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Elliptic deformation of $\mathcal{W}_N$-algebras

by J. Avan, L. Frappat, E. Ragoucy

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Submission summary

Authors (as registered SciPost users): Luc FRAPPAT · Eric Ragoucy
Submission information
Preprint Link:  (pdf)
Date submitted: 2018-10-30 01:00
Submitted by: FRAPPAT, Luc
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Quantum Algebra
  • Mathematical Physics
Approach: Theoretical


We construct $q$-deformations of quantum $\mathcal{W}_N$ algebras with elliptic structure functions. Their spin $k+1$ generators are built from $2k$ products of the Lax matrix generators of ${\mathcal{A}_{q,p}(\widehat{gl}(N)_c)}$). The closure of the algebras is insured by a critical surface condition relating the parameters $p,q$ and the central charge $c$. Further abelianity conditions are determined, either as $c=-N$ or as a second condition on $p,q,c$. When abelianity is achieved, a Poisson bracket can be defined, that we determine explicitly. One connects these structures with previously built classical $q$-deformed $\mathcal{W}_N$ algebras and quantum $\mathcal{W}_{q,p}(\mathfrak{sl}_N)$.

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Reports on this Submission

Anonymous Report 2 on 2019-4-11 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1810.11410v1, delivered 2019-04-11, doi: 10.21468/SciPost.Report.906


The paper clarifies the q-deformation of W_N algebras by relating them to well known quadratic algebra structures. The quite complicated exchange relations follow from more fundamental relations following directly from RLL type relations. It is a rather elegant approach and clearly relates q-Virasoro to elliptic quantum algebras.


It is written in the continuation of other papers. Although each paper can be read independently, at some point it would be interesting to have a synthetic paper gathering the main results.
The algebra looks very nice, but its practical use for physical applications is elusive at that point. Are the t^k_mn generators useful for some physical application ?. Could they be useful in connection with the spectrum of the Ruisjenaars-Schneider chains ?.
The algebra is much wider than the W_N (or Virasoro) algebra and contains a lot more generators, but no interpretation of these generators is given. Do they have a geometric interpretation in the classical limit? Are they connected in some ways to the bigger algebras such as Ding Iohara Miki?


The paper is one among a long series of works (in particular 17) by the authors to unravel the connections between q-W algebras and elliptic algebras of QISM. More specifically, in this paper, the authors are able using their approach to recover the quadratic relations of the W algebra obtained by Feigin and Frenkel. This is an interesting progress.

Requested changes

It would be nice if the authors clarify the relation between their paper and ref [2]. I can guess that t^j_2,-1 is related to F.F. T_j but the authors should make it clear. At the top of page 5, the t^k_mn are claimed to be defined in [2], could the authors be more precise.

  • validity: high
  • significance: high
  • originality: good
  • clarity: good
  • formatting: good
  • grammar: excellent

Anonymous Report 1 on 2019-3-15 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1810.11410v1, delivered 2019-03-15, doi: 10.21468/SciPost.Report.869


1.This paper develops an approach to a construction of q-deformations of quantum $W_N$ algebras based on the Lax operator of $A_{q,p}({\hat{gl}}(N)_c)$.
2. The main result is a derivation for exchange relations for generators satisfied on a special surface in the space of parameters.
3. Authors compare their construction with a previously known approach to construct quantum $W$-algebra generators.
4. They also analyze a critical level case in details.


1. Section 5 contains a lot of technical details which could be moved to another appendix.
2. A notation $\hat R_{12}(z)$ in (2.3) is not the best one. It can be easily confused with the standard
notation $\hat R_{12}(z)=P_{12}R_{12}(z)$. At least, formulas (B.14-B.15) should be quoted immediately after (2.3).


I have read the article "Elliptic deformation of $W_N$ algebras" by J.Avan et al. The paper addresses important problems in the theory of quantum elliptic algebras. It relies on several previous results by the authors and is quite challenging technically but this is probably due to the very nature of the subject.
The main result is the Theorem 3.1 where exchange relations for quantum generators are derived provided that parameters $q,p,c$ lie on a special surface. The generators are constructed in terms L-operators for the $A_{q,p}({\hat{gl}}(N)_c)$ algebra. Then the authors match their construction to the construction derived by Feigin and Frenkel and discuss their differences.
In Chapter 4 they introduce Abelian subalgebras with the required restrictions on parameters and derive Poisson structures on such subalgebras. Section 5 is dedicated to the case of critical level where they present in Proposition 5.2 structural constants of a closed Poisson algebra for the algebra generators.
Section 5 exlpains a reduction to $U_q(\hat{gl}_N)$ at ctirical level.
Overall, the paper is well written and opens some interesting new directions for a future research.
As noted in Conclusion, explicit realizations of these algebras are required similar to the case of
$W_{q,p}(A_N)$ algebras. A comparison with other approaches to elliptis $W$-algebras is also would be desirable.

Requested changes

Overall the paper is well written and I recommend it for publication in the current form.
I notices a couple of points above which could improve the paper but they are not compulsory.

  • validity: high
  • significance: high
  • originality: high
  • clarity: good
  • formatting: good
  • grammar: good

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