A tale of two shuffle algebras

Andrei Negu\c{t}

REVIEW 2 major objections 4 minor 21 references

A tale of two shuffle algebras

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Top and bottom shuffle algebras decompose quantum toroidal gl_n

desk verdict Strong new shuffle realization of the quantum toroidal algebra, but the key pairing in Prop 6.6 is not fully proved as written. read the letter →

arxiv 1908.08395 v4 pith:THE3475J submitted 2019-08-22 math.QA math.RT

Andrei Neguţ This is my paper · ORCID

classification math.QAmath.RT MSC 17B3716T25

keywords quantumtoroidalalgebrashuffleDrinfelddoubleR-matrixtopologicalcoproductPBWbasisaffinewheelconditions

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 sets out to prove that the quantum toroidal algebra of $\mathfrak{gl}_n$, usually built from its left and right halves, can equally be decomposed into top and bottom halves. Both new halves are shuffle algebras: spaces of matrix-valued rational functions in many variables, multiplied using the R-matrix of the evaluation representation of the quantum affine group. The payoff is a new topological coproduct on the whole algebra that extends the usual Drinfeld-Jimbo coproduct on the horizontal quantum affine subalgebra, together with a realization of the whole algebra as the Drinfeld double of the two extended halves. If the construction is right, the quantum toroidal algebra carries a second, genuinely different triangular decomposition, parallel to the classical left-right one.

What carries the argument

The central object is a matrix-valued shuffle algebra $A^+$ with spectral parameter. Its elements are $\mathrm{End}(V^{\otimes k})$-valued rational functions in variables $z_1,\dots,z_k$, symmetric in a braided sense, with only simple poles at $z_a = z_b q^2$ and with iterated residues constrained by the wheel conditions. The product is assembled from the R-matrix $R(x)$ and its mate $\tilde{R}(x)$, and associativity follows from Reidemeister-type moves; the extended algebra $\tilde{A}^+$ adds generating series $S(x)$ and $T(x)$, a topological coproduct, and a bialgebra pairing defined by iterated residues. The slope subalgebras $B^+_\mu$, with PBW generators $F^\mu_{[i;j)}$ and $\bar{F}^\mu_{[i;j)}$, organize the comparison with the explicit algebra $D$ that is already known to be isomorphic to $U_{q,q}(\ddot{gl}_n)$.

What would settle it

Compute the trace identity behind Proposition 6.6 for $k=4$ and a permutation $\sigma$ that is not the longest element, take the iterated residue of the integrand at $z_i q^2 = z_j$ for an inversion pair $(i,j)$ of $\sigma$, and check whether the residue vanishes. A single nonzero residue for any such $\sigma$ would show that the claimed bialgebra pairing is not well-defined, and with it the realization of the algebra as a Drinfeld double.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.5: there exist injective algebra homomorphisms $A^+ \hookrightarrow U_{q,q}(\ddot{gl}_n)$ and $A^{-,\mathrm{op}} \hookrightarrow U_{q,q}(\ddot{gl}_n)$, whose images $U^{\mathrm{up}}$ and $U^{\mathrm{down}}$ satisfy $U_{q,q}(\ddot{gl}_n) \cong U^{\mathrm{up}} \otimes U^{\mathrm{down}}$. The extended shuffle algebras $\tilde{A}^+ = A^+ \otimes U^{\geq}_q(\dot{gl}_n)$ and $\tilde{A}^{-,\mathrm{op}} = (A^- \otimes U^{\leq}_q(\dot{gl}_n))^{\mathrm{op}}$ carry topological coproducts, and $U_{q,q}(\ddot{gl}_n)$ is their Drinfeld double. This new coproduct extends the Drinfeld-Jimbo coproduct on the horizontal subalgebra $U_q(\dot{gl}_n)\subset U_{q,q}(\ddot{gl}_n)$.

Load-bearing premise

The load-bearing premise is that a certain contour-reordering step in the bialgebra pairing never picks up leftover residues; the paper verifies this vanishing only for the longest permutation and leaves the general case to the reader. If any such residue is nonzero, the pairing is not well-defined and the Drinfeld-double realization of the quantum toroidal algebra collapses.

Editorial extensions

If this is right

The quantum toroidal algebra gains a top-bottom triangular decomposition alongside the usual left-right one; unlike the left-right halves, the top half in degree $\mathbb{Z}^n \times \{1\}$ is generated by elements indexed by all roots of $U_q(\dot{sl}_n)$, not just positive roots.
The new topological coproduct makes $U_{q,q}(\ddot{gl}_n)$ into a Drinfeld double of two extended shuffle algebras, so the pairing between the halves is encoded in explicit commutation relations among the generators.
The shuffle presentation yields a PBW basis: ordered products of slope-$\mu$ generators over increasing $\mu$ form a linear basis, with dimension controlled by the number of unordered interval collections.
For any quantum group with a representation $V$ and a unitary R-matrix, the Section 2 machinery produces a double shuffle algebra; specializing to $V = \mathbb{C}^n(z)$ with the standard R-matrix recovers the quantum toroidal algebra.

Reading between the lines

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

Editorial inference: the general R-matrix shuffle construction opens a concrete testbed for other affine types: feed in a unitary R-matrix of a representation, form the double shuffle algebra, and check whether it is isomorphic to the corresponding quantum affinization; the paper itself floats a connection to q-deformed extended Yangians outside type A.
Editorial inference: the success of the whole argument is concentrated in the contour-reordering step of the pairing; a natural stress test is to compute the residue for a small non-longest permutation, such as $k=4$, before relying on the Drinfeld-double statement.
Editorial inference: in the $n=1$ limit the top-bottom decomposition is the left-right decomposition conjugated by the $SL_2(\mathbb{Z})$ rotation, whereas for $n>1$ the two decompositions are genuinely non-isomorphic; this suggests the new coproduct is a new structure rather than a reindexing of the old one.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Natasha, here's the take. The paper is a solid, creative piece: it gives the first top/bottom shuffle presentations for U_{q,q}(gl_n), builds the extended algebras, and packages them into a Drinfeld double with a topological coproduct extending Drinfeld-Jimbo. I believe the main architecture is correct and the result is significant. The general shuffle-with-R-matrix framework in Section 2 is clean, and Section 5's PBW basis is a genuine technical achievement.

Where I'd push back: the well-definedness of the pairing in Proposition 6.6 is not fully demonstrated. The stress-test note is on point. Equality of the two integral formulas requires changing contours from |z1| << ... << |zk| to any permuted order, and the only obstruction is nonzero residues at z_i = z_j q_+^2. The author proves vanishing for the longest permutation, then leaves general sigma to the reader with 'We leave the visual depiction of this fact to the interested reader.' That's a load-bearing identity, not a side remark—if some residue survives, the pairing is double-valued and the Drinfeld double collapses. I think the vanishing is probably true: the braid argument for sigma = omega_k is explicit, and the generalization only needs the green and blue strands to be unlinked except at the two specified points, which does look right from the figures. But in a paper this ambitious, a statement like that should be proved, not delegated. A referee should insist on a complete argument.

The other soft spot is structural: the chain of isomorphisms D \cong S \cong U goes through the author's preprint [17]. The circularity burden is low, and I agree it's not circular—[17] is an independent PBW construction. But it is still unpublished, and some of the relations in Section 3.26 are only stated 'sufficient' with a proof sketched. That's a verification burden for the community, not a flaw.

The many 'exercises to the interested reader' are mostly routine, but a few (e.g., the sigma = - analogues of Prop 6.7) are less trivial than the label suggests. These are proportional concerns; the paper is not hand-wavy in its main conceptual steps.

Who benefits? Anyone working on shuffle algebras, quantum toroidal algebras, or Drinfeld doubles. It deserves a serious referee. My recommendation: send it to peer review, and require the author to fill the Prop 6.6 gap and specify exactly which parts of [17] are being used.

Referee Report

2 major / 4 minor

Summary. The paper introduces a new family of matrix-valued shuffle algebras A^+ and A^- built from the R-matrix with spectral parameter of U_q(gl_n), and claims that these algebras are isomorphic to the 'up' and 'down' halves D^+ and D^- of the quantum toroidal algebra U_{q,q}(gl_n double dot) that appear in a new triangular decomposition. The author constructs extended algebras ~A^+ and ~A^- with topological coproducts, defines a bialgebra pairing between them, forms the Drinfeld double, and asserts that this double is isomorphic to the full quantum toroidal algebra. The main theorem, Theorem 1.5, also states that the resulting topological coproduct extends the Drinfeld-Jimbo coproduct on the horizontal subalgebra U_q(gl_n). The proof is largely a comparison with the author's earlier PBW presentation of U_{q,q}(gl_n double dot) from [17].

Significance. If the main theorem is correct, the paper gives a genuinely new shuffle-algebra description of the 'top' and 'bottom' halves of the quantum toroidal algebra, complementary to the known left/right shuffle presentations. The new topological coproduct extending the Drinfeld-Jimbo coproduct is a concrete and falsifiable structural claim. The construction itself is explicit and parameter-free: the shuffle algebras are defined directly from the universal R-matrix, and the claimed isomorphisms are reduced to a finite set of algebraic checks. The paper is well organized and the overall architecture is coherent. Its main weakness is that a load-bearing contour-reordering step in the construction of the bialgebra pairing is not proved in general, and the proof of the Drinfeld double realization inherits this gap.

major comments (2)

[§6.6, Eqs. (6.14)–(6.15)] The pairing (6.13) is not proved to be well-defined. To show that the two defining formulas (6.14) and (6.15) agree on elements of the form (6.12), the proof must justify changing the integration contour from |z1|≪...≪|zk| to |zσ(1)|≪...≪|zσ(k)| in (6.20)–(6.21). The only poles that can be met are at zi q_+^2 = zj and the analogous q_- poles. The paper explicitly shows the vanishing of the residue only for the longest permutation σ=ω_k, using identity (6.23). For general σ, the text says: 'We leave the visual depiction of this fact to the interested reader' and asserts that the relevant blue and green strands do not cross except at two points. This is not a cosmetic omission: if for some σ and some i<j with σ^{-1}(i)>σ^{-1}(j) the residue at zi q_+^2 = zj is nonzero, then formulas (6.14) and (6.15) define different functionals, the pairing (6.13) is not single-valued, and the Drinfeld double construction in (6.30) collapses. Proposition 6.7, the double (6.30), and the realization of U_{q,q}(gl_n double dot) in Theorem 1.5 all depend on this pairing. A complete proof of the residue vanishing for all σ, or an alternative contour-independent definition of the pairing, is required.
[§6.7–6.9] The construction of the Drinfeld double relies on several substantial checks that are delegated rather than proved. The proof of Proposition 6.7 says it 'follows that of Proposition 2.11 very closely' and leaves the verification of (2.29) and several cases to the reader; the displayed verification of (6.27) itself ends with the sentence 'We may move R_{ω_k} to the very right of the expression above', which hides exactly the kind of braid manipulation that is problematic in Proposition 6.6. Likewise, relations (6.31)–(6.33) are asserted with proofs left as exercises. Since these relations are what match the commutation relations of D and thereby produce the algebra isomorphism in the final proof of Theorem 1.5, the double realization is only as solid as the unproved pairwise checks. They should be written out or given precise references to where they are proved.

minor comments (4)

[§6.6, proof of Proposition 6.6] The proof refers to 'Braid 1' through 'Braid 6' on the previous page, but these braid diagrams are not present in the text under review; without them, the displayed braid moves cannot be checked by the reader. If the diagrams exist in the published version, this is not an issue, but they should be included.
[Throughout] Several statements are justified with 'left as an exercise to the interested reader', including Proposition 2.11, formulas (3.115)–(3.117), Proposition 3.39, and parts of Propositions 5.17 and 6.11. While many of these are routine, some are non-trivial and are used later in load-bearing positions; the author should indicate which of these exercises are genuinely routine and which are needed for the main theorem.
[§4.3] The nonstandard residue convention, by which (α−x)^{-1} has residue 1 at x=α, is stated only in passing. Since all later contour computations depend on it, this convention should be prominently displayed and used consistently.
[§5.22] The author notes that no closed formula is known for the imaginary generators P^μ_{lδ,r} in (5.56). This is acceptable, but it should be stated more prominently, since the isomorphism Υ_μ : E^+_μ → B^+_μ is only defined through the existence of these generators.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the shuffle algebras are defined from the R-matrix and independently matched to the known presentation D, with the only notable gap being an omitted contour-reordering verification in Proposition 6.6, which is a correctness risk rather than a circularity.

full rationale

The paper's construction is definition-theorem, not fit. A+ and A− are new shuffle algebras built from the fixed R-matrix (3.87) by the shuffle product (4.9) and the wheel conditions (4.13); no parameter is fitted to the desired conclusion. Theorem 5.25 is proved by explicit maps F^μ_{[i;j)} ↦ p^μ, by computing α-functionals, and by dimension bounds obtained inside the paper (Lemmas 5.10, 5.15, Claim 5.29), while the identification of the abstract algebra D with U_{q,q}(gln) is imported from [16,17]. That import is a self-citation, but it is an independent prior structure theorem: its assumptions are about the standard shuffle algebra S and the PBW presentation D, not about the new A±, so it does not make the target result true by definition. The same applies to the use of the D± PBW basis from [17] in the surjectivity part of Theorem 5.25. The genuinely fragile point is Proposition 6.6: the equality of (6.14) and (6.15) is reduced to contour reordering, and for general σ the paper states 'We leave the visual depiction of this fact to the interested reader' after proving the residue vanishing only for the longest permutation. If a residue survived, the pairing (6.13) would be ill-defined and the Drinfeld double realization would collapse. This is a load-bearing rigor gap, but it is not circularity: the gap is in verifying a well-definedness property of a newly constructed integral, not in defining the answer to match the input. No step in the claimed derivation chain equates the conclusion with its inputs by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

No parameters are fitted; the central claim rests on standard Yang-Baxter and R-matrix axioms and on the author's prior PBW result [17]. The new shuffle algebra is explicitly constructed and anchored to the previously-defined D+.

assumptions (4)

standard math Yang-Baxter equation for the R-matrix R(x) of (3.87), and the unitarity identity R_{12}(x)R_{21}(1/x) = f(x) Id
Assumed in (4.3) and (4.6) and used throughout to construct the shuffle product and braid equivalences.
domain assumption The quantum toroidal algebra U_{q,q}(gl_n) is isomorphic to the double shuffle algebra S (theorem of [16]) and to the algebra D of [17] with its PBW basis
Invoked in Theorem 3.29 and Corollary 3.30 to transfer the new shuffle construction to U; [17] is a preprint by the same author and supplies the load-bearing PBW basis.
standard math The subalgebra E of U_q(gl_n) is the Drinfeld double of its positive and negative halves with the pairing (3.25)
Used to set up the horizontal subalgebra and the series S± and T±.
domain assumption The evaluation representation V = C^n(z) of U_q(gl_n) has the R-matrix (3.87) with the stated analytic properties
The specific form of R(x) underlies the wheel conditions and the computations of residues; if a different R were chosen, the shuffle algebra A+ would change.

invented entities (1)

Matrix-valued shuffle algebra A+ with spectral parameters independent evidence
purpose: Presents the subalgebra U^up_{q,q}(gl_n) of the quantum toroidal algebra
Defined in Definition 4.8 via wheel conditions; Theorem 5.25 identifies it with D+ of [17], giving an external anchor in prior literature.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A tale of two shuffle algebras." pith.science (2026). https://pith.science/paper/THE3475J

@misc{pith2026190808395,
  author       = {Pith},
  title        = {Pith review of: A tale of two shuffle algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/THE3475J}},
  note         = {Machine review of arXiv:1908.08395}
}

read the original abstract

As a quantum affinization, the quantum toroidal algebra is defined in terms of its "left" and "right" halves, which both admit shuffle algebra presentations. In the present paper, we take an orthogonal viewpoint, and give shuffle algebra presentations for the "top" and "bottom" halves instead, starting from the evaluation representation of the quantum affine group and its usual R-matrix. An upshot of this construction is a new topological coproduct on the quantum toroidal algebra which extends the Drinfeld-Jimbo coproduct on the horizontal quantum affine subalgebra.

Figures

Figures reproduced from arXiv: 1908.08395 by the authors.

**Figure 1.** Various crossings The strands are represented either as straight or squiggly, because we wish to indicate whether the picture in question refers to either R or Re. Compositions are always read left-to-right, for example the following equivalence of braids underlies the Yang-Baxter relations (2.1) [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Reidemeister III move - version 1 while the following equivalences underlie equations (2.2) and (2.3), respectively [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Reidemeister III move - version 2 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (17 more)

**Figure 4.** Figure 4: Reidemeister III move - version 3 We will equivalate braids connected by the Reidemeister III type moves above. 2.3. We will now recall the construction of Section 5.2 of [14] (itself a dual version of the construction of [7]) and present it in the language of shuffle …

**Figure 5.** Figure 5: A ∗ B as a braid The proof of Proposition 2.4, namely that the multiplication defined above is associative, is a straightforward consequence of the following equivalence of braids [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: (A ∗ B) ∗ C [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: A ∗ (B ∗ C) Indeed, in the top picture, one can pull the straight red strands to the left of the blue-green crossings, and the squiggly red strands below the blue-green crossings [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: A braid representation of Rσ · (σXσ−1 ) · R−1 σ Choosing one braid lift of σ over another is just the ambiguity of choosing Rab over R −1 ba for any crossing between the strands labeled a and b. Since (2.4) says that these two endomorphisms differ by a scalar, the ambi…

**Figure 9.** Figure 9: Rµ · (µΦµ −1 ) · R−1 µ it is easy to see that the definition (2.6) can be restated as: (2.12) A ∗ B = µ goes over X (k,l)–shuffles Rµ · (µΦµ −1 ) · R −1 µ where: (2.13) Φ = h Rk,k+1...R1,k+l i A1...kh Re1,k+l ...Rek,k+1i Bk+1...k+l For any τ ∈ S(k) × S(l) ⊂ S(k + l), w…

**Figure 10.** Figure 10: Rτ · (τΦτ −1 ) · R−1 τ is equivalent to Φ (since we can cancel the braids representing Rτ and R−1 τ by pulling them through the symmetric tensors A and B). Then (2.12) implies: A ∗ B = 1 k!l! µ goes over X (k,l)–shuffles X τ∈S(k)×S(l) Rµ · µ [PITH_FULL_IMAGE:figures/…

**Figure 11.** Figure 11: Braids decorated with variables represent the following compositions ∈ End(V ⊗2 )(z1, z2): R12 z1 z2 A1(z1)Re12 z1 z2 B2(z2) and A2(z2)Re21 z2 z1 B1(z1)R21 z2 z1 respectively. The variable does not change along a strand, except at a box. 4.3. We make a…

**Figure 12.** Figure 12: Black dots can slide past arbitrary strands The equality of braids depicted in [PITH_FULL_IMAGE:figures/full_fig_p040_12.png]

**Figure 13.** Figure 13: Changing a crossing [PITH_FULL_IMAGE:figures/full_fig_p040_13.png]

**Figure 14.** Figure 14: The RHS of (4.13) for u = 2, λ1 = 4, λ2 = 3 Note the symbol “blue over blue” to the right of [PITH_FULL_IMAGE:figures/full_fig_p043_14.png]

**Figure 15.** Figure 15: The black dots in the middle of the braid appear because the variables on the braids in question are set equal to each other in the iterated residue. By sliding the black dots as far to the right as possible (which is allowed, due to [PITH_FULL_IMAGE:figures/full_fig…

**Figure 16.** Figure 16 [PITH_FULL_IMAGE:figures/full_fig_p044_16.png]

**Figure 17.** Figure 17: Note that the black dots on the right side of the braid above yield the same permutation as the black dots on the right side of the braid in [PITH_FULL_IMAGE:figures/full_fig_p045_17.png]

**Figure 18.** Figure 18: (we ignore the scalar-valed rational functions f in the diagrams above, as they commute with all the braids involved). The braid called Rσ interchanges the two collections of λs = λt braids corresponding to the variables ysq 2∗ and ytq 2∗ . Although we could choose t…

**Figure 19.** Figure 19: Then we pull the red strands as far up as possible, and notice that the blue strands are all unlinked, thus yielding the braid in [PITH_FULL_IMAGE:figures/full_fig_p046_19.png]

**Figure 20.** Figure 20: The red strands in [PITH_FULL_IMAGE:figures/full_fig_p047_20.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[17]

Negut, A., PBW basis for Uq,q(¨gln), arχiv:1905.06277

work page Pith review arXiv 1905
[1]

Beck J., Braid group action and quantum aﬃne algebras , Commun. Math. Physics vol. 165 (1994), 555568

work page 1994
[2]

Burban I., Schiﬀmann O., On the Hall algebra of an elliptic curve, I , Duke Math. J. 161 (2012), no. 7, 1171–1231

work page 2012
[3]

Ding J., Frenkel I., Isomorphism of two realizations of quantum aﬃne algebra Uq(ˆgln), Comm. Math. Phys. 156 (1993), no. 2, 277–300

work page 1993
[4]

Ding J., Iohara K., Generalization of Drinfeld quantum aﬃne algebras , Lett. Math. Phys., 41 (1997), no. 2, 181–193

work page 1997
[5]

G., A new realization of Yangians and quantized aﬃne algebras , Soviet Math

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

work page 1988
[6]

Groups 5 (2000), no

Enriquez B., On correlation functions of Drinfeld currents and shuﬄe algebras, Transform. Groups 5 (2000), no. 2, 111 - 120

work page 2000
[7]

J.1(1990) 193–226

Faddeev L., Reshetikhin N., Takhtajan L., Quantization of Lie groups and Liealgebras , Leningrad Math. J.1(1990) 193–226

work page 1990

Show all 21 references

[8]

Feigin B., Hashizume K., Hoshino A., Shiraishi J., Yanagida S., A commutative algebra on degenerate CP1 and MacDonald polynomials , J. Math. Phys. 50 (2009), no. 9

2009
[9]

380 (2013), 78–108 80 ANDREI NEGUT ,

Feigin B., Jimbo M., Miwa T., Mukhin E., Representations of quantum toroidal gln, Journal of Algebra vol. 380 (2013), 78–108 80 ANDREI NEGUT ,

2013
[10]

Feigin B., Odesskii A., Vector bundles on elliptic curve and Sklyanin algebras , Topics in Quantum Groups and Finite-Type Invariants, Amer. Math. Soc. Transl. Ser. 2, 185 (1998), Amer. Math. Soc., 65–84

1998
[11]

Jing N., Zhang H., Hopf algebraic structures of quantum toroidal algebras , arχiv:1604.05416

arXiv
[12]

Maulik D., Okounkov A., Quantum groups and quantum cohomology , Ast´ erisque, Volume 408 (2019) 212 pp

2019
[13]

Miki K., A (q,γ ) analog of the W1+∞ algebra, J. Math. Phys., 48 (2007), no. 12

2007
[14]

I., Reﬂection equation and twisted Yangians , Journal of Mathematical Physics 48, 093501 (2007)

Mudrov A. I., Reﬂection equation and twisted Yangians , Journal of Mathematical Physics 48, 093501 (2007)

2007
[15]

Negut, A., Shuﬄe algebra revisited , Int. Math. Res. Not., Volume 2014, Issue 22, 2014, 6242–6275

2014
[16]

Math., Volume 372 (2020), 107288

Negut, A., Quantum toroidal and shuﬄe algebras , Adv. Math., Volume 372 (2020), 107288

2020
[18]

Negut, A., DeformedW –algebras in type A for rectangular nilpotent, arχiv:2004.02737

2004 arXiv
[19]

Okounkov A., Smirnov A., Quantum diﬀerence equation for Nakajima varieties , arχiv:1602.09007

arXiv
[20]

Schiﬀmann O., Drinfeld realization of the elliptic Hall algebra , Journal of Algebraic Com- binatorics, vol 35 (2012), no 2, 237–262

2012
[21]

Wendlandt C., The R-Matrix Presentation for the Yangian of a Simple Lie Algebra , Comm. Math. Phys., October 2018, Volume 363, Issue 1, 289–332 MIT, Department of Mathematics, Cambridge, MA, USA Simion Stoilow Institute of Mathematics, Bucharest, Romania E-mail address: andrei...

2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [17]

Negut, A., PBW basis for Uq,q(¨gln), arχiv:1905.06277

work page Pith review arXiv 1905

[2] [1]

Beck J., Braid group action and quantum aﬃne algebras , Commun. Math. Physics vol. 165 (1994), 555568

work page 1994

[3] [2]

Burban I., Schiﬀmann O., On the Hall algebra of an elliptic curve, I , Duke Math. J. 161 (2012), no. 7, 1171–1231

work page 2012

[4] [3]

Ding J., Frenkel I., Isomorphism of two realizations of quantum aﬃne algebra Uq(ˆgln), Comm. Math. Phys. 156 (1993), no. 2, 277–300

work page 1993

[5] [4]

Ding J., Iohara K., Generalization of Drinfeld quantum aﬃne algebras , Lett. Math. Phys., 41 (1997), no. 2, 181–193

work page 1997

[6] [5]

G., A new realization of Yangians and quantized aﬃne algebras , Soviet Math

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

work page 1988

[7] [6]

Groups 5 (2000), no

Enriquez B., On correlation functions of Drinfeld currents and shuﬄe algebras, Transform. Groups 5 (2000), no. 2, 111 - 120

work page 2000

[8] [7]

J.1(1990) 193–226

Faddeev L., Reshetikhin N., Takhtajan L., Quantization of Lie groups and Liealgebras , Leningrad Math. J.1(1990) 193–226

work page 1990