The Gaudin model for the general linear Lie superalgebra and the completeness of the Bethe ansatz

Ngau Lam; Wan Keng Cheong

The Gaudin model for the general linear Lie superalgebra and the completeness of the Bethe ansatz

T0 review · 0 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The Gaudin algebra of gl(m|n) is cyclic, maximal, and diagonalizable on singular spaces.

desk verdict Main results are new and the reduction to the non-super Gaudin algebra is sound; worth sending to a strong referee with a minor request to fix the cyclicity argument in Theorem 4.7(i). read the letter →

arxiv 2501.00401 v1 pith:PRIYMN6D submitted 2024-12-31 math.RT math-phmath.MP

Wan Keng Cheong , Ngau Lam This is my paper · ORCID

classification math.RTmath-phmath.MP MSC 17B1017B8181R12

keywords GaudinmodelLiesuperalgebragl(m|n)BerezinianBetheansatzcompletenessFrobeniusalgebraFuchsiandifferentialoperatorspolynomialmodulesoddreflections

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

The paper proves that the Gaudin algebra of the general linear Lie superalgebra $\mathfrak{gl}_{m|n}$ completely controls the singular space of any tensor product of irreducible polynomial modules: that singular space is generated from a single vector, and for generic distinct site parameters the algebra acts with one-dimensional eigenspaces. This gives a superalgebra analogue of the strongest completeness statements known for ordinary Gaudin models, where the Bethe ansatz is replaced by a separation-of-variables description through differential operators. The proof embeds the super Gaudin model into a larger ordinary general-linear Gaudin model, then uses odd reflections and a Berezinian reduction identity to transfer cyclicity and diagonalizability downward. If correct, the Gaudin algebra on the singular space has dimension equal to that space and is a maximal commutative subalgebra of its endomorphisms.

What carries the argument

The load-bearing object is the Berezinian of the Lax matrix $L_{m|n}(z)$, a $(m+n)\times(m+n)$ matrix of super pseudo-differential operators whose coefficients generate the Gaudin algebra $\mathcal{B}_{m|n}(z)$. The Berezinian is the super-analogue of the determinant that makes the transfer-matrix construction commutative; the matrix is of Manin type, meaning its entries satisfy the super-commutation relations that make the Berezinian well behaved. The transfer step is Proposition 3.16: on a $\sigma_p$-singular weight space whose last $m-p$ even diagonal weights vanish, the Berezinian of $L_{m|n}(z)$ acts exactly like the Berezinian of the smaller matrix $L_{p|n}(z)$ followed by a power of the formal derivative $\partial_z$, and a dual statement truncates odd coordinates. Odd reflections — changes of Borel subalgebra that permute the ordering of even and odd roots — provide isomorphisms between the relevant singular weight spaces, so the known cyclicity and diagonalizability of ordinary $\mathfrak{gl}_{m+r}$ Gaudin algebras can be transported down to $\mathfrak{gl}_{m|n}$.

What would settle it

In the smallest nontrivial super case, $\mathfrak{gl}(2|1)$ with two sites and fixed polynomial tensor factors, compute the common eigenvectors of the Gaudin Hamiltonians at a generic pair $(z_1,z_2)$; if their number is less than $\dim M^{\mathrm{sing}}$, or if some eigenspace has dimension greater than one, the theorem fails. A cheaper check is the reduction identity itself: on a $\sigma_2$-singular weight space with vanishing third-even weight, verify that $\operatorname{Ber}(L_{3|1}(z))v = \operatorname{Ber}(L_{2|1}(z))\partial_z v$ for every $v$ in that space.

Watch

Extended reading notes

Core claim

The central claim, stated as Theorem 4.7 and Theorem 4.10, is that for any pairwise distinct complex parameters $z=(z_1,\ldots,z_\ell)$ and any $\ell$-fold tensor product $M$ of irreducible polynomial $\mathfrak{gl}_{m|n}$-modules, the singular space $M^{\mathrm{sing}}$ is cyclic as a module over the Gaudin algebra $\mathcal{B}_{m|n}(z)$, and the image of $\mathcal{B}_{m|n}(z)$ on $M^{\mathrm{sing}}$ is a Frobenius algebra. Consequently this image has dimension $\dim M^{\mathrm{sing}}$, is a maximal commutative subalgebra of $\operatorname{End}(M^{\mathrm{sing}})$, and every eigenspace is one-dimensional; for generic $z$ the algebra is diagonalizable with a simple spectrum. The paper also constructs an eigenbasis: it is pulled back, through an odd-reflection isomorphism, from the known eigenbasis of an ordinary $\mathfrak{gl}_{m+r}$ Gaudin algebra, and the corresponding eigenvalues are encoded by monic Fuchsian differential operators with polynomial kernels. The authors state that this should be read as the completeness of a reformulation of the Bethe ansatz for $\mathcal{B}_{m|n}(z)$ acting on $M^{\mathrm{sing}}$.

Load-bearing premise

The load-bearing premise is a block-matrix calculation: on a singular weight space whose last even coordinate weights vanish, the Berezinian of the large Gaudin Lax matrix acts exactly like the Berezinian of a smaller matrix composed with a power of the formal derivative; if that equality fails, the transfer argument from the known ordinary-linear case collapses.

Editorial extensions

If this is right

The Gaudin algebra image on $M^{\mathrm{sing}}$ has dimension exactly $\dim M^{\mathrm{sing}}$ and is a maximal commutative subalgebra of $\operatorname{End}(M^{\mathrm{sing}})$.
Every eigenspace of the Gaudin algebra on $M^{\mathrm{sing}}$ is one-dimensional, so the Gaudin Hamiltonians have a joint spectrum with no accidental degeneracy.
For generic parameters the algebra is diagonalizable with a simple spectrum, giving a simultaneous eigenbasis for all Gaudin Hamiltonians on $M^{\mathrm{sing}}$.
An eigenbasis can be obtained from Fuchsian differential operators of appropriate order with polynomial kernels, and the eigenvalues are read off from their coefficients; this is a super version of the geometric Langlands correspondence for Gaudin models.
If the ordinary Bethe ansatz is complete for the embedded $\mathfrak{gl}_{m+r}$ model, then Bethe vectors give an explicit eigenbasis for the super Gaudin algebra as well.

Reading between the lines

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

The reduction identity in Proposition 3.16 suggests the completeness statement may extend beyond polynomial modules, provided the singular weights satisfy the same vanishing condition; testing the smallest non-polynomial case, such as a $\mathfrak{gl}(1|1)$ factor with a non-polynomial highest weight, would separate the hook-partition assumption from the Berezinian mechanism.
One can test Theorem 4.10 numerically without building the Gaudin algebra: for small $m,n$, count the monic Fuchsian operators with polynomial kernels and prescribed exponents at the marked points, and compare that count with $\dim M^{\mathrm{sing}}$ at a generic $z$; equality is exactly the paper's picture.
The maximality of the image on $M^{\mathrm{sing}}$ is evidence for the paper's Conjecture 5.5 that the full affine vertex algebra center is generated by derivatives of the Berezinian coefficients, because a larger center would still act through the same maximal commutative algebra on singular spaces.

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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

Short version: this paper proves the right thing and the proof hangs together. The main results—cyclicity, Frobenius property, and generic diagonalizability with simple spectrum for the Gaudin algebra B_{m|n}(z) on the singular space of tensor products of irreducible polynomial gl(m|n)-modules—are new and not in the cited literature. Prior work covered gl(m) (Rybnikov, Mukhin–Tarasov–Varchenko), the cubic Hamiltonians in [CCL], and gl(1|1) (Lu). The paper closes the gap for all gl(m|n).

The proof strategy is the interesting part: lift to gl(m+r|n), use odd reflections to convert a singular weight space into a sigma_m-singular space with zero weights on the last r even coordinates, then reduce to the non-super gl(m+r) Gaudin algebra via the Berezinian identity in Proposition 3.16. That identity is the load-bearing step, and it is a direct block-elimination calculation. I checked the two places a skeptical reader might distrust: the claim that E_{i,m}(z) acts trivially on the relevant weight space (justified by polynomial weights plus Lemma 3.9), and the inverse Schur complement in part (ii). Both hold. The stress-test note agrees, and I think it is right.

The soft spots are minor. The proof of Theorem 4.7(i) says a direct summand of a cyclic B_{m+r}(z)-module is cyclic because the algebra preserves singular weight spaces. That implication is false in general. Here the conclusion still follows because the Frobenius property (already established via Lemma 4.2 and the Shapovalov form) makes the full singular space free of rank one over the Gaudin algebra, so each weight-space summand is cyclic. This is an expository gap, not a gap in the mathematics, and it should be patched.

The word "generic" in Theorem 4.10 is inherited from the non-super diagonalization theorem; the paper does not quantify it. That is standard but could be said more explicitly. The final section on the Feigin–Frenkel center is speculative but clearly marked as such, and it does not affect the main results.

Citation pattern: the paper leans on published results (Rybnikov, Mukhin–Tarasov–Varchenko, Molev–Ragoucy, Cheng–Lam), and the self-citations are for background lemmas that are available elsewhere. No red flag.

Who it is for: anyone working on Gaudin models, Bethe ansatz, or superintegrable systems. It deserves a serious referee; I would send it out and expect a quick accept after minor revision.

Referee Report

0 major / 5 minor

Summary. The paper studies the Gaudin algebra B_{m|n}(z) of the general linear Lie superalgebra gl_{m|n} acting on tensor products of irreducible polynomial modules. The main results are Theorem 4.7, asserting that the singular space M^sing is a cyclic B_{m|n}(z)-module and that the image of B_{m|n}(z) in End(M^sing) is a Frobenius algebra, and Theorem 4.10, asserting diagonalizability with a simple spectrum for generic z. The proof proceeds by lifting M to a gl_{m+r|n}-module, truncating to gl_{m+r}, and transferring the known cyclicity and diagonalizability results for gl_{m+r} down to gl_{m|n} using the Berezinian calculus and the key identity Proposition 3.16. The final section relates the diagonalization to Bethe vectors and Fuchsian differential operators with polynomial kernels, yielding a completeness statement for a reformulated Bethe ansatz.

Significance. If correct, the paper establishes a substantial super analogue of the Gaudin Bethe ansatz completeness theorem, extending the results of Rybnikov and Mukhin-Tarasov-Varchenko from gl_m to gl_{m|n}. The reduction strategy is elegant: rather than developing a new superalgebraic diagonalization method, the authors transfer the known non-super results through a carefully chosen odd reflection and truncation. The key technical identity, Proposition 3.16, is a direct Berezinian calculation, and the stress-test scrutiny found it correct; this lends credibility to the main theorems. The paper also gives explicit connections to Bethe vectors and Fuchsian differential operators, and it identifies a plausible conjecture about the Feigin-Frenkel center, which adds context. The proofs rely on published external results rather than circular reasoning, and the manuscript is careful in citing the sources for the structural facts it uses.

minor comments (5)

[Section 4.3, proof of Theorem 4.7(i)] The sentence 'as B_{m+r} preserves singular weight spaces' does not by itself imply that the direct summand N of a cyclic module is cyclic; this implication is false in general. The conclusion nevertheless follows from the already established Frobenius property of the Gaudin algebra on the full singular space (Remark 4.5) together with Lemma 4.2(iii), which implies that every weight-space direct summand is cyclic. Please replace the given reason with this argument.
[Section 3.4, Proposition 3.16] The equality N := ⊕_μ tr_{p|n}(M)^sing_μ = ⊕_μ M^{σ_p-sing}_μ is slightly ambiguous because the two direct sums are indexed by different sets of weights; the first is over singular weights of tr_{p|n}(M) and the second over σ_p-singular weights of M. The intended identification via Corollary 3.13 should be stated more explicitly.
[Section 5.2, Theorem 5.4] In the statement of Theorem 5.4, the notation Δ_{η,ᵇ́γ,z} is used without repeating the definitions of η and ᵇ́γ; the surrounding text defines them, but restating them in the theorem statement would improve readability.
[Section 3.2] There is a typo: '[MR, Corallary 3.6]' should read '[MR, Corollary 3.6]'.
[Section 4.3, proof of Theorem 4.7] The word 'isomophism' should be 'isomorphism'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are derived from external gl(m) results and in-paper Berezinian reductions, with no step assuming its conclusion.

full rationale

The central claim is that the singular space M^sing is cyclic over the Gaudin algebra B_{m|n}(z) and that B_{m|n}(z)_{M^sing} is Frobenius (Theorem 4.7), with generic diagonalizability (Theorem 4.10). The derivation is self-contained in the relevant sense: it lifts M to a gl(m+r)|n-module, identifies singular weight spaces with sigma_m-singular spaces via odd reflection results cited from Cheng-Lam and Cheng-Lam-Wang, then uses the in-paper Proposition 3.16 to show that on those spaces the higher-rank Berezinian reduces to the lower-rank gl(m+r) action. Cyclicity and diagonalizability are then imported from Rybnikov's theorem and Mukhin-Tarasov-Varchenko's theorem, which are external results for ordinary gl(m+r) Gaudin algebras. No parameter is fitted to the target data, no quantity is defined in terms of the claimed output, and no uniqueness theorem from the authors' prior work is invoked to force a choice. The self-citations [CL], [CLW1], [CLW2], [CCL], and [ChL] are used for background structural facts or as a stated special case, not as the load-bearing proof of the main theorems. The only presentational gap is the assertion in Theorem 4.7(i) that a direct summand of a cyclic B_{m+r}-module is cyclic; that assertion is not justified by the reason given, but it is repaired by the already-proved Frobenius property via Lemma 4.2, so it is an expository omission rather than a circular step. Overall, no circular reduction of the claimed results to their own inputs was found.

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

The paper is a proof-based mathematics preprint. It introduces no free parameters or invented entities. Its arguments rest on a web of published structural results about Lie superalgebras and Gaudin algebras (semisimplicity, odd reflections, Berezinian calculus, and the non-super Gaudin theorems of Rybnikov and Mukhin-Tarasov-Varchenko). These are external benchmarks rather than assumptions tailored to the target result.

assumptions (6)

domain assumption The category of polynomial gl(m|n)-modules is semisimple and every object decomposes into irreducible highest weight modules corresponding to (m|n)-hook partitions.
Invoked as Proposition 3.8 in Section 3.3 to decompose tensor products into irreducibles, a necessary prerequisite for reducing to the non-super case.
domain assumption Rybnikov's theorem: for gl(m), the singular space of a tensor product of finite-dimensional irreducible modules is cyclic over the Gaudin algebra B_m(z).
Used as Theorem 4.4 in Section 4.2 as the base result that is transferred to the super setting in Theorem 4.7.
domain assumption Mukhin-Tarasov-Varchenko's theorems: for generic z, B_m(z) acts diagonally with simple spectrum on the singular space, and eigenspaces correspond bijectively to Fuchsian differential operators with polynomial kernels.
Used as Theorems 4.6 and 5.3 to obtain generic diagonalizability and the differential operator description for gl(m|n) through Proposition 4.9.
domain assumption The Berezinian of the matrix L_{m|n}(z) generates the Gaudin algebra B_{m|n}(z), and this algebra is commutative.
Stated in Section 3.2 following Molev-Ragoucy; this is the definition of the object of study.
domain assumption Odd reflections and the truncation functor satisfy the properties in Propositions 3.10, 3.11, 3.13 and 3.14, including the identification of singular weight spaces under truncation.
These structural results, cited from Cheng-Lam and Cheng-Lam-Wang, allow the paper to move singular vectors between different Borel subalgebras and to lift modules from gl(m|n) to gl(m+r|n).
standard math The tensor Shapovalov form is nondegenerate on the relevant singular spaces, and the non-super Gaudin algebra is symmetric with respect to it.
Used in Section 4.2 and in the proof of Theorem 4.7(ii) to establish the Frobenius property via Lemma 4.1.

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Pith. "Pith review of The Gaudin model for the general linear Lie superalgebra and the completeness of the Bethe ansatz." pith.science (2026). https://pith.science/paper/PRIYMN6D

@misc{pith2026250100401,
  author       = {Pith},
  title        = {Pith review of: The Gaudin model for the general linear Lie superalgebra and the completeness of the Bethe ansatz},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PRIYMN6D}},
  note         = {Machine review of arXiv:2501.00401}
}

abstract

Let $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})$ be the Gaudin algebra of the general linear Lie superalgebra $\mathfrak{gl}_{m|n}$ with respect to a sequence $\underline{\boldsymbol{z}} \in \mathbb{C}^\ell$ of pairwise distinct complex numbers, and let $M$ be any $\ell$-fold tensor product of irreducible polynomial modules over $\mathfrak{gl}_{m|n}$. We show that the singular space $M^{\rm sing}$ of $M$ is a cyclic $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})$-module and the Gaudin algebra $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})_{M^{\rm sing}}$ of $M^{\rm sing}$ is a Frobenius algebra. We also show that $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})_{M^{\rm sing}}$ is diagonalizable with a simple spectrum for a generic $\underline{\boldsymbol{z}}$ and give a description of an eigenbasis and its corresponding eigenvalues in terms of the Fuchsian differential operators with polynomial kernels. This may be interpreted as the completeness of a reformulation of the Bethe ansatz for $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})_{M^{\rm sing}}$.

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Forward citations

Cited by 1 Pith paper

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

Bethe algebras for unitarizable modules over classical Lie (super)algebras and a duality
math.RT 2025-05 conditional novelty 6.0 of 10

The Bethe algebra for gl_{p+m|q+n} is diagonalizable with simple spectrum on weight spaces of tensor products of certain infinite-dimensional unitarizable modules, via a duality with the Bethe algebra for gl_d.

Reference graph

Works this paper leans on

9 extracted references · 8 canonical work pages · cited by 1 Pith paper

[1]

[BF] H. M. Babujian, R. Flume, Oﬀ-shell Bethe ansatz equation for Gaudin magnets and soluti ons of Knizhnik-Zamolodchikov equations , Modern Phys. Lett. A 9 (1994), no. 22, 2029–2039. [Ber] F. A. Berezin, Introduction to superanalysis , Mathematical Physics and Applied Mathematics,

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[MV1] E. Mukhin, A. Varchenko, Critical points of master functions and ﬂag varieties , Commun. Con- temp. Math. 6 (2004), no. 1, 111–163. [MV2] E. Mukhin, A. Varchenko, Norm of a Bethe vector and the Hessian of the master function , Compos. Math. 141 (2005), no. 4, 1012–1028. [MV3] E. Mukhin, A. Varchenko, Multiple orthogonal polynomials and a counterexam...

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[Bet] H. Bethe, Zur Theorie der Metalle I. Eigenwerte und Eigenfunktionen de r linearen Atomkette , Z. Phys. 71 (1931) 205–226. 22 W. K. CHEONG AND N. LAM [CCL] B. Cao, W. K. Cheong, N. Lam, Quadratic and cubic Gaudin Hamiltonians and super Knizhnik - Zamolodchikov equations for general linear Lie superalgebr as, arXiv:2310.17484. [CL] S.-J. Cheng, N. Lam...

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[Ry] L. Rybnikov, A proof of the Gaudin Bethe ansatz conjecture , Int. Math. Res. Not. IMRN (2020), no. 22, 8766–8785. [Sh] N. N. Shapovalov, A certain bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra , Funkcional. Anal. i Priloˇ zen. 6 (1972), no. 4, 65–70. [Sk] E. K. Sklyanin, Separation of variables in the Gaudin mo...

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[KM] P. P. Kulish, N. Manojlovi´ c, Bethe vectors of the osp(1|2) Gaudin model , Lett. Math. Phys. 55 (2001), no. 1, 77–95. [Lu1] K. Lu, Perfect integrability and Gaudin models , SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), Paper No. 132, 10 pp. [Lu2] K. Lu, Completeness of Bethe ansatz for Gaudin models associated w ith gl(1|1), Nuclear Ph...

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arXiv 2024

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[MM1] A. I, Molev, E. Mukhin, Invariants of the vacuum module associated with the Lie supe ralgebra gl(1|1), J. Phys. A 48 (2015), no. 31, 314001, 20 pp. [MM2] A. I, Molev, E. Mukhin, Eigenvalues of Bethe vectors in the Gaudin model , Teoret. Mat. Fiz. 192 (2017), no. 3, 369–3...

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

[BF] H. M. Babujian, R. Flume, Oﬀ-shell Bethe ansatz equation for Gaudin magnets and soluti ons of Knizhnik-Zamolodchikov equations , Modern Phys. Lett. A 9 (1994), no. 22, 2029–2039. [Ber] F. A. Berezin, Introduction to superanalysis , Mathematical Physics and Applied Mathematics,

work page 1994

[2] [8]

Mukhin, A

[MV1] E. Mukhin, A. Varchenko, Critical points of master functions and ﬂag varieties , Commun. Con- temp. Math. 6 (2004), no. 1, 111–163. [MV2] E. Mukhin, A. Varchenko, Norm of a Bethe vector and the Hessian of the master function , Compos. Math. 141 (2005), no. 4, 1012–1028. [MV3] E. Mukhin, A. Varchenko, Multiple orthogonal polynomials and a counterexam...

work page 2004

[3] [1983]

Gelfand, S

[GGR W] I. Gelfand, S. Gelfand, V. Retakh, R. L. Wilson, Quasideterminants. Adv. Math. 193 (2005), no. 1, 56–141. [GW] R. Goodman. N. Wallach, Higher-order Sugawara operators for aﬃne Lie algebras , Trans. Amer. Math. Soc. 315 (1989), no. 1, 1–55. [Ha] T. Hayashi, Sugawara operators and Kac–Kazhdan conjecture , Invent. Math. 94 (1988), no. 1, 13—52. [HM] ...

work page 2005

[4] [1987]

Quadratic and cubic Gaudin Hamiltonians and super Knizhnik-Zamolodchikov equations for general linear Lie superalgebras

[Bet] H. Bethe, Zur Theorie der Metalle I. Eigenwerte und Eigenfunktionen de r linearen Atomkette , Z. Phys. 71 (1931) 205–226. 22 W. K. CHEONG AND N. LAM [CCL] B. Cao, W. K. Cheong, N. Lam, Quadratic and cubic Gaudin Hamiltonians and super Knizhnik - Zamolodchikov equations for general linear Lie superalgebr as, arXiv:2310.17484. [CL] S.-J. Cheng, N. Lam...

work page Pith review arXiv 1931

[5] [1995]

Rybnikov, A proof of the Gaudin Bethe ansatz conjecture , Int

[Ry] L. Rybnikov, A proof of the Gaudin Bethe ansatz conjecture , Int. Math. Res. Not. IMRN (2020), no. 22, 8766–8785. [Sh] N. N. Shapovalov, A certain bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra , Funkcional. Anal. i Priloˇ zen. 6 (1972), no. 4, 65–70. [Sk] E. K. Sklyanin, Separation of variables in the Gaudin mo...

work page 2020

[6] [1997]

[KM] P. P. Kulish, N. Manojlovi´ c, Bethe vectors of the osp(1|2) Gaudin model , Lett. Math. Phys. 55 (2001), no. 1, 77–95. [Lu1] K. Lu, Perfect integrability and Gaudin models , SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), Paper No. 132, 10 pp. [Lu2] K. Lu, Completeness of Bethe ansatz for Gaudin models associated w ith gl(1|1), Nuclear Ph...

work page 2001

[7] [2004]

Gaudin, Diagonalisation d’une classe d’Hamiltoniens de spin , J

[G1] M. Gaudin, Diagonalisation d’une classe d’Hamiltoniens de spin , J. Physique 37 (1976), no. 10, 1089–1098. [G2] M. Gaudin, La fonction d’onde de Bethe , Collection du Commissariat a‘ l’E’nergie Atomique: Se’rie Scientiﬁque, Masson, Paris,

work page 1976

[8] [2012]

[ChL] W. K. Cheong, N. Lam, Gaudin Hamiltonians on unitarizable modules over classica l Lie (su- per)algebras, J. Algebra 642 (2024), 400–431. [CF] A. Chervov, G. Falqui, Manin matrices and Talalaev’s formula , J. Phys. A 41 (2008), no. 19, 194006, 28 pp. [CFR] A. Chervov, G. Falqui, V. Rubtsov, Algebraic properties of Manin matrices , I. Adv. in Appl. Ma...

arXiv 2024