Skip to main content
SpringerLink
Log in

Bethe subalgebras of \(U_q(\widehat{\mathfrak {gl}}_n)\) via shuffle algebras

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

In this article, we construct certain commutative subalgebras of the big shuffle algebra of type \(A^{(1)}_{n-1}\). This can be considered as a generalization of the similar construction for the small shuffle algebra, obtained in Feigin et al. (J Math Phys 50(9):42, 2009). We present a Bethe algebra realization of these subalgebras. The latter identifies them with the Bethe subalgebras of \(U_q(\widehat{\mathfrak {gl}}_n)\).

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

Notes

  1. Our notation are consistent with that of [17], but following [15] we add the elements \(q^{\pm d_1}, q^{\pm d_2}\) satisfying (T0.3, T0.4). This update is essential for our discussion of the Drinfeld double and the universal R-matrix.

  2. It is easy to see that the space of solutions of this system is 1-dimensional if q is not a root of unity.

  3. Actually, one can consider the whole category of highest weight \(\ddot{U}^{'}_{q,d}(\mathfrak {sl}_n)\)-representations, see [12].

References

  1. Ding, J., Iohara, K.: Generalization of Drinfeld quantum affine algebras. Lett. Math. Phys. 41(2), 181–193 (1997). arXiv:q-alg/9608002

  2. Feigin, B., Jimbo, M., Miwa, T., Mukhin, E.: Representations of quantum toroidal \({\mathfrak{gl}}_n\). J. Algebra 380, 78–108 (2013). arXiv:1204.5378

    Article  MathSciNet  MATH  Google Scholar 

  3. Feigin, B., Jimbo, M., Miwa, T., Mukhin, E.: Quantum toroidal \({\mathfrak{gl}}_1\) and Bethe ansatz. J. Phys. A 48(24), 244001 (2015). arXiv:1502.07194

  4. Feigin, B., Hashizume, K., Hoshino, A., Shiraishi, J., Yanagida, S.: A commutative algebra on degenerate \({\mathbb{CP}}^1\) and Macdonald polynomials. J. Math. Phys. 50(9), 095215 (2009). arXiv:0904.2291

  5. Feigin, B., Kojima, T., Shiraishi, J., Watanabe, H.: The integrals of motion for the deformed W-algebra \(W_{q,t}({\mathfrak{\widehat{s}l}}_{N})\). In: Proceedings of Representation Theory 2006, Atami, Japan, pp. 102–114 (2006). arXiv:0705.0627

  6. Feigin, B., Odesskii, A.: Sklyanin’s elliptic algebras. Funct. Anal. Appl. 23(3), 207–214 (1989)

  7. Feigin, B., Odesskii, A.: Elliptic deformations of current algebras and their representations by difference operators. Funct. Anal. Appl. 31(3), 193–203 (1997)

  8. Feigin, B., Odesskii, A.: Quantized moduli spaces of the bundles on the elliptic curve and their applications. Integrable structures of exactly solvable two-dimensional models of quantum field theory (Kiev, 2000), 123–137; NATO Sci. Ser. II Math. Phys. Chem., 35, Kluwer Acad. Publ., Dordrecht (2001) arXiv:math/9812059

  9. Feigin, B., Tsymbaliuk, A.: Equivariant \(K\)-theory of Hilbert schemes via shuffle algebra. Kyoto J. Math. 51(4), 831–854 (2011). arXiv:0904.1679

    Article  MathSciNet  MATH  Google Scholar 

  10. Hernandez, D.: Representations of quantum affinizations and fusion product. Transform. Groups 10(2), 163–200 (2005). arXiv:math/0312336

    Article  MathSciNet  MATH  Google Scholar 

  11. Kassel, C., Rosso, M., Turaev, V.: Quantum groups and knot invariants. Panoramas et Synthèses [Panoramas and Syntheses], 5. Société Mathématique de France, Paris, vi+115pp (1997)

  12. Miki, K.: Toroidal braid group action and an automorphism of toroidal algebra \(U_q({\mathfrak{sl}}_{n+1, tor}) (n\ge 2)\). Lett. Math. Phys. 47(4), 365–378 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Negut, A.: The shuffle algebra revisited. Int. Math. Res. Not. (22), 6242–6275 (2014) arXiv:1209.3349

  14. Negut, A.: Quantum toroidal and shuffle algebras, R-matrices and a conjecture of Kuznetsov. arXiv:1302.6202

  15. Saito, Y.: Quantum toroidal algebras and their vertex representations. Publ. Res. Inst. Math. Sci. 34(2), 155–177 (1998). arXiv:q-alg/9611030

  16. Schiffmann, O., Vasserot, E.: The elliptic Hall algebra and the \(K\)-theory of the Hilbert scheme of \(\mathbb{A}^2\). Duke Math. J. 162(2), 279–366 (2013). arXiv:0905.2555

    Article  MathSciNet  MATH  Google Scholar 

  17. Varagnolo, M., Vasserot, E.: Schur duality in the toroidal setting. Commun. Math. Phys. 182(2), 469–483 (1996). arXiv:q-alg/9506026

Download references

Acknowledgments

We are grateful to A. Negut and J. Shiraishi for stimulating discussions. We are indebted to B. Enriquez for useful comments on the first version of the paper, which led to a better exposition of the material. A. T. is grateful to P. Etingof and H. Nakajima for their interest and support. A. T. thanks the Research Institute for Mathematical Sciences (Kyoto) and the Japan Society for the Promotion of Science for support during the main stage of the project. A. T. also gratefully acknowledges support from the Simons Center for Geometry and Physics, Stony Brook University, at which some of the research for this paper was performed. The work of A. T. was partially supported by the NSF Grant DMS-1502497. B. F. gratefully acknowledges the financial support of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program.

Author information

Authors and Affiliations

  1. International Laboratory of Representation Theory and Mathematical Physics, National Research University Higher School of Economics, Russian Federation, Moscow, Russia

    Boris Feigin

  2. Simons Center for Geometry and Physics, Stony Brook, NY, 11794, USA

    Alexander Tsymbaliuk

Authors
  1. Boris Feigin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Tsymbaliuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Tsymbaliuk.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feigin, B., Tsymbaliuk, A. Bethe subalgebras of \(U_q(\widehat{\mathfrak {gl}}_n)\) via shuffle algebras. Sel. Math. New Ser. 22, 979–1011 (2016). https://doi.org/10.1007/s00029-015-0212-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-015-0212-z

Keywords

Mathematics Subject Classification

Search

Navigation