Modular orthogonal Yangians

Hao Chang; Hongmei Hu

REVIEW 2 major objections 5 minor 17 references

Modular orthogonal Yangians

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The center of modular orthogonal Yangians is exactly described

desk verdict A useful extension of the Brundan–Topley program to orthogonal types, but the characteristic-p Drinfeld presentation is assumed rather than proved, and it may fail at primes dividing 2κ. read the letter →

arxiv 2506.00395 v2 pith:CUVAC2I6 submitted 2025-05-31 math.QA math.RT

Hao Chang , Hongmei Hu This is my paper

classification math.QAmath.RT MSC 17B3717B50

keywords modularYangiansorthogonalp-centercenterpositivecharacteristicDrinfeldpresentationHarish-Chandra

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 determines the center of the orthogonal Yangians of types B and D over fields of characteristic p > 2. It defines a p-center generated by p-th powers of certain Drinfeld generators and proves that the center of the extended orthogonal Yangian is generated by the Harish-Chandra center together with this p-center. It then shows that the center of the orthogonal Yangian Y(o_N) itself coincides with its p-center. The upshot is an explicit free-polynomial description of both centers.

What carries the argument

The argument rests on the triangular (quasideterminant) decomposition of the RTT generator matrix T(u) = F(u)H(u)E(u), which converts the RTT presentation into the Drinfeld presentation with generators h_i(u), e_i(u), f_i(u). In characteristic p the key mechanism is that (ad e_i(u))^p acts trivially on the e_j, h_j and f_j generators, and likewise for f_i, so the p-th powers (e_{i,j}^{(r)})^p and (f_{j,i}^{(r)})^p are central. Transposition and permutation automorphisms reduce all computations to the simple generators, and the Harish-Chandra center is obtained from the scalar-matrix identity T(u - kappa) T^t(u) = c(u) 1.

What would settle it

For p = 3 and N = 3, compute whether the coefficient of $u^{{-6}}$ in e_1(u)^3 commutes with h_2(v) using only the stated relations; a nonzero commutator would disprove Theorem 4.9. Alternatively, check whether the associated graded image of (e_i(u))^p equals (F_{i,j} $t^{{r-1}}$)^p for a small example; a single mismatch would falsify the lifting statement.

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Extended reading notes

Core claim

Theorem 4.9 states that in characteristic p > 2 the center Z(X(o_N)) of the extended orthogonal Yangian is generated by the Harish-Chandra center Z_HC(X(o_N)) and the p-center Z_p(X(o_N)). More specifically, Z_p(X(o_N)) is freely generated by the families {$b_i^{{(rp)}}$, (e_{i,j}^{(r)})^p, (f_{j,i}^{(r)})^p} for the stated index sets and r > 0, lifting the p-central generators of U(o_N[t]). For the Yangian Y(o_N), Theorem 4.10 states that the p-center Z_p(Y(o_N)) equals the full center Z(Y(o_N)). The proof passes to the associated graded algebra, where the claimed generators become algebraically independent generators of Z_p(o_N[t]), and combines this with direct centrality computations using the Drinfeld presentation.

Load-bearing premise

The load-bearing premise is that the Drinfeld presentation of the modular orthogonal Yangian, with the extra cubic relations needed for p = 3, is exactly the characteristic-zero presentation; if that presentation fails after reduction modulo p, the centrality computations collapse.

Editorial extensions

If this is right

The center Z(X(o_N)) is a free polynomial algebra with explicit generators b_i^{(rp)}, c^{(r)}, and the p-th powers (e_{i,j}^{(r)})^p, (f_{j,i}^{(r)})^p for the declared index sets.
The center Z(Y(o_N)) equals its p-center, so every central element of the modular orthogonal Yangian is a polynomial in the p-power generators (4.18).
The associated graded algebra of the center is Z_p(o_N[t]) tensor k[zeta_1, zeta_2, ...], matching the center of the enveloping algebra of the current Lie algebra.
These explicit centers provide the precise structural information needed to classify finite-dimensional irreducible modules of the restricted Yangian Y^{[p]}(o_N) in positive characteristic.

Reading between the lines

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

The same mechanism, if the presentation assumption holds for all p > 2, should extend to the modular symplectic Yangian of type C once an analogous Drinfeld presentation is established; the paper notes this requires genuinely new calculations.
The characteristic-p vanishing (ad e_i(u))^p = 0 on generators is reminiscent of the p-center phenomenon in enveloping algebras of restricted Lie algebras, so the construction here may serve as a template for centers of other quantum algebras at roots of unity.
For p = 3 the paper must add cubic Serre relations beyond the usual ones, suggesting that a uniform presentation for all p may require a whole hierarchy of higher-order relations; testing centrality of the p-th powers for p = 5 would probe whether such relations are needed generally.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the short version. Chang and Hu prove that, for p>2, the center of the extended orthogonal Yangian X(o_N) is generated by the Harish-Chandra center and an explicitly described p-center, and that Z(Y(o_N)) = Z_p(Y(o_N)). This is the expected type B/D analogue of Brundan–Topley (type A) and Chang–Hu (super type A), and it's the first explicit center description for modular orthogonal Yangians. The proof architecture is genuinely good: Section 2 proves the p-center description for U(o_N[t]) with a clean triangularity argument; Lemmas 4.2 and 4.6 show p-centrality of the explicit candidates by direct computation; and Theorem 4.9(3) uses an associated-graded comparison, not a restatement, to prove generation. The type B short-root/long-root split and the p=3 cubic Serre relations (Lemma 3.5) are real new technical content.

The soft spot is Theorem 3.1. The Drinfeld presentation is imported from JLM with the one-line claim that 'the proof works perfectly' in positive characteristic. That is load-bearing: every centrality computation in Section 4 runs inside this presentation. The stress-test concern is legitimate. In the RTT relation (3.2), the denominator is u−v−κ with κ=N/2−1. When κ ≡ 0 mod p, for example type B with N=5 and p=3, or type D with N=8 and p=3, that denominator becomes u−v. Reductions of Gauss-decomposition isomorphisms can change drastically when a parameter vanishes; the JLM argument may not survive. The paper gives no argument for these primes. There is also a tension the authors themselves flag: the Serre relations (3.26)–(3.27) are said to be inadequate for p=3, yet Theorem 3.1 asserts that the algebra is 'subject only to' those relations. The additional cubic relations of Lemma 3.5 are presumably true in X(o_N), but their existence undercuts the 'only' in the presentation claim.

The fix is likely manageable. The authors need to prove Theorem 3.1 directly over Z or Z[1/2], or carefully check the JLM isomorphism step by step at primes dividing 2κ; alternatively, they can exclude those primes, though that weakens the main theorem. The rest of the paper is probably correct conditional on that.

This is a solid paper for the modular Yangians community, and it deserves referee time. The gap is a missing proof, not a wrong idea. I'd send it to a referee who really knows the JLM/Mol2 machinery, and ask them to verify the base change. If the presentation gap is closed, the results stand.

Referee Report

2 major / 5 minor

Summary. The paper studies the extended orthogonal Yangian X(o_N) and the Yangian Y(o_N) over an algebraically closed field of characteristic p>2. It defines a p-center for these algebras and claims that Z(X(o_N)) is generated by the Harish-Chandra center and the p-center, with explicit free polynomial generators, and that Z(Y(o_N)) equals its p-center. The strategy is to use a Drinfeld presentation for X(o_N) (Theorem 3.1), compute centrality of explicit series using a battery of identities in Section 3, and then pass to the associated graded algebra, which for X(o_N) is identified with U(o_N[t]) tensor k[zeta] and for Y(o_N) with U(o_N[t]). The main theorems are Theorem 4.9 (structure of Z(X(o_N))) and Theorem 4.10 (Z(Y(o_N)) = Z_p(Y(o_N))).

Significance. If the main theorems are correct, the paper gives a complete and explicit description of the centers of modular orthogonal Yangians of types B and D, extending in a natural way the Brundan-Topley results for gl_N. The associated-graded arguments are a genuine triangularity check, and the proposed generators are explicit and natural lifts of the p-center of U(o_N[t]). The centrality proofs in Section 4 are detailed and use relations imported from [JLM] and [BT] as external benchmarks. The main caveat is that virtually the whole Section 4 operates inside the Drinfeld presentation of Theorem 3.1, whose characteristic-p validity is asserted but not demonstrated.

major comments (2)

[Section 3.4, Theorem 3.1] The one-sentence proof that the Drinfeld presentation of [JLM, Theorem 5.14] 'works perfectly in positive characteristic' is not sufficient. The RTT relation (3.2) contains the denominator u-v-kappa with kappa = N/2 - 1. When p divides 2kappa (for example N=5 and p=3, or N=8 and p=3), kappa specializes to 0 and the defining relations of the RTT algebra change. The JLM isomorphism between the RTT and Drinfeld presentations relies on shifted denominators and cancellations involving kappa; those steps do not automatically survive reduction modulo p. Since every centrality computation in Section 4 (Lemma 4.2, Lemma 4.6, Theorem 4.7, Theorem 4.9) uses the presentation of Theorem 3.1, the main results are not established for such primes. The paper must either prove the isomorphism for all p>2, or explicitly exclude the primes p dividing N-2 and state which results remain valid in the exceptional cases.
[Section 1 and Lemma 3.5] The introduction states that for p=3 the usual Serre relations (3.26)-(3.27) are inadequate and additional cubic relations are needed. This is in tension with Theorem 3.1, which describes X(o_N) as being 'subject only to' the listed relations including the Serre relations. The proof of Lemma 3.5 is also very terse, referring to [BK, Lemma 5.7] without details. Please clarify whether the cubic relations in Lemma 3.5 are consequences of the relations listed in Theorem 3.1 or whether they must be added as extra defining relations. If they are extra, the statement of Theorem 3.1 must be modified, and the conclusions of the subsequent centrality arguments in Section 4 must be reassessed.

minor comments (5)

[Throughout] There are numerous typographical errors, for example 'cannonical' (p.1), 'algerba' (p.1), 'CorD' in the phrase 'B, CorD Lie algebras' (p.1), 'beacuse' (p.14), and 'epresentations' in reference [CHL]; the manuscript needs a careful copyedit.
[Section 2, Theorem 2.1] The sentence defining Z(o_N[t]) has an unbalanced parenthesis: 'Z(U(o_N[t])))' appears before 'be the center'; please correct the punctuation.
[Theorem 4.9(3)] The generating set (4.13) omits b_{n+1}^{(rp)} for N=2n+1. Since bc(u)=b_{n+1}(u)^2 and bc^{(r)} lies in Z_HC, the coefficients b_{n+1}^{(r)} are polynomials in the bc^{(s)} (obtained by inverting 2 recursively), and hence in the c^{(r)}. The authors should state this explicitly so that the omission of b_{n+1} from (4.13) is justified.
[Lemma 4.2 proof] The chain of equalities for (ad e_i(u))^p(e_i(v)) is abbreviated in a way that is hard to follow; consider displaying the recursive step explicitly.
[Section 4.4] The notation bc(u) is introduced after Lemma 4.8 but used in (4.10); it would be clearer to define bc(u) before Theorem 4.9.

Circularity Check

0 steps flagged · score 2.0 of 10

Not circular: p-center generators are verified central by direct computation inside an externally sourced Drinfeld presentation (JLM), and generation follows by a graded lifting argument (in-paper Theorem 2.1). Self-citations ([CH], [CHL]) are non-load-bearing; the one-line JLM transfer is a soundness risk at primes with κ≡0 mod p, not a circular step.

full rationale

Walking the derivation chain: X(o_N) is defined by the RTT relation (3.2); Theorem 3.1 imports the Drinfeld presentation verbatim from [JLM, Theorem 5.14] with a one-line proof; Sections 3-4 then derive the centrality of the p-center candidates ((e_{i,j}^{(r)})^p, (f_{j,i}^{(r)})^p, b_i^{(rp)}=h_i(u)...h_i(u-p+1)) directly from those relations (Lemmas 4.2 and 4.6), and prove generation by comparing associated graded rings with Z(gr X(o_N)) = Z_p(U(o_N[t])) ⊗ k[ζ] (Theorems 4.3, 4.7, 4.9(3), 4.10, with Theorem 2.1 proved in the paper). Each step is a genuine check rather than a restatement of the conclusion: the definition of Z_p in (4.9) and (4.18) does not presuppose the equality Z = Z_HC · Z_p, and the graded inclusions gr Z ⊆ Z(gr X) as well as Z_p(U(o_N[t])) ⊗ k[ζ] ⊆ gr Z are verified from the computed leading terms (4.6), (4.8), (4.11), not assumed. The load-bearing external imports ([JLM, Theorem 5.14], [BT, Lemmas 2.1 and 2.9], [AMR, Corollary 3.9], [BK, Lemma 5.7]) are not the authors' own work; the self-citations [CH] and [CHL] appear only as "see also" method pointers and background motivation, so no load-bearing argument reduces to a self-citation. Two flagged passages deserve mention but are correctness risks, not circularity: Theorem 3.1's one-sentence proof ("The proof in [JLM, Theorem 5.14] works perfectly in positive characteristic") skips the specialization of the RTT denominator u-v-κ when κ = N/2-1 ≡ 0 mod p, and Section 1 concedes the Serre relations (3.26)-(3.27) are inadequate at p=3, requiring new cubic relations (Lemma 3.5). Both concern whether the externally imported presentation survives reduction modulo p; if the import fails, the Section 4 computations would be unsupported, but nothing in the paper's argument assumes the theorem it aims to prove, so the derivation is self-contained rather than circular, and the score reflects only the peripheral self-citation [CH] in Lemma 3.10 and the introduction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters: the paper proves structural theorems about explicit algebras; the characteristic p is part of the input, not a fitted quantity. The axioms are imported characteristic-zero results assumed to survive specialization to p > 2, plus standard restricted-Lie-algebra facts. No new entities are postulated: the series b_i(u), a_i(u), c(u), H^m_{v,u} and the sets I, J are all constructed inside X(o_N) or Y(o_N) from the Drinfeld generators, and none requires external falsifiable evidence.

assumptions (6)

domain assumption The PBW theorem for the extended orthogonal Yangian over C and the graded isomorphism gr X(o_N) = U(o_N[t]) tensor k[zeta_1, zeta_2, ...] (3.4) remain valid after specialization to characteristic p > 2.
Section 3.2, equation (3.4); the cited source [AMR, Corollary 3.10] is over C. This identification underpins the graded comparisons in Theorems 4.3, 4.7, 4.9 and 4.10, including the zeta_r terms in (4.8) and (4.11).
domain assumption The Drinfeld presentation of X(o_N) in characteristic p > 2 is exactly the characteristic-zero presentation of [JLM, Theorem 5.14] (Theorem 3.1).
Section 3.4; the proof is a one-sentence assertion that the JLM proof 'works perfectly in positive characteristic'. All centrality computations in Section 4 use these relations.
standard math o_N[t] is a restricted Lie algebra with p-map (F_{i,j} t^r)^[p] = delta_{i,j} F_{i,j} t^{rp}, so Z(U(o_N[t])) = Z_p(o_N[t]) with free generators (2.4) (Theorem 2.1).
Theorem 2.1 is proved in the paper for types B and D, with the type B case handled separately; the restricted structure is inherited from gl_N[t], a standard fact (cf. [Hum], [Jan]). This is the base case for all graded comparisons.
domain assumption The identity e_{n+1,n+2}(u) = -e_n(u - 1/2) of [JLM, Proposition 5.7] persists in characteristic p > 2, and the corrections from [Mol2, page 487] to [JLM] are incorporated.
Section 4.2, proofs of Lemma 4.2 and Corollary 4.4; this reduces the exceptional B-type generator to the standard series e_n(u) with shifted argument. No char-p re-derivation is given.
domain assumption The filtered-algebra facts [BT, Lemma 2.1 and Lemma 2.9] on p-th powers (degrees of (e^(r))^p, vanishing of b_i^(r) for 1 < r < p, and polynomial dependence of the p-not-divide-r coefficients) apply verbatim to the orthogonal Yangian generators.
Used in Theorems 4.3, 4.7 and 4.9 to compute degrees and algebraic independence; these facts are stated in [BT] for the type A Yangian.
domain assumption The Harish-Chandra center Z_HC(X(o_N)) is the free polynomial algebra generated by {c^(r); r > 0} (from [AMR, Corollary 3.9]), and c(u) has the product form of Theorem 4.1 ([JLM, Theorem 5.8], [Mol2, Theorem 5.3]) in characteristic p > 2.
Theorem 4.1 and Theorem 4.9(1) cite characteristic-zero results; they are reused without a modular proof, although c(u) is defined by the same RTT relation (3.3) in char p.

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Pith. "Pith review of Modular orthogonal Yangians." pith.science (2026). https://pith.science/paper/CUVAC2I6

@misc{pith2026250600395,
  author       = {Pith},
  title        = {Pith review of: Modular orthogonal Yangians},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CUVAC2I6}},
  note         = {Machine review of arXiv:2506.00395}
}

abstract

We study the (extended) orthogonal Yangians associated to the Lie algebras types $B$ and $D$ over a field of positive characteristic. We define the $p$-center for the Yangians and obtain an explicit description of the center in terms of Drinfeld generators, showing that the center is generated by its Harish-Chandra center together with a large $p$-center.

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Works this paper leans on

17 extracted references · 16 canonical work pages

[1]

Arnaudon, J

D. Arnaudon, J. Avan, N. Cramp\' e , L. Frappat and E. Ragoucy, R -matrix presentation for super-Yangians Y(osp(m|2n)) . J. Math. Phys. 44 (2003), 302--308

work page 2003
[2]

Arnaudon, A

D. Arnaudon, A. Molev and E. Ragoucy, On the R -matrix realization of Yangians and their representations . Ann. Henri Poincar\' e 7 (2006), 1269--1325

work page 2006
[3]

Brundan and L

J. Brundan and L. Topley, The p -centre of Yangians and shifted Yangians . Mosc. Math. J. 18 (2018), no. 4, 617-657

work page 2018
[4]

Brundan and A

J. Brundan and A. Kleshchev, Parabolic presentations of the Yangian Y( gl _n) . Comm. Math. Phys. 254 (2005), 191--220

work page 2005
[5]

Chang and H.-M

H. Chang and H.-M. Hu, The centre of the modular super Yangian Y_ m|n . J. Lond. Math. Soc. (2) 107 (2023), 1074--1109

work page 2023
[6]

Chang, J.-X

H. Chang, J.-X. Hu and L. Topley, Modular epresentations of the Yangian Y_2 . J. London Math. Soc. (2) 111 (2025), Paper No. e70056

work page 2025
[7]

V. G. Drinfeld, Hopf algebras and the quantum Yang--Baxter equation . Soviet Math. Dokl. 32 (1985), 254--258

work page 1985
[8]

V. G. Drinfeld, A new realization of Yangians and quantized affine algebras . Soviet Math. Dokl. 36 (1988), 212--216

work page 1988

Show all 17 references

[9]

Gelfand and V

I. Gelfand and V. Retakh, Quasideterminants. I . Selecta Math. 3 (1997), 517--546

1997
[10]

N. Guay, V. Regelskis and C. Wendlandt, Equivalences between three presentations of orthogonal and symplectic Yangians . Lett. Math. Phys. 109 (2019), 327--379

2019
[11]

J. E. Humphreys, Conjugacy classes in semisimple algebraic groups. Math. Surv. Monogr., 43, American Mathematical Society (AMS), Providence, RI, 1995

1995
[12]

J. C. Jantzen, Representations of Lie algebras in prime characteristic . in: A. Broer (Ed.), Representation Theories and Algebraic Geometry, in: Proceedings, Montreal, NATO ASI Series, vol. C 514, Kluwer, Dordrecht, 1998, pp. 185--235

1998
[13]

N. Jing, M. Liu and A. Molev, Isomorphism between the R-matrix and Drinfeld presentations of Yangian in types B , C and D . Comm. Math. Phys. 361 (2018), 827--872

2018
[14]

Molev, Yangians and classical Lie algebras

A. Molev, Yangians and classical Lie algebras . Mathematical Surveys and Monographs, 143. AMS, Providence, RI, 2007

2007
[15]

Molev, A Drinfeld-type presentation of the orthosymplectic Yangians

A. Molev, A Drinfeld-type presentation of the orthosymplectic Yangians . Algebr. Represent. Theory 27 (2024), 469--494

2024
[16]

Molev, M

A. Molev, M. Nazarov, and G. Olshanskii, Yangians and classical Lie algebras . Russ. Math. Serv. 51 (1996), 205--282

1996
[17]

Takhtadzhyan and L

L. Takhtadzhyan and L. Faddeev, The quantum method of the inverse problem and the Heisenberg XYZ -model . Russ. Math. Serv. 34 (1979), 11--68

1979

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Arnaudon, J

D. Arnaudon, J. Avan, N. Cramp\' e , L. Frappat and E. Ragoucy, R -matrix presentation for super-Yangians Y(osp(m|2n)) . J. Math. Phys. 44 (2003), 302--308

work page 2003

[2] [2]

Arnaudon, A

D. Arnaudon, A. Molev and E. Ragoucy, On the R -matrix realization of Yangians and their representations . Ann. Henri Poincar\' e 7 (2006), 1269--1325

work page 2006

[3] [3]

Brundan and L

J. Brundan and L. Topley, The p -centre of Yangians and shifted Yangians . Mosc. Math. J. 18 (2018), no. 4, 617-657

work page 2018

[4] [4]

Brundan and A

J. Brundan and A. Kleshchev, Parabolic presentations of the Yangian Y( gl _n) . Comm. Math. Phys. 254 (2005), 191--220

work page 2005

[5] [5]

Chang and H.-M

H. Chang and H.-M. Hu, The centre of the modular super Yangian Y_ m|n . J. Lond. Math. Soc. (2) 107 (2023), 1074--1109

work page 2023

[6] [6]

Chang, J.-X

H. Chang, J.-X. Hu and L. Topley, Modular epresentations of the Yangian Y_2 . J. London Math. Soc. (2) 111 (2025), Paper No. e70056

work page 2025

[7] [7]

V. G. Drinfeld, Hopf algebras and the quantum Yang--Baxter equation . Soviet Math. Dokl. 32 (1985), 254--258

work page 1985

[8] [8]

V. G. Drinfeld, A new realization of Yangians and quantized affine algebras . Soviet Math. Dokl. 36 (1988), 212--216

work page 1988