Theoretical and Mathematical Physics

, Volume 192, Issue 2, pp 1218–1229 | Cite as

Rings of h-deformed differential operators



We describe the center of the ring Diff h (n) of h-deformed differential operators of type A. We establish an isomorphism between certain localizations of Diff h (n) and the Weyl algebra W n , extended by n indeterminates.


reduction algebra oscillatory realization ring of differential operators Gelfand–Kirillov conjecture dynamical Yang–Baxter equation 


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  1. 1.
    J. Mickelsson, “Step algebras of semisimple subalgebras of Lie algebras,” Rep. Math. Phys., 4, 307–318 (1973).ADSCrossRefMATHGoogle Scholar
  2. 2.
    R. M. Asherova, Yu. F. Smirnov, and V. N. Tolstoy, “Projection operators for simple lie groups,” Theor. Math. Phys., 15, 392–401 (1973).CrossRefGoogle Scholar
  3. 3.
    V. N. Tolstoy, “Fortieth anniversary of extremal projector method for Lie symmetries,” in: Noncommutative Geometry and Representation Theory in Mathematical Physics (Contemp. Math., Vol. 391, J. Fuchs, J. Mickelsson, G. Rozenblioum, A. Stolin, and A. Westerberg, eds.), Amer. Math. Soc., Providence, R. I. (2005), pp. 371–384.CrossRefGoogle Scholar
  4. 4.
    D. Zhelobenko, Representations of Reductive Lie Algebras [in Russian], Nauka, Moscow (1994).MATHGoogle Scholar
  5. 5.
    O. V. Ogievetskii and S. M. Khoroshkin, “Diagonal reduction algebras of gl type,” Funct. Anal. Appl., 44, 182–198 (2010).MathSciNetCrossRefGoogle Scholar
  6. 6.
    S. Khoroshkin and O. V. Ogievetsky, “Structure constants of diagonal reduction algebras of gl type,” SIGMA, 7, 064 (2011).MATHGoogle Scholar
  7. 7.
    S. Khoroshkin and O. Ogievetsky, “Diagonal reduction algebra and reflection equation,” arXiv:1510.05258v1 [math.RT] (2015).MATHGoogle Scholar
  8. 8.
    J. Wess and B. Zumino, “Covariant differential calculus on the quantum hyperplane,” Nucl. Phys. B Proc. Suppl., 18, 302–312 (1990).ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    G. Felder, “Conformal field theory and integrable systems associated to elliptic curves,” in: Proc. Intl. Congr. Math. (Zürich, Switzerland, 3–11 August 1994, S. D. Chatterji, ed.), Birkhäuser, Basel (1995), pp. 1247–1255.CrossRefGoogle Scholar
  10. 10.
    J.-L. Gervais and A. Neveu, “Novel triangle relation and absence of tachyons in Liouville string field theory,” Nucl. Phys. B, 238, 125–141 (1984).ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    P. Etingof and O. Schiffmann, “Lectures on the dynamical Yang–Baxter equations,” in: Quantum Groups and Lie Theory (London Math. Soc. Lect. Note Ser., Vol. 290, A. Pressley, ed.), Cambridge Univ. Press, Cambridge (2001), pp. 89–129.Google Scholar
  12. 12.
    S. Khoroshkin and O. Ogievetsky, “Rings of fractions of reduction algebras,” Algebras and Representation Theory, 17, 265–274 (2014).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    I. M. Gelfand and A. A. Kirillov, “Sur les corps liés aux algèbres enveloppantes des algèbres de Lie,” Inst. Hautes Études Sci. Publ. Math., 31, 5–19 (1966).CrossRefMATHGoogle Scholar
  14. 14.
    O. Ogievetsky, “Differential operators on quantum spaces for GL q(n) and SO q(n),” Lett. Math. Phys., 24, 245–255 (1992).ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    S. Khoroshkin and O. Ogievetsky, “Mickelsson algebras and Zhelobenko operators,” J. Algebra, 319, 2113–2165 (2008).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    B. Herlemont and O. Ogievetsky, “Differential calculus on h-deformed spaces,” arXiv:1704.05330v1 [math.RA] (2017).Google Scholar
  17. 17.
    O. Ogievetsky and T. Popov, “R-matrices in rime,” Adv. Theor. Math. Phys., 14, 439–505 (2010).MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    M. Artin, “Noncommutative rings: Lecture notes,” (1999).Google Scholar
  19. 19.
    I. V. Cherednik, “Factorizing particles on a half-line and root systems,” Theor. Math. Phys., 61, 977–983 (1984).MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    E. K. Sklyanin, “Boundary conditions for integrable quantum systems,” J. Phys. A: Math. Gen., 21, 2375–2389 (1988).ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    N. Yu. Reshetikhin and M. A. Semenov-Tian-Shansky, “Central extensions of quantum current groups,” Lett. Math. Phys., 19, 133–142 (1990).ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    P. P. Kulish and E. K. Sklyanin, “Algebraic structures related to the reflection equations,” J. Phys. A, 25, 5963–5975 (1992).ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    A. P. Isaev and O. V. Ogievetsky, “On baxterized solutions of reflection equation and integrable chain models,” Nucl. Phys. B, 760, 167–183 (2007); arXiv:math-ph/0510078v2 (2005).ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    A. Isaev, O. Ogievetsky, and P. Pyatov, “Generalized Cayley–Hamilton–Newton identities,” Czechoslovak J. Phys., 48, 1369–1374 (1998).ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    A. P. Isaev, A. I. Molev, and O. V. Ogievetsky, “A new fusion procedure for the Brauer algebra and evaluation homomorphisms,” Int. Math. Res. Notices, 2012, 2571–2606 (2012).MathSciNetMATHGoogle Scholar
  26. 26.
    A. P. Isaev, A. I. Molev, and O. V. Ogievetsky, “Idempotents for Birman–Murakami–Wenzl algebras and reflection equation,” Adv. Theor. Math. Phys., 18, 1–25 (2014).MathSciNetCrossRefMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Aix Marseille UnivUniversité de Toulon, CNRS, CPTMarseilleFrance
  2. 2.Kazan Federal UniversityKazanRussia
  3. 3.Lebedev Physical InstituteRASMoscowRussia

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