Drinfeld–Sokolov reduction in quantum algebras: canonical form of generating matrices

Article

Abstract

We define the second canonical forms for the generating matrices of the Reflection Equation algebras and the braided Yangians, associated with all even skew-invertible involutive and Hecke symmetries. By using the Cayley–Hamilton identities for these matrices, we show that they are similar to their canonical forms in the sense of Chervov and Talalaev (J Math Sci (NY) 158:904–911, 2008).

Keywords

Reflection Equation algebra Braided Yangian Second canonical form Quantum elementary symmetric functions Quantum power sums Cayley–Hamilton identity 

Mathematics Subject Classification

81R50 

Notes

Acknowledgements

The work of D.T. was partially supported by the grant of the Simons foundation and the RFBR Grant 17-01-00366 A. The work of P.S. has been funded by the Russian Academic Excellence Project ’5-100’ and was also partially supported by the RFBR Grant 16-01-00562.

References

  1. 1.
    Chervov, A., Talalaev, D.: The KZ equation, G-opers, the quantum Drinfeld–Sokolov reduction and the quantum Cayley–Hamilton identity. J. Math. Sci. (N.Y.) 158, 904–911 (2008)MATHGoogle Scholar
  2. 2.
    Drinfeld, V., Sokolov, V.: Lie algebras and equations of Korteweg-de Vries type (Russian) Current problems in math, vol. 24. Moscow, pp. 81–180 (1984)Google Scholar
  3. 3.
    Feigin, B., Frenkel, E.: Quantization of the Drinfeld–Sokolov reduction. Phys. Lett. B 246, 75–81 (1990)ADSCrossRefMATHGoogle Scholar
  4. 4.
    Frenkel, E., Reshetikhin, N., Semenov-Tian-Shansky, M.: Drinfeld–Sokolov reduction for difference operators and deformations of W-algebras. I. The case of Virasoro algebra. Commun. Math. Phys. 192, 605–629 (1998)ADSCrossRefMATHGoogle Scholar
  5. 5.
    Frenkel, E., Reshetikhin, N.: Quantum affine algebras and deformations of the Virasoro and W-algebras. CMP 178, 237–264 (1996)MATHGoogle Scholar
  6. 6.
    Gurevich, D., Pyatov, P., Saponov, P.: Hecke symmetries and characteristic relations on reflection equation algebras. Lett. Math. Phys. 41(3), 255–264 (1997)CrossRefMATHGoogle Scholar
  7. 7.
    Gurevich, D., Pyatov, P., Saponov, P.: Representation theory of (modified) reflection equation algebra of the \(GL(m|n)\) type. Algebra Anal. 20, 70–133 (2008)Google Scholar
  8. 8.
    Gurevich, D., Pyatov, P., Saponov, P.: Cayley–Hamilton theorem for quantum matrix algebras of \(GL(m|n)\) type. St Petersburg Math. J. 17, 157–179 (2005)MATHGoogle Scholar
  9. 9.
    Gurevich, D., Saponov, P.: Generic super-orbits in \(gl(m|n)^*\) and their braided counterparts. J. Geom. Phys. 60, 1411–1423 (2010)ADSCrossRefMATHGoogle Scholar
  10. 10.
    Gurevich, D., Saponov, P.: Braided Yangians. arXiv:1612.05929
  11. 11.
    Gurevich, D., Saponov, P.: Generalized Yangians and their Poisson counterparts. Theor. Math. Phys. 192, 1243–1257 (2017)CrossRefMATHGoogle Scholar
  12. 12.
    Gurevich, D., Saponov, P.: From Reflection Equation Algebra to Braided Yangians. In: Proc. International Conference in Mathematical Physics, Kezenoy-Am 2016 (Russia). Springer (to appear) (2018)Google Scholar
  13. 13.
    Isaev, A., Ogievetsky, O.: Half-quantum linear algebra. In: Symmetries and groups in contemporary physics, Nankai Ser. Pure Appl. Math. Theoret. Phys., vol. 11. World Sci. Publ., Hackensack, NJ, pp. 479–486 (2013)Google Scholar
  14. 14.
    Isaev, A., Ogievetsky, O., Pyatov, P.: On quantum matrix algebras satisfying the Cayley–Hamilton–Newton identities. J. Phys. A Math. Gen. 32(9), L115–L121 (1999)ADSCrossRefMATHGoogle Scholar
  15. 15.
    Kirillov, A.: Introduction to family algebras. Moscow Math. J. 1, 49–63 (2001)MATHGoogle Scholar
  16. 16.
    Ogievetsky, O.: Uses of Quantum Spaces, 3rd cycle. Bariloche, Argentine, pp. 72, cel-00374419 (2000)Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Université de Valenciennes, EA 4015-LAMAVValenciennesFrance
  2. 2.Interdisciplinary Scientific Center Poncelet (ISCP, UMI 2615 du CNRS)MoscowRussian Federation
  3. 3.National Research University Higher School of EconomicsMoscowRussian Federation
  4. 4.Institute for High Energy PhysicsNRC “Kurchatov Institute”ProtvinoRussian Federation
  5. 5.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussian Federation

Personalised recommendations