras, every element of the center of the enveloping algebra is the projection of an invariant of the tensor algebra, and it gives a unified construction using Schur-Weyl duality. It also provides a new combinatorial proof that the periplectic superalgebra p_n has trivial center.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main theorem leans on the imported surjectivity of Ψ_k (Theorem 2.9); for p_n this is the deep periplectic FFT, and the paper does not pin down its exact hypotheses.","rationale":"The reader's weakest assumption correctly identifies the surjectivity of Ψ_k as the load-bearing external input. I found no internal contradiction in the paper's main chain of argument: the split projection End(V)→g, the isomorphism Ω, the symmetrization comparison, and the sign computations in Key Lemma 3.19 are coherent, and the osp_{2m|2n} exception is handled consistently. The p_n case is the most delicate because it depends on the full first fundamental theorem for the periplectic supergroup, a deep result that is cited rather than proved. However, this is a known theorem and the paper's conclusion for p_n is independently known (Scheunert), so the risk is not that the theorem is false, but that the paper's proof may silently inherit an unstated hypothesis. If the cited FFT is as strong as stated, Theorem 3.23 follows. Accordingly, my concern does not change the reader's ACCEPT verdict; it strengthens the case for moderate confidence rather than high confidence.","tokens_in":31878,"tokens_out":41316,"duration_ms":366673,"concrete_test":"Check the original statements: (a) In Coulembier, Proc. LMS 117 (2018), Lemma 8.1.4, and Deligne–Lehrer–Zhang, Adv. Math. 327 (2018), Sec. 5.3, determine whether B_k^-(0) → End_{p_n}(V^{⊗k}) is asserted surjective for all n,k or only in a stable range. (b) As an independent computational spot check, compute the dimension of End_{p_2}(V^{⊗3}) from the defining equations of p_2-invariants and compare with dim B_3^-(0)=15; a mismatch would refute the spanning claim in §3.4, while a match supports it.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.23 works by showing that the image of η' contains a spanning set of Z(g). That spanning set is manufactured as η'∘π∘Ω∘Ψ(a); hence every step in the chain must be surjective on invariants for the conclusion to follow. For gl_{m|n}, q_n and osp_{2m+1|2n} the surjectivity of Ψ_k is a classical Schur-Sergeev duality fact, cited as Theorem 2.9. For p_n, however, the required statement is that the Periplectic Brauer algebra B_k^-(0) maps onto End_{p_n}(V^{⊗k}) for every k; this is the first fundamental theorem for the periplectic supergroup, imported from [5, Lemma 8.1.4] and [7, Section 5.3]. The paper does not state the exact rank/degree conditions under which those sources prove the theorem. If the periplectic FFT has a hidden stability range (e.g., k ≤ n), or if it only holds for the algebraic supergroup rather than the Lie superalgebra p_n, then the set {σ·c^{⊗k}} in Section 3.4 may fail to span (V^{⊗2k})^{p_n}. In that event the proof of Theorem 3.18 would not establish triviality of Z(p_n), and the p_n case of Theorem 3.23 would not be supported by the paper's argument. The rest of the chain—splitness of End(V)→g, the isomorphism Ω, and the sign bookkeeping in Key Lemma 3.19—appears internally consistent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a uniform Schur-Weyl framework for the classical Lie superalgebras g in {gl_{m|n}, q_n, osp_{m|2n}, p_n}. Using the split surjection pi~: End(V) -> g of Proposition 2.1 and the canonical isomorphism Omega_k: End_g(V^{⊗k}) ≅ [End(V)^{⊗k}]^g, each element of the relevant centralizer algebra A_k (symmetric group algebra, Hecke-Clifford algebra, Brauer algebra, or periplectic Brauer algebra) is converted into a g-invariant of T(g), S(g), and U(g). The main results are: (i) Theorem 3.4 and Proposition 3.15, relating the supersymmetrization of invariants to sums of z_σ = η'∘π∘Ω∘Ψ(σ); (ii) Theorem 3.23, asserting that the canonical projection η': T(g)^g -> Z(g) is surjective for gl_{m|n}, q_n, osp_{2m+1|2n}, and p_n, with osp_{2m|2n} as the sole exception; (iii) generators of Z(g) in the non-exceptional cases (Str E-hat^k for gl_{m|n}; Z_k with k odd for q_n; Str F-hat^{2k} for osp_{2m+1|2n}), verified through the Harish-Chandra isomorphism; and (iv) Theorem 3.18, a new proof, based on the combinatorial Key Lemma 3.19 about Brauer diagrams, of Scheunert's theorem that Z(p_n) is trivial.","tokens_in":69,"tokens_out":59483,"duration_ms":606705,"significance":"If the results hold, the paper provides a genuinely unified construction of Gelfand-type invariants for four families of classical Lie superalgebras, and its explicit generators reproduce and tie together known theorems (Sergeev for Z(q_n), Scheunert for Z(p_n), Molev-type invariants for gl_{m|n} and osp_{2m+1|2n}); the honest scoping (osp_{2m|2n} excluded, in agreement with the super-Pfaffian phenomenon) is a strength. The approach is also cross-validated: the Harish-Chandra isomorphism is used as an independent check on generation, and Example 3.24 gives a fully explicit sign computation in the p_n case. The principal new technical content is the reduction of the triviality of Z(p_n) to a symmetric-group/Brauer-diagram lemma; this is the part that most needs careful verification (see Major comments 1 and 3). The p_n argument's reliance on the imported periplectic FFT qualifies the paper's 'purely algebraic' framing, since the completeness of the spanning set comes from a deep external theorem rather than from the elementary part of the proof.","major_comments":[{"comment":"The p_n clauses of Theorems 3.18 and 3.23 rest on the assertion that {σ·c^{⊗k}}_{σ∈S_{2k}} spans (V^{⊗2k})^{p_n} for every k (Section 3.4), a deep result: the first fundamental theorem (FFT) for the periplectic supergroup, imported from [5, Lemma 8.1.4] and [7, Section 5.3]. The manuscript does not state the exact hypotheses under which those sources prove the theorem, in particular whether k is unrestricted or a stability range k ≤ n is required, and whether the cited results are for the supergroup P(n) rather than the Lie superalgebra p_n (with the equality of the two invariant spaces on V^{⊗2k} left implicit). Theorem 2.9 packages the same surjectivity claim as a black box labeled '[Schur-Sergeev duality]' with no citation. Since the spanning claim is needed in every degree of S(p_n) for the triviality conclusion, this load-bearing import must be stated precisely: the exact theorem, its hypotheses, and the identification of supergroup and Lie-superalgebra invariants. If the periplectic FFT had a hidden restriction (for instance n ≥ k), the proof of Theorem 3.18 as written would not establish the full result.","section":"Section 3.4 / Theorem 2.9"},{"comment":"The statement 'By [7, Section 3], the set {σ·c^{⊗k}} spans (V^{⊗2k})^{OSP(V)}' is made for the ortho-symplectic supergroup without qualification. This conflicts with the super-Pfaffian exception for osp_{2m|2n}, which the paper itself acknowledges in the Introduction and in Theorem 2.9; for those algebras Brauer-type spanning is known to fail. The argument in Section 3.3 is used only to reach the conclusion for osp_{2m+1|2n}, so the theorem is not endangered, but the spanning claim should be stated in the precise form that is actually true and actually used (odd first index), with the citation matched to that case.","section":"Section 3.3"},{"comment":"Key Lemma 3.19 is the technical core of the advertised new proof of the triviality of Z(p_n), and Theorem 3.18 reduces to it. Its proof, however, is completed by several steps that are only asserted: the reduction of the general case to a single circle, the claim that the lemma propagates from S_{2l} × S_l to S_{2k} × S_k when λ_l > 0, and the final 'one can check' verification for σ = (2l 2l-1···3 2). The statements (I)-(III) about the actions on Brauer diagrams and the double-coset decomposition (3.42) also need justification. The lemma may well be true (the small cases k = 1, 2 check out), but the written proof is not complete enough for a reader to verify, which is a real defect in a paper whose central selling point is this elementary reduction; please provide full details or cite a published proof.","section":"Key Lemma 3.19 / Theorem 3.18"}],"minor_comments":[{"comment":"The paragraph beginning 'Moreover, End(V) = q_n ⊕ q_n^⊥ ...' is duplicated verbatim; please delete the repetition.","section":"Section 2.1.2"},{"comment":"The root systems listed for the two ortho-symplectic families are scrambled; for example, the even roots given for osp_{2m+1|2n} include ±2ǫ_p, which is not a root of type B_m, and the second positive system likewise needs correction (the correct data are D_m × C_n for osp_{2m|2n} and B_m × C_n for osp_{2m+1|2n}).","section":"Section 2.1.3"},{"comment":"In the conclusion 'the set {ψ∘η∘π(θ_σ) | σ ∈ S_k, k ∈ Z+} spans the center Z(osp_{2m+1|2n})', the index set should be σ ∈ S_{2k}, since θ_σ is defined for σ ∈ S_{2k} in this section.","section":"Section 3.3"},{"comment":"The statement writes (1/k!)Σ z_{τ^{-1}στ} while the proof derives (1/k!)Σ z_{τ^{-1}σ^{-1}τ}; the two forms are equivalent as σ ranges over S_k, but statement and proof should be aligned for readability.","section":"Theorem 3.4"},{"comment":"In the sentence 'spans all the invariants in S(p_n)' (and its analogue in Section 3.3), the invariant subalgebra should be written S(p_n)^{p_n} explicitly, and the sentence should indicate that the FFT input discussed in Major comment 1 is being used at that point.","section":"Sections 3.3-3.4"},{"comment":"The step from 'top-degree components of the Harish-Chandra images generate S(h)^W' to 'the elements generate Z(g)' uses the standard associated-graded filtration argument; please state it in one sentence for completeness.","section":"Corollaries 3.9 and 3.17"},{"comment":"The exception in Theorem 2.9 and the formula in (3.14) would be clearer stated in terms of the Brauer parameter: for g = osp_{m|2n} the parameter is δ = m − 2n, and the surjectivity of Ψ_k fails precisely when δ is an even integer (equivalently, the first index is even).","section":"Theorem 2.9 / equation (3.14)"},{"comment":"Proposition 2.1 is referred to in Section 3 as 'Lemma 2.1'; please fix the cross-reference, and please render the commutative diagrams in the Introduction with standard diagram macros, as the current arrows do not display.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"I was not able to consult [5, Lemma 8.1.4] and [7, Section 5.3] independently; if the periplectic FFT holds unconditionally as the manuscript's Section 3.4 requires, the mathematical conclusions appear sound and consistent with known theorems, and the paper is publishable after revision. The editor may wish to have the combinatorial Key Lemma 3.19 checked by an expert in diagram algebras, since it is the centerpiece of the advertised new proof of Scheunert's theorem and the written proof is compressed. Also note that a large part of the output (generators of Z(q_n), triviality of Z(p_n), Gelfand invariants for gl_{m|n} and osp_{2m+1|2n}) consists of known results recovered by a new method; the durable novelty is the uniform construction and the new p_n proof, whose 'elementary' framing should be tempered by its dependence on the deep periplectic FFT."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid, useful paper. It sets up a single Schur-Weyl framework that turns centralizer elements into invariant elements of T(g), S(g), and U(g), and proves that the map η' from T(g)^g to Z(g) is surjective for gl_{m|n}, q_n, osp_{2m+1|2n}, and p_n, with osp_{2m|2n} left out for the right reason. The p_n part is the most substantial: a purely algebraic proof that Z(p_n) is trivial, replacing Scheunert's geometric argument with the combinatorial Key Lemma 3.19. The sign bookkeeping in Example 3.24 checks out, and the generators for the center of q_n and osp_{2m+1|2n} are consistent with Sergeev and others.\n\nWhat is genuinely new is the unifying diagram and the surjectivity theorem as a package. The individual generator systems were known; the paper makes the connection explicit and shows how all central elements come from the centralizer algebras.\n\nThe main caveat is that the p_n case rests on the surjectivity of the Periplectic Brauer algebra action, imported from [5] and [7]. The paper states this as part of Theorem 2.9 but does not list the exact hypotheses—all k, or only k ≤ n? If those sources only prove the FFT for a restricted range, the p_n argument would not cover all degrees. I did not find evidence of such a restriction in the literature; Coulembier's paper appears to give the full centralizer theorem. Still, the authors should state the precise conditions and verify that k is arbitrary. This is a clarification, not a fatal gap.\n\nThere are also some typographical blemishes and a duplicated paragraph in Section 2.1.2, and the proof of Key Lemma 3.19 is dense enough to need careful checking. These are minor.\n\nOverall, I think this deserves a serious referee. The central argument is coherent, the results are consistent with known theorems, and the unified perspective is worth publishing. I would send it to peer review with a request to strengthen the citation of the p_n FFT and to polish the text.","headline":"A genuinely useful unified Schur-Weyl machine for central elements of four classical Lie superalgebras; the p_n triviality proof is a real alternative, though the imported periplectic FFT needs its hypotheses made explicit.","tokens_in":96,"tokens_out":9829,"would_cite":true,"duration_ms":152413,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17B35","17B10","20C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for the classical Lie superalgebras $\\mathfrak{gl}_{m|n}$, $\\mathfrak{q}_n$, $\\mathfrak{osp}_{2m+1|2n}$, and $\\mathfrak{p}_n$, every central element of the universal enveloping algebra is the image of a…","keywords":["Classical Lie superalgebras","Schur-Weyl duality","Brauer diagrams","Casimir elements","Gelfand invariants","center of the enveloping algebra","periplectic Lie superalgebra","queer Lie superalgebra"],"falsifier":"A direct low-degree computation for $\\mathfrak{g}=\\mathfrak{gl}_{1|1}$ would settle the main claim: list the $\\mathfrak{g}$-invariants of $T(\\mathfrak{g})$ through degree 4, apply the projection $\\eta'$, and compare the resulting span with the two-variable supersymmetric polynomials that form $\\mathcal{Z}(\\mathfrak{gl}_{1|1})$ by the Harish-Chandra isomorphism; any central element outside the span disproves Theorem 3.23.","tokens_in":31673,"feed_emoji":"🧮","tokens_out":16717,"duration_ms":136001,"temperature":0.7,"pith_summary":"This paper asks whether the Schur-Weyl machine produces every central element of the enveloping algebra of a classical Lie superalgebra. The answer proposed here is yes for $\\mathfrak{gl}_{m|n}$, $\\mathfrak{q}_n$, $\\mathfrak{osp}_{2m+1|2n}$, and $\\mathfrak{p}_n$: the canonical projection from the tensor algebra $T(\\mathfrak{g})$ to the enveloping algebra $\\mathrm{U}(\\mathfrak{g})$ stays surjective after passing to $\\mathfrak{g}$-invariants. Consequently every Casimir element can be built from elements of the corresponding centralizer algebra, giving a uniform source of central generators. The same machinery also yields explicit Gelfand invariants, recovers the known generators of the centers for the general linear and queer cases, and supplies an elementary proof that the periplectic superalgebra $\\mathfrak{p}_n$ has trivial center.","feed_headline":"Schur-Weyl duality yields all Casimirs for four superalgebra families","feed_subtitle":"For four classical Lie superfamilies, every central element of the enveloping algebra is a projected g-invariant tensor.","key_machinery":"The load-bearing mechanism is the super Schur-Weyl duality map $\\Psi_k\\colon A_k\\to\\operatorname{End}_{\\mathfrak{g}}(V^{\\otimes k})$, where $A_k$ is the symmetric group algebra for $\\mathfrak{gl}_{m|n}$, the Hecke-Clifford algebra for $\\mathfrak{q}_n$, the Brauer algebra for $\\mathfrak{osp}_{2m+1|2n}$, and the periplectic Brauer algebra for $\\mathfrak{p}_n$. Its surjectivity (imported as Theorem 2.9, with $\\mathfrak{osp}_{2m|2n}$ excluded) guarantees that every equivariant endomorphism of the tensor space is accessible from the diagram algebra, and the split projection $\\tilde{\\pi}$ converts each such endomorphism into a $\\mathfrak{g}$-invariant tensor. The supersymmetrization isomorphism from the supersymmetric algebra to the enveloping algebra then lifts these tensor invariants to central elements. For $\\mathfrak{p}_n$, the decisive step is Key Lemma 3.19, a sign identity for permutations in $S_{2k}$ established by drawing the closure of a Brauer diagram; it forces every $\\mathfrak{p}_n$-invariant in the supersymmetric algebra to equal its negative, so it vanishes.","core_discovery":"The central claim is Theorem 3.23: for $\\mathfrak{g}=\\mathfrak{gl}_{m|n}, \\mathfrak{q}_n, \\mathfrak{osp}_{2m+1|2n}, \\mathfrak{p}_n$, the restriction of the projection $\\eta'\\colon T(\\mathfrak{g})\\to \\mathrm{U}(\\mathfrak{g})$ to $T(\\mathfrak{g})^{\\mathfrak{g}}$ is surjective onto the center $\\mathcal{Z}(\\mathfrak{g})$; the family $\\mathfrak{osp}_{2m|2n}$ is the sole case left unresolved. The proof routes every element of the centralizer algebra $\\operatorname{End}_{\\mathfrak{g}}(V^{\\otimes k})$ through a split projection $\\tilde{\\pi}\\colon \\operatorname{End}(V)\\to\\mathfrak{g}$, turning it into a $\\mathfrak{g}$-invariant of $T(\\mathfrak{g})$, and then shows that supersymmetrization converts these without loss into central elements. Along the way the paper constructs explicit Gelfand invariants for $\\mathfrak{gl}_{m|n}$ and $\\mathfrak{osp}_{m|2n}$, identifies the odd-degree elements $Z_k$ as generators of $\\mathcal{Z}(\\mathfrak{q}_n)$, and proves $\\mathcal{Z}(\\mathfrak{p}_n)=0$ using a Brauer-diagram sign lemma.","pith_inferences":["The authors leave open the dimension of $(\\mathfrak{g}^{\\otimes k})^{\\mathfrak{g}}$; the same comparison between tensor, supersymmetric, and enveloping invariants suggests an upper bound in terms of the dimension of the relevant centralizer algebra, which would make the open problem computable.","For $\\mathfrak{osp}_{2m|2n}$, the obstruction is the known failure of the Schur-Weyl map to be surjective; a natural extension is to add the missing endomorphisms to the construction and test whether the surjectivity of $\\eta'$ still holds for that family.","The Brauer-diagram sign lemma proving the triviality of $\\mathcal{Z}(\\mathfrak{p}_n)$ is a self-contained statement about $S_{2k}$; similar sign identities could control invariants of other superalgebras with an odd invariant bilinear form and could be checked on small $k$ before seeking a general theorem.","The coexistence of nontrivial $\\mathfrak{p}_n$-invariants in the tensor algebra with a trivial center suggests that tensor invariants encode module-theoretic or categorical information invisible to Casimir elements, a distinction that the construction makes sharp."],"forward_implications":["For $\\mathfrak{gl}_{m|n}$, the elements $\\operatorname{Str}(\\hat E^k)$ for $k\\ge1$ generate the center $\\mathcal{Z}(\\mathfrak{gl}_{m|n})$, giving a direct bridge from $k$-cycles in the symmetric group to Casimir generators.","For $\\mathfrak{q}_n$, the elements $Z_k$ with odd $k$ generate $\\mathcal{Z}(\\mathfrak{q}_n)$, while even-degree cycles collapse to zero; this recovers the known center of the queer enveloping algebra.","For $\\mathfrak{osp}_{2m+1|2n}$, the elements $\\operatorname{Str}(\\hat F^{2k})$ for $k\\ge1$ generate the center, extending the orthosymplectic Gelfand-invariant picture.","The center of the periplectic enveloping algebra $\\mathrm{U}(\\mathfrak{p}_n)$ is trivial, even though $T(\\mathfrak{p}_n)^{\\mathfrak{p}_n}$ contains many nonzero invariants.","The same construction uniformly produces the central generators for the four families, showing that the surjectivity of $\\eta'$ is not a case-by-case accident but a consequence of the super Schur-Weyl duality."],"supporting_citations":[{"why":"Standard reference for double centralizer theory and Schur-Weyl-Sergeev duality for Lie superalgebras; supplies the background for the surjectivity theorem used throughout.","marker":"[3]"},{"why":"Establishes Schur-Weyl duality for the Brauer algebra and the ortho-symplectic Lie superalgebra; supplies the surjectivity of the action map needed for the orthosymplectic case.","marker":"[6]"},{"why":"First fundamental theorem of invariant theory for the orthosymplectic supergroup; gives the spanning sets of invariants from which the Brauer-diagram construction starts.","marker":"[7]"},{"why":"Shows how to build Casimir elements from the Brauer-Schur-Weyl duality; the pattern of turning centralizer-algebra cycles into central elements is extended here.","marker":"[9]"},{"why":"Provides the Gelfand invariants and Harish-Chandra image computations for classical Lie algebras that the paper adapts to the superalgebra setting.","marker":"[21]"},{"why":"Identifies the periplectic Brauer algebra as the centralizer of the periplectic Lie superalgebra action; supplies the surjectivity needed in that case.","marker":"[23]"},{"why":"Source for the Harish-Chandra isomorphism and the description of invariant supersymmetric polynomials used to prove that the constructed elements generate the center.","marker":"[24]"},{"why":"Centralizer construction of the Yangian of the queer Lie superalgebra; gives the elements whose odd-degree members generate the queer center.","marker":"[25]"},{"why":"Earlier result on invariant supersymmetric multilinear forms proving the triviality of the center of the periplectic enveloping algebra; the new proof here replaces that argument.","marker":"[29]"},{"why":"Earlier computation of the center of the enveloping algebra of the queer superalgebra; the generators obtained here match that description.","marker":"[30]"}],"fun_headline_variants":["All centers for four Lie superalgebras from Schur-Weyl duality","Schur-Weyl duality pinpoints all Casimirs for four superalgebras","Every center of four superfamilies arises from one duality","Schur-Weyl duality: all invariants for four Lie superalgebras","Four superalgebras: all centers from Schur-Weyl duality"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction assumes that the Schur-Weyl action map $\\Psi_k$ is surjective, meaning every linear endomorphism of $V^{\\otimes k}$ commuting with the superalgebra is realized by the relevant diagram algebra; this assumption is known to fail for $\\mathfrak{osp}_{2m|2n}$, which is exactly why that family is excluded, and for $\\mathfrak{p}_n$ it is a nontrivial theorem rather than a formality.","fun_headline_variants_meta":{"raw":{"variants":["All centers for four Lie superalgebras from Schur-Weyl duality","Schur-Weyl duality pinpoints all Casimirs for four superalgebras","Every center of four superfamilies arises from one duality","Schur-Weyl duality: all invariants for four Lie superalgebras","Four superalgebras: all centers from Schur-Weyl duality"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000906,"raw_usage":{"total_tokens":3933,"prompt_tokens":1017,"completion_tokens":2916,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":633,"completion_tokens_details":{"reasoning_tokens":2817}},"tokens_in":633,"tokens_out":2916,"duration_ms":20731,"temperature":1.0,"reasoning_tokens":2817,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:32:05.453575+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct low-degree computation for $\\mathfrak{g}=\\mathfrak{gl}_{1|1}$ would settle the main claim: list the $\\mathfrak{g}$-invariants of $T(\\mathfrak{g})$ through degree 4, apply the projection $\\eta'$, and compare the resulting span with the two-variable supersymmetric polynomials that form $\\mathcal{Z}(\\mathfrak{gl}_{1|1})$ by the Harish-Chandra isomorphism; any central element outside the span disproves Theorem 3.23.","supporting_citations":[{"cited_title":"Cheng, W","cited_arxiv_id":null,"evidence_quote":"Standard reference for double centralizer theory and Schur-Weyl-Sergeev duality for Lie superalgebras; supplies the background for the surjectivity theorem used throughout."},{"cited_title":"Ehrig, C","cited_arxiv_id":null,"evidence_quote":"Establishes Schur-Weyl duality for the Brauer algebra and the ortho-symplectic Lie superalgebra; supplies the surjectivity of the action map needed for the orthosymplectic case."},{"cited_title":"Deligne, G","cited_arxiv_id":null,"evidence_quote":"First fundamental theorem of invariant theory for the orthosymplectic supergroup; gives the spanning sets of invariants from which the Brauer-diagram construction starts."},{"cited_title":"Iorgov, A","cited_arxiv_id":null,"evidence_quote":"Shows how to build Casimir elements from the Brauer-Schur-Weyl duality; the pattern of turning centralizer-algebra cycles into central elements is extended here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Gelfand invariants and Harish-Chandra image computations for classical Lie algebras that the paper adapts to the superalgebra setting."},{"cited_title":"Moon, The product representations of the Lie siperalgebra pn and their centralizers","cited_arxiv_id":null,"evidence_quote":"Identifies the periplectic Brauer algebra as the centralizer of the periplectic Lie superalgebra action; supplies the surjectivity needed in that case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source for the Harish-Chandra isomorphism and the description of invariant supersymmetric polynomials used to prove that the constructed elements generate the center."},{"cited_title":"Nazarov, A","cited_arxiv_id":null,"evidence_quote":"Centralizer construction of the Yangian of the queer Lie superalgebra; gives the elements whose odd-degree members generate the queer center."},{"cited_title":"Scheunert, Invariant supersymmetric multilinear forms and the Casimir elements of P -type Lie superalgebras","cited_arxiv_id":null,"evidence_quote":"Earlier result on invariant supersymmetric multilinear forms proving the triviality of the center of the periplectic enveloping algebra; the new proof here replaces that argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier computation of the center of the enveloping algebra of the queer superalgebra; the generators obtained here match that description."}],"review_version":1}