Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 082, 28 pages      arXiv:1704.05330
Contribution to the Special Issue on Recent Advances in Quantum Integrable Systems

Differential Calculus on h-Deformed Spaces

Basile Herlemont a and Oleg Ogievetsky abc
a) Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France
b) Kazan Federal University, Kremlevskaya 17, Kazan 420008, Russia
c) On leave of absence from P.N. Lebedev Physical Institute, Leninsky Pr. 53, 117924 Moscow, Russia

Received April 18, 2017, in final form October 17, 2017; Published online October 24, 2017

We construct the rings of generalized differential operators on the ${\bf h}$-deformed vector space of ${\bf gl}$-type. In contrast to the $q$-deformed vector space, where the ring of differential operators is unique up to an isomorphism, the general ring of ${\bf h}$-deformed differential operators $\operatorname{Diff}_{{\bf h},\sigma}(n)$ is labeled by a rational function $\sigma$ in $n$ variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system and describe some properties of the rings $\operatorname{Diff}_{{\bf h},\sigma}(n)$.

Key words: differential operators; Yang-Baxter equation; reduction algebras; universal enveloping algebra; representation theory; Poincaré-Birkhoff-Witt property; rings of fractions.

pdf (546 kb)   tex (29 kb)


  1. Alekseev A.Y., Faddeev L.D., $(T^*G)_t$: a toy model for conformal field theory, Comm. Math. Phys. 141 (1991), 413-422.
  2. Bergman G.M., The diamond lemma for ring theory, Adv. Math. 29 (1978), 178-218.
  3. Bokut' L.A., Embeddings into simple associative algebras, Algebra Logic 15 (1976), 73-90.
  4. Bytsko A.G., Faddeev L.D., $(T^*{\mathcal B})_q$, $q$-analog of model space and the Clebsch-Gordan coefficients generating matrices, J. Math. Phys. 37 (1996), 6324-6348, q-alg/9508022.
  5. Furlan P., Hadjiivanov L.K., Isaev A.P., Ogievetsky O.V., Pyatov P.N., Todorov I.T., Quantum matrix algebra for the ${\rm SU}(n)$ WZNW model, J. Phys. A: Math. Gen. 36 (2003), 5497-5530, hep-th/0003210.
  6. Gel'fand I.M., Tsetlin M.L., Finite-dimensional representations of the group of unimodular matrices, Dokl. Akad. Nauk USSR 71 (1950), 825-828, English translation in Gelfand I.M., Collected papers, Vol. II, Springer-Verlag, Berlin, 1988, 653-656.
  7. Hadjiivanov L.K., Isaev A.P., Ogievetsky O.V., Pyatov P.N., Todorov I.T., Hecke algebraic properties of dynamical $R$-matrices. Application to related quantum matrix algebras, J. Math. Phys. 40 (1999), 427-448, q-alg/9712026.
  8. Khoroshkin S., Nazarov M., Mickelsson algebras and representations of Yangians, Trans. Amer. Math. Soc. 364 (2012), 1293-1367, arXiv:0912.1101.
  9. Khoroshkin S., Ogievetsky O., Mickelsson algebras and Zhelobenko operators, J. Algebra 319 (2008), 2113-2165, math.QA/0606259.
  10. Khoroshkin S., Ogievetsky O., Diagonal reduction algebras of ${\bf gl}$ type, Funct. Anal. Appl. 44 (2010), 182-198, arXiv:0912.4055.
  11. Khoroshkin S., Ogievetsky O., Structure constants of diagonal reduction algebras of ${\bf gl}$ type, SIGMA 7 (2011), 064, 34 pages, arXiv:1101.2647.
  12. Khoroshkin S., Ogievetsky O., Rings of fractions of reduction algebras, Algebr. Represent. Theory 17 (2014), 265-274.
  13. Khoroshkin S., Ogievetsky O., Diagonal reduction algebra and the reflection equation, Israel J. Math. 221 (2017), 705-729, arXiv:1510.05258.
  14. Mickelsson J., Step algebras of semi-simple subalgebras of Lie algebras, Rep. Math. Phys. 4 (1973), 307-318.
  15. Ogievetsky O., Uses of quantum spaces, in Quantum Symmetries in Theoretical Physics and Mathematics (Bariloche, 2000), Contemp. Math., Vol. 294, Amer. Math. Soc., Providence, RI, 2002, 161-232.
  16. Ogievetsky O., Herlemont B., Rings of $\bf h$-deformed differential operators, Theoret. and Math. Phys. 192 (2017), 1218-1229, arXiv:1612.08001.
  17. Tolstoy V.N., Fortieth anniversary of extremal projector method for Lie symmetries, in Noncommutative Geometry and Representation Theory in Mathematical Physics, Contemp. Math., Vol. 391, Amer. Math. Soc., Providence, RI, 2005, 371-384, math-ph/0412087.
  18. van den Hombergh A., Harish-Chandra modules and representations of step algebra, Ph.D. Thesis, Katolic University of Nijmegen, 1976, available at
  19. Wess J., Zumino B., Covariant differential calculus on the quantum hyperplane, Nuclear Phys. B Proc. Suppl. 18 (1990), 302-312.
  20. Zhelobenko D.P., Classical groups. Spectral analysis of finite-dimensional representations, Russian Math. Surveys 17 (1962), no. 1, 1-94.
  21. Zhelobenko D.P., Extremal cocycles on Weyl groups, Funct. Anal. Appl. 21 (1987), 183-192.
  22. Zhelobenko D.P., Representations of reductive Lie algebras, Nauka, Moscow, 1994.

Previous article  Next article   Contents of Volume 13 (2017)