Vogel's universality and Macdonald dimensions

Liudmila Bishler

REVIEW 3 major objections 5 minor 50 references

Vogel's universality and Macdonald dimensions

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The main result is a universal rational formula, eq. (161), giving the adjoint Macdonald dimension for every simply laced Lie algebra (A_n, D_n, E6, E7, E8) from the three Vogel parameters and q and t, reducing to the known universal…

desk verdict Careful, honest extension of the group's own earlier results; the new mixed dual Macdonald formulas are the real content, and the main claim rides on Macdonald's cited conjecture without a check. read the letter →

arxiv 2507.11414 v2 pith:NR6PJMFQ submitted 2025-07-15 hep-th math-phmath.COmath.MP

Liudmila Bishler This is my paper · ORCID

classification hep-thmath-phmath.COmath.MP MSC 17B1017B2233D5205E05

keywords VogeluniversalityMacdonaldpolynomialsdimensionsquantumrefinedChern-Simonstheorysimplylacedrootsystemsadjointrepresentationdual

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 claims that Vogel's algebraic universality survives the refinement from quantum dimensions to Macdonald dimensions, but only for simply laced root systems. Its main result is a single rational formula, eq. (161), that reproduces the adjoint Macdonald dimension for A_n, D_n, E6, E7, and E8 from three Vogel parameters and the two refinement parameters q and t. When t = q the formula collapses to the previously known universal quantum dimension of the adjoint representation. A sympathetic reader should care because this is the simplest refined quantity in Chern-Simons theory, so the formula gives the cleanest test of where refined universality begins and ends, and it explains that the divide sits exactly at simply laced versus non-simply-laced root systems.

What carries the argument

The load-bearing objects are the admissible pairs of root systems, the associated Macdonald polynomials, the refined Weyl vectors and their duals, and the three Vogel parameters a, b, c with t = a + b + c. The factorization identity at the dual refined Weyl vector converts a dual Macdonald dimension into a finite product over positive roots; in simply laced cases that same point is the refined Weyl vector itself, so the ordinary Macdonald dimension factorizes too. The paper rewrites each factorized adjoint answer in Vogel parameters and identifies the single rational expression (161) that interpolates between the five simply laced families. It also isolates u = q^a, v = t^b, and w = t^c as the genuinely universal refined parameters, since the formula is symmetric under their permutations.

What would settle it

The falsifier is a direct independent computation: evaluate the adjoint Macdonald polynomial for one simply laced root system, say E8, at the refined Weyl vector using the defining triangular expansion and an eigenoperator check at generic numerical q and t, and compare with eq. (161). A single mismatch at generic values, or a counterexample to the factorization identity (91) for any root-system pair used in the derivation, would settle the claim negatively.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the Macdonald polynomial in the adjoint representation of any simply laced algebra, evaluated at the refined Weyl vector, factorizes and then collapses into one universal expression. The paper defines Macdonald dimensions as Macdonald polynomials at the refined Weyl vector, and dual Macdonald dimensions as the same polynomials at the dual refined Weyl vector. A factorization identity gives closed products for the dual versions, and for root systems A_n, D_n, E6, E7, and E8 the two Weyl vectors coincide, so the ordinary Macdonald dimensions also factorize. Those factorized answers are all equal to the right-hand side of eq. (161), which is symmetric in u = q^a, v = t^b, and w = t^c, and which matches the universal quantum dimension formula at t = q.

Load-bearing premise

The load-bearing premise is an unproved factorization property: evaluated at a certain dual distinguished point, the refined polynomials split into a product of simple factors. The universal formula rests on that property for every root-system pair and representation used, and for simply laced algebras the ordinary and dual evaluations coincide at that same point.

Editorial extensions

If this is right

Equation (161) replaces the separate adjoint Macdonald dimension formulas for A_n, D_n, E6, E7, and E8 with one rational expression in the Vogel parameters and q and t.
At t = q the same expression becomes the known universal quantum dimension of the adjoint representation, so the refined formula has the correct unrefined limit.
Universality after refinement holds for the simply laced algebras only; the non-simply-laced families carry an extra root-length parameter, which is why a single Vogel-type formula does not cover them.
For simply laced root systems the ordinary and dual refined Weyl vectors coincide, so the factorized dual Macdonald dimensions directly give the ordinary Macdonald dimensions in the adjoint representation.
Mixed Macdonald dimensions for pairs such as (B_n, C_n) have their own factorization formulas and interpolate between different classical quantum dimension series in special limits.

Reading between the lines

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

A testable extension is to insert eq. (161) as the unknot factor in refined adjoint-colored knot invariants: if the refined Hopf-link and torus-knot invariants share the same u, v, w parameters, the formula should re-emerge there as the color-one building block.
The derivation's gate is the factorization property (91); if that property is proved for all admissible pairs, the dual Macdonald dimensions of B_n, C_n, F4, and G2 would also fit a universal description, even though their ordinary Macdonald dimensions do not.
The permutation symmetry in u = q^a, v = t^b, w = t^c suggests refined universality may be governed by these exponentiated parameters rather than the classical a, b, c lines; scanning (161) along the B- and C-lines at generic q and t could reveal accidental coincidences with non-simply-laced data.

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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

Quick take: this is a careful, readable extended version of the group's own earlier work, not a new universal mechanism. The headline object, formula (161), is openly stated to coincide with eq. (26) of arXiv:2505.16569 [28], so the actual novelty sits in section 4 — the mixed dual Macdonald dimensions for (Bn,Cn), (Cn,Bn), (BCn,Bn) and (BCn,Cn). Those explicit formulas look new and pass the t→q consistency checks against known quantum dimensions.

The good parts: the setup of root-system Macdonald polynomials via densities, the Koornwinder parameter dictionary, and the careful discussion of root-normalization dependence (section 3.4) are useful and plainly written. The A_n formula (102) checks by direct substitution, and the reduction of (161) at t→q to the universal quantum dimension (158) is a genuine anchor. The citation pattern is honest: the paper says plainly that (161) comes from [28], and the self-citations there are legitimate.

The soft spot is the one the stress-test flags, and it is load-bearing. The simply-laced Macdonald dimensions are identified with their factorized duals because ρk = rk, and the factorization itself is quoted from Macdonald's conjecture 12.10 (eqs. (91)/(128)). The paper explicitly labels it a conjecture — credit for that — but then builds (102)-(119) and ultimately (161) on it without proof and without numerical verification for the specific pairs and weights used. Strictly, the main result is conditional on a cited conjecture. I would temper the worry in one direction: this is a central conjecture of the subject, not an exotic assumption, and for the adjoint simply-laced pairs the factorization is exactly the kind of thing that direct symbolic computation should settle. The gap is real; the fix is routine. No code or data ships, which would have closed the issue cleanly.

The reader's CONDITIONAL verdict is about right. The paper deserves a serious referee: send it to review, and require a verification of the factorization for the pairs actually used — either a proof sketch of the needed cases of Macdonald's conjecture or explicit numerical checks — plus a sentence in the introduction flagging that (161) rests on that conjecture.

Who gets value: people in Vogel universality, refined Chern-Simons, or Macdonald/Koornwinder specializations, as a reference and for the new mixed formulas.

Referee Report

3 major / 5 minor

Summary. The paper defines Macdonald dimensions and dual Macdonald dimensions for Macdonald polynomials associated with arbitrary (admissible) pairs of root systems, lists explicit adjoint-representation formulas for all simple Lie algebras, and proposes a Vogel-universal formula, eq. (161), that unifies the simply-laced adjoint Macdonald dimensions for A_n, D_n, E6, E7, E8. It also treats mixed Macdonald dimensions for pairs such as (B_n,C_n), (C_n,B_n), (BC_n,B_n), and (BC_n,C_n), and argues that refined Vogel universality is restricted to simply-laced algebras. The main claim is that eq. (161) is the universal simply-laced adjoint Macdonald dimension, reducing to the known universal quantum dimension at t=q.

Significance. If the underlying factorization statement is valid, eq. (161) is a clean, parameter-free unification of five otherwise independent-looking formulas, and I have verified by direct substitution that it reproduces eqs. (102), (113), (115), (117), and (119) for the appropriate Vogel lines. The paper also gives a useful conceptual explanation of why universality survives only for simply-laced root systems, namely the coincidence of the refined and dual refined Weyl vectors. The value of the paper, however, is conditional: the case-by-case formulas rest on a factorization formula that the paper explicitly identifies as Macdonald's conjecture 12.10, and the main universal formula is presented by assertion rather than derivation. The contribution is therefore useful and likely correct, but not fully self-contained in its current form.

major comments (3)

[§3.3, eq. (91); §4.1, eq. (128)] The factorization of Macdonald polynomials at the dual refined Weyl vector is the load-bearing step for the entire derivation. The paper explicitly labels the formula as Macdonald's conjecture 12.10 and gives no proof or reference to a proof, yet it uses the formula as a theorem to obtain the adjoint Macdonald dimensions (102), (113), (115), (117), and (119), and hence the main result (161). Since rho_k = r_k for simply-laced root systems, the conclusion that simply-laced Macdonald dimensions coincide with the factorized dual Macdonald dimensions depends entirely on this conjecture. The authors should either supply a proof or a precise citation to a proof for the specializations used here, or explicitly state in the abstract and conclusion that the main result is conditional on this conjecture.
[§5.2, eq. (161)] The main universal formula is asserted rather than derived. The text says the individual formulas 'can be unified' and that (161) coincides with eq. (26) of [28], but it does not show how each of (102), (113), (115), (117), and (119) follows by substitution of the corresponding Vogel parameters. This is the central claim of the paper, so it should be supported by an explicit verification table or a short derivation. Such a table would also make the paper more useful to readers who want to check the claimed universality independently.
[§4, eqs. (132), (138), (141), (148), (154)] The mixed Macdonald dimension formulas inherit exactly the same conjecture dependence as the simply-laced formulas. They are all obtained by applying eq. (128), which is the restatement of the conjectured factorization used in the paper. The section should indicate which of these results are theorems in the Koornwinder polynomial setting and which remain conditional on the conjecture, so that the reader can distinguish established facts from conjectural statements.

minor comments (5)

[§4.4–4.5 and Appendix B] The symbols a, b, c, d are used both for Vogel parameters and for Koornwinder parameters, which is confusing in the mixed-dimension sections. Consider renaming the Koornwinder parameters or adding an explicit warning whenever the two notations overlap.
[§2.2, eq. (44)] The integration domain T in the scalar product is not defined. It becomes clear from eq. (45) that the constant-term interpretation is intended, but the torus and measure should be specified explicitly for completeness.
[§5.1] The heuristic argument against universalization of dual quantum dimensions would be clearer if written as follows: if a universal formula for ∨qD existed, it would have to agree with qD on the D_n line 2a+b=0 but differ from qD on the B_n line, even though both lines lie in 2a+b=0.
[References] The citation to Macdonald's conjecture should include the precise equation number in the published source, and the spelling of the author name in ref. [25] should be checked.
[§3.5.3, eq. (110)] The overline notation in \overline{Md}^{C_n}_{Adj} is used before the normalization convention is explained in §3.4. Consider moving the definition of the overline, or adding a one-line reminder at the point of first use.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: eq. (161) is a direct unification of the case-by-case simply-laced Macdonald dimensions, not a fitted prediction; reliance on Macdonald's conjecture is an unproven external premise rather than a circular reduction.

full rationale

The derivation chain is: define Macdonald dimensions (eq. 90); invoke Macdonald's factorization formula (91)/(128) at the dual refined Weyl vector; for simply laced R=S, rho_k = r_k, so Macdonald and dual Macdonald dimensions coincide (Section 3.3); specialize to the adjoint representation to obtain the five formulas (102), (113), (115), (117), (119); then observe that these five expressions are unified by the rational formula (161) in Vogel parameters, which at t=q also reduces to the known universal quantum dimension (158). I find no step in which an output is fed back into an input by definition, and no fitted parameter is relabeled as a prediction. In particular, eq. (161) is explicitly introduced as a unification ('can be unified with a universal formula'), not as a quantity derived from a separate universal principle and then forced to match the cases; it is algebraically equivalent to the listed simply-laced formulas under the Vogel parameter substitutions. The load-bearing external ingredient is Macdonald's conjecture 12.10, quoted at eq. (91) and restated at eq. (128); the paper provides no proof, so the case formulas and therefore eq. (161) are conditional on that conjecture. That is a foundational/correctness risk rather than circularity, because the conjecture is an independent external statement and is used also for non-simply-laced and mixed admissible pairs, outside the simply-laced target of eq. (161). The paper does contain self-citations to [27] and [28], including 'This particular form for the universal Macdonald dimension coincides with (26) in [28]'; these are provenance notes, and the main formula's support also comes from direct reduction to the external unrefined quantum-dimension formula (158) and to the five case formulas, so the self-citations are not load-bearing. Hence no reportable circular step is present; the score of 2 reflects the presence of minor, non-load-bearing self-citation rather than any circular reduction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No fitted numerical constants are introduced; q, t and the Vogel parameters are fixed inputs from deformation theory and the Vogel table. The paper introduces mathematical definitions, Macdonald dimensions and dual Macdonald dimensions, from existing Macdonald polynomials. The central formula rests on standard background plus one cited conjecture, but no new physical entities are postulated.

assumptions (3)

domain assumption Macdonald's dual factorization: P^(R,S)_lambda at q^(2 r*_k) equals the product over positive roots and j shown in eqs. (91) and (128).
Quoted from Macdonald [29] as conjecture 12.10. The dual Macdonald dimension formulas, and the claim that simply laced Macdonald dimensions factorize, rest on this unproved statement.
standard math Weyl character factorization at q^(2 rho), eq. (88), from reference [38].
Standard background used to define quantum dimensions and to anchor the t equals q limit of the universal formula.
standard math For admissible pairs (R, S), Macdonald polynomials exist, are unique, and are orthogonal with respect to the density in eqs. (35)-(46).
Background from Macdonald's theory, used throughout to define Macdonald dimensions and dual Macdonald dimensions.

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Cite this review

Pith. "Pith review of Vogel's universality and Macdonald dimensions." pith.science (2026). https://pith.science/paper/NR6PJMFQ

@misc{pith2026250711414,
  author       = {Pith},
  title        = {Pith review of: Vogel's universality and Macdonald dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NR6PJMFQ}},
  note         = {Machine review of arXiv:2507.11414}
}

read the original abstract

We discuss algebraic universality in the sense of P. Vogel for the simplest refined quantity, the Macdonald dimensions. The main known source of universal quantities is given by Chern-Simons theory. Refinement of Chern-Simons theory means introducing additional parameters. At the level of symmetric functions, the refinement is the transition from the Schur functions to the Macdonald polynomials. We consider the Macdonald polynomials associated with the simple Lie algebras, define Macdonald dimensions and dual Macdonald dimensions, and present a universal formula for them that unifies these quantities for algebras associated with simply laced root systems. We also consider mixed Macdonald dimensions that depend on two different root systems.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 21 canonical work pages

[28]

Bishler, A

L. Bishler, A. Mironov, A. Morozov, Phys.Lett. B868 (2025) 139695, arXiv:2505.16569

arXiv 2025
[1]

Vogel, Algebraic structures on modules of diagrams , Preprint (1995), available at https://webusers

P. Vogel, Algebraic structures on modules of diagrams , Preprint (1995), available at https://webusers. imj-prg.fr/~pierre.vogel/diagrams.pdf, J. Pure Appl. Algebra, 215 (2011) 1292-1339

work page 1995
[2]

Vogel, The Universal Lie algebra , Preprint (1999), available at https://webusers.imj-prg.fr/ ~pierre.vogel/grenoble-99b.pdf

P. Vogel, The Universal Lie algebra , Preprint (1999), available at https://webusers.imj-prg.fr/ ~pierre.vogel/grenoble-99b.pdf

work page 1999
[3]

Mkrtchyan, A.P

R.L. Mkrtchyan, A.P. Veselov, JHEP 08 (2012) 153, arXiv:1203.0766

arXiv 2012
[4]

Mkrtchyan, JHEP 09 (2013) 54, arXiv:1302.1507

R.L. Mkrtchyan, JHEP 09 (2013) 54, arXiv:1302.1507

arXiv 2013
[5]

Krefl, R

D. Krefl, R. Mkrtchyan, JHEP 10 (2015) 45, arXiv:1506.03907

arXiv 2015
[6]

Mkrtchyan, J

R.L. Mkrtchyan, J. Geom. Phys. 129 (2018) 186-191, arXiv:1709.03261

arXiv 2018
[7]

Westbury, Proceedings of the Tenth Oporto Meeting on Geometry, Topology and Physics, 18 (2001) suppl

B. Westbury, Proceedings of the Tenth Oporto Meeting on Geometry, Topology and Physics, 18 (2001) suppl. 2003, pp. 49-82

work page 2001

Show all 50 references

[8]

Mkrtchyan, Nucl

R.L. Mkrtchyan, Nucl. Phys. B921 (2017) 236-249, arXiv:1610.09910

2017 arXiv
[9]

Landsberg, L

J.M. Landsberg, L. Manivel, Adv. Math. 201 (2006) 379-407

2006
[10]

Mkrtchyan, A.N

R.L. Mkrtchyan, A.N. Sergeev, A.P. Veselov, J. Math. Phys. 53 (2012) 102106, arXiv:1105.0115

2012 arXiv
[11]

Avetisyan, A.P

M.Y. Avetisyan, A.P. Isaev, S.O. Krivonos, R.L. Mkrtchyan, Russian Journal of Mathematical Physics, 31 (2024) 379-388, arXiv:2311.05358

2024 arXiv
[12]

Isaev, A.A

A.P. Isaev, A.A. Provorov, Theor.Math.Phys. 206 (2021) 3-22, arXiv:2012.00746

2021 arXiv
[13]

Isaev, S.O

A.P. Isaev, S.O. Krivonos, J. Math. Phys. 62 (2021) 083503, arXiv:2102.08258

2021 arXiv
[14]

Isaev, S.O

A.P. Isaev, S.O. Krivonos, A.A. Provorov, Int. J. Mod. Phys. A38 (2023) 235003, arXiv:2212.14761

2023 arXiv
[15]

Khudaverdian, R.L

H.M. Khudaverdian, R.L. Mkrtchyan, Lett. Math. Phys. 107 (2017) 1491-1514, arXiv:1602.00337

2017 arXiv
[16]

Mironov, R

A. Mironov, R. Mkrtchyan, A. Morozov, JHEP 02 (2016) 078, arXiv:1510.05884

2016 arXiv
[17]

A.Mironov, A.Morozov, Phys. Lett. B755 (2016) 47-57, arXiv:1511.09077

2016 arXiv
[18]

Torus knots in adjoint representation and Vogel’s universality,

L. Bishler, A. Mironov, “Torus knots in adjoint representation and Vogel’s universality,” arXiv:2506.06219

arXiv
[19]

Khudoteplov, E

D. Khudoteplov, E. Lanina, A. Sleptsov, Construction of Lie algebra weight system kernel via Vogel algebra, arXiv:2411.14417

arXiv
[20]

Aganagic and S

M. Aganagic and S. Shakirov, Commun. Math. Phys. 333 (2015) 187-228, arXiv:1105.5117

2015 arXiv
[21]

Aganagic and S

M. Aganagic and S. Shakirov, Proc. Symp. Pure Math. 85 (2012) 3-32, arXiv:1202.2489

2012 arXiv
[22]

Krefl, A

D. Krefl, A. Schwarz, Journal of Geometry and Physics, 74 (2013) 119-129, arXiv:1304.7873

2013 arXiv
[23]

Avetisyan, R.L

M.Y. Avetisyan, R.L. Mkrtchyan, JHEP 10 (2021) 033, arXiv:2107.08679

2021 arXiv
[24]

Avetisyan, Vogel’s Universality and its Applications , arXiv:2207.04302

M. Avetisyan, Vogel’s Universality and its Applications , arXiv:2207.04302

arXiv
[25]

Cherednik, R

I. Cherednik, R. Elliot, Refined composite invariants of torus knots via DAHA , arXiv:1503.01441

arXiv
[26]

Deligne, B

P. Deligne, B. Gross, Comptes Rendus Acad. Sci. Paris, Ser I, 335 (2002) 877-881

2002
[27]

Bishler, A

L. Bishler, A. Mironov, Phys. Lett. B867 (2025) 139596, arXiv:2504.13831

2025 arXiv
[29]

Macdonald, S` eminaire Lotharingien Combin.45 (2000), Article B45a, 40 pp, arXiv:math/0011046

I.G. Macdonald, S` eminaire Lotharingien Combin.45 (2000), Article B45a, 40 pp, arXiv:math/0011046

2000 arXiv
[30]

Macdonald, SIAM J.Math

I.G. Macdonald, SIAM J.Math. Anal. 13:6 (1982) 988-1007

1982
[31]

Cherednik, Inventiones mathematicae, 125 (1996) 391, arXiv:q-alg/9412016 28

I. Cherednik, Inventiones mathematicae, 125 (1996) 391, arXiv:q-alg/9412016 28

1996 arXiv
[32]

Cherednik, The Annals of Mathematics, Second Series, 141 (1995) 191-216

I. Cherednik, The Annals of Mathematics, Second Series, 141 (1995) 191-216

1995
[33]

Koornwinder, in: Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Appli- cations (Tampa, FL, 1991), Contemp

T.H. Koornwinder, in: Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Appli- cations (Tampa, FL, 1991), Contemp. Math. 138, Amer. Math. Soc., Providence, RI, 1992, 189-204

1991
[34]

Macdonald, A new class of symmetric functions , Publ.I.R.M.A., Strasbourg, Actes 20-e Seminaire Lotharingen, (1988) 131-171

I.G. Macdonald, A new class of symmetric functions , Publ.I.R.M.A., Strasbourg, Actes 20-e Seminaire Lotharingen, (1988) 131-171

1988
[35]

Macdonald, Symmetric functions and Hall polynomials (2nd edition) , Clarendon Press, Oxford, 1995, 475 pp

I.G. Macdonald, Symmetric functions and Hall polynomials (2nd edition) , Clarendon Press, Oxford, 1995, 475 pp

1995
[36]

Balantekin, P

A.B. Balantekin, P. Cassak, J.Math.Phys. 43 (2002) 604-620, arXiv:hep-th/0108130

2002 arXiv
[37]

Markus Fulmek and Christian Krattenthaler, J. Combin. Theory, Ser. A, 77:3-50, 1997

1997
[38]

Di Francesco, P

P. Di Francesco, P. Mathieu, D. S´ en´ echal,Conformal Field Theory , Springer-Verlag, New York, 1997

1997
[39]

Avetisyan, R.L

M.Y. Avetisyan, R.L. Mkrtchyan, Uniqueness of universal dimensions and configurations of points and lines, arXiv:2101.10860

arXiv
[40]

King, Can

R.C. King, Can. J. Math. 23 (1971) 176

1971
[41]

Mkrtchyan, Phys

R.L. Mkrtchyan, Phys. Lett. B105 (1981) 174-176

1981
[42]

Cvitanovic, Group Theory, Princeton Univ

P. Cvitanovic, Group Theory, Princeton Univ. Press, Princeton, NJ, 2004

2004
[43]

Mkrtchyan, A.P

R.L. Mkrtchyan, A.P. Veselov, J. Math. Phys. 52 (2011) 083514, arXiv:1011.0151

2011 arXiv
[44]

Mkrtchyan, Phys

R.L. Mkrtchyan, Phys. Lett. b808 (2020) 135692, arXiv:2007.09346

2020 arXiv
[45]

Avetisyan, R.L

M.Y. Avetisyan, R.L. Mkrtchyan, JHEP 2022 (2022) 97, arXiv:2205.12832

2022 arXiv
[46]

Avetisyan, R.L

M.Y. Avetisyan, R.L. Mkrtchyan, J. Geom. Phys. 191 (2023) 104907, arXiv:2302.14319

2023 arXiv
[47]

Avetisyan, R.L

M.Y. Avetisyan, R.L. Mkrtchyan, Phys. Part. Nucl. 54 (2023) 1059-1062, arXiv:2304.05184

2023 arXiv
[48]

Boris Feigin, Ayumu Hoshino, Masatoshi Noumi, Jun Shibahara, Jun’ichi Shiraishi, SIGMA 11 (2015), 100, arXiv:1412.8001

2015 arXiv
[49]

Hoshino, Ayumu, and Jun’ichi Shiraishi, SIGMA 14 (2018): 101, arXiv:1801.09939

2018 arXiv
[50]

1998, Vol 3, Num 2, pp 181-207, arXiv:q-alg/9611011 29

Andrei Okounkov, Transformation groups. 1998, Vol 3, Num 2, pp 181-207, arXiv:q-alg/9611011 29

1998 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [28]

Bishler, A

L. Bishler, A. Mironov, A. Morozov, Phys.Lett. B868 (2025) 139695, arXiv:2505.16569

arXiv 2025

[2] [1]

Vogel, Algebraic structures on modules of diagrams , Preprint (1995), available at https://webusers

P. Vogel, Algebraic structures on modules of diagrams , Preprint (1995), available at https://webusers. imj-prg.fr/~pierre.vogel/diagrams.pdf, J. Pure Appl. Algebra, 215 (2011) 1292-1339

work page 1995

[3] [2]

Vogel, The Universal Lie algebra , Preprint (1999), available at https://webusers.imj-prg.fr/ ~pierre.vogel/grenoble-99b.pdf

P. Vogel, The Universal Lie algebra , Preprint (1999), available at https://webusers.imj-prg.fr/ ~pierre.vogel/grenoble-99b.pdf

work page 1999

[4] [3]

Mkrtchyan, A.P

R.L. Mkrtchyan, A.P. Veselov, JHEP 08 (2012) 153, arXiv:1203.0766

arXiv 2012

[5] [4]

Mkrtchyan, JHEP 09 (2013) 54, arXiv:1302.1507

R.L. Mkrtchyan, JHEP 09 (2013) 54, arXiv:1302.1507

arXiv 2013

[6] [5]

Krefl, R

D. Krefl, R. Mkrtchyan, JHEP 10 (2015) 45, arXiv:1506.03907

arXiv 2015

[7] [6]

Mkrtchyan, J

R.L. Mkrtchyan, J. Geom. Phys. 129 (2018) 186-191, arXiv:1709.03261

arXiv 2018

[8] [7]

Westbury, Proceedings of the Tenth Oporto Meeting on Geometry, Topology and Physics, 18 (2001) suppl

B. Westbury, Proceedings of the Tenth Oporto Meeting on Geometry, Topology and Physics, 18 (2001) suppl. 2003, pp. 49-82

work page 2001