Skip to main content
SpringerLink
Log in

Difference operators via GKLO-type homomorphisms: shuffle approach and application to quantum Q-systems

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

We present a shuffle realization of the GKLO-type homomorphisms for shifted quantum affine, toroidal, and quiver algebras in the spirit of Feigin and Odesskii (Funktsional. Anal. Prilozhen. 31(3):57–70, 1997), thus generalizing its rational version of Frassek and Tsymbaliuk (Commun. Math. Phys. 392:545–619, 2022) and the type A construction of Finkelberg and Tsymbaliuk (Arnold Math. J. 5(2–3):197–283, 2019). As an application, this allows us to construct large families of commuting and q-commuting difference operators, in particular, providing a convenient approach to the Q-systems where it proves a conjecture of Di Francesco and Kedem (Commun. Math. Phys. 369(3):867–928, 2019).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Notes

  1. There seems to be a sign typo in [4, (2.23)] making it actually incompatible with [4, (2.25)].

References

  1. Braverman, A., Finkelberg, M., Nakajima, H.: Towards a mathematical definition of Coulomb branches of \(3\)-dimensional \(\cal{N} =4\) gauge theories, II. Adv. Theor. Math. Phys. 22(5), 1071–1147 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. Braverman, A., Finkelberg, M., Nakajima, H.; Coulomb branches of \(3d\)\(\cal{N}=4\) quiver gauge theories and slices in the affine Grassmannian (with appendices by A. Braverman, M. Finkelberg, J. Kamnitzer, R. Kodera, H. Nakajima, B. Webster, A. Weekes), Adv. Theor. Math. Phys. 23(1), 75–166 (2019)

  3. Drinfeld, V.: A new realization of Yangians and quantized affine algebras. Sov. Math. Dokl. 36(2), 212–216 (1988)

    Google Scholar 

  4. Di Francesco, P., Kedem, R.: Quantum \(Q\)-systems: from cluster algebras to quantum current algebras. Lett. Math. Phys. 107(2), 301–341 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Di Francesco, P., Kedem, R.: \((t, q)\)-deformed \(Q\)-systems, DAHA and quantum toroidal algebras via generalized Macdonald operators. Commun. Math. Phys. 369(3), 867–928 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Di Francesco, P., Kedem, R.: Macdonald operators and quantum \(Q\)-systems for classical types. Representation theory, mathematical physics, and integrable systems. Prog. Math. 340, 163–199 (2021)

    Article  MATH  Google Scholar 

  7. Feigin, B., Feigin, E., Jimbo, M., Miwa, T., Mukhin, E.: Quantum continuous \({\mathfrak{gl} }_{\infty }\): semiinfinite construction of representations. Kyoto J. Math. 51(2), 337–364 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Feigin, B., Hashizume, K., Hoshino, A., Shiraishi, J., Yanagida, S.: A commutative algebra on degenerate \({\mathbb{C} }{\mathbb{P} }^1\) and Macdonald polynomials. J. Math. Phys. 50(9), 095215 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Feigin, B., Odesskii, A.: Elliptic deformations of current algebras and their representations by difference operators (Russian). Funktsional. Anal. i Prilozhen. 31(3), 57–70 (1997); translation in Funct. Anal. Appl. 31(3), 193–203 (1998)

  10. Feigin, B., Tsymbaliuk, A.: Bethe subalgebras of \(U_q({\widehat{{\mathfrak{gl} }}}_n)\) via shuffle algebras. Sel. Math. (N. S.) 22(2), 979–1011 (2016)

    Article  MATH  Google Scholar 

  11. Finkelberg, M., Tsymbaliuk, A.: Multiplicative slices, relativistic Toda and shifted quantum affine algebras. In: Gorelik, M., Hinich, V., Melnikov, A. (eds.) Representations and Nilpotent Orbits of Lie Algebraic Systems (Special volume in honour of the 75th birthday of Anthony Joseph). Progress in Mathematics, vol. 330, pp. 133–304. Birkhäuser, Cham (2019)

    MATH  Google Scholar 

  12. Finkelberg, M., Tsymbaliuk, A.: Shifted quantum affine algebras: integral forms in type \(A\) (with appendices by A. Tsymbaliuk, A. Weekes). Arnold Math. J. 5(2–3), 197–283 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Frassek, R., Tsymbaliuk, A.: Rational Lax matrices from antidominantly shifted extended Yangians: BCD types. Commun. Math. Phys. 392, 545–619 (2022)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Gerasimov, A., Kharchev, S., Lebedev, D., Oblezin, S.: On a Class of Representations of Quantum Groups, Noncommutative Geometry and Representation Theory in Mathematical Physics. Contemporary Mathematics, vol. 391, pp. 101–110. American Mathematical Society, Providence (2005)

    MATH  Google Scholar 

  15. Neguţ, A.: The shuffle algebra revisited. Int. Math. Res. Not. IMRN 22, 6242–6275 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Neguţ, A.: Quantum toroidal and shuffle algebras. Adv. Math. 372, 107288 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Neguţ, A.: Shuffle algebras for quivers and wheel conditions. arXiv:2108.08779

  18. Neguţ, A., Sala, F., Schiffmann, O.: Shuffle algebras for quivers as quantum groups. arXiv:2111.00249

  19. Neguţ, A., Tsymbaliuk, A.: Quantum loop groups and shuffle algebras via Lyndon words. arXiv:2102.11269

  20. Orr, D., Shimozono, M.: Difference operators for wreath Macdonald polynomials. arXiv:2110.08808

  21. Tsymbaliuk, A.: Several realizations of Fock modules for toroidal \(\ddot{U}_{q, d}({\mathfrak{sl} }_n)\). Algebr. Represent. Theory 22(1), 177–209 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tsymbaliuk, A.: Shuffle approach towards quantum affine and toroidal algebras. arXiv:2209.04294

Download references

Acknowledgements

I am indebted to Boris Feigin, Michael Finkelberg, and especially Andrei Neguţ for many enlightening discussions about shuffle algebras and related structures; to Philippe Di Francesco and Rinat Kedem for a correspondence about their work [4,5,6] on quantum Q-systems; to the anonymous referees for useful suggestions that improved the exposition. I am gratefully acknowledging the support from NSF Grants DMS-1821185 and DMS-2037602.

Author information

Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Alexander Tsymbaliuk

Authors
  1. Alexander Tsymbaliuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Tsymbaliuk.

Ethics declarations

Conflict of interest

The author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tsymbaliuk, A. Difference operators via GKLO-type homomorphisms: shuffle approach and application to quantum Q-systems. Lett Math Phys 113, 22 (2023). https://doi.org/10.1007/s11005-023-01639-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11005-023-01639-1

Keywords

Mathematics Subject Classification

Search

Navigation