Geodesic
Differential Calculus on $\mathbf{h}$-Deformed Spaces
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We construct the rings of generalized differential operators on the $\mathbf{h}$-deformed vector space of $\mathbf{gl}$-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 $\mathbf{h}$-deformed differential operators $\operatorname{Diff}_{\mathbf{h},\sigma}(n)$ 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 $\operatorname{Diff}_{\mathbf{h},\sigma}(n)$.
Keywords: differential operators; Yang–Baxter equation; reduction algebras; universal enveloping algebra; representation theory; Poincaré–Birkhoff–Witt property; rings of fractions. 
@article{SIGMA_2017_13_a81,
     author = {Basile Herlemont and Oleg Ogievetsky},
     title = {Differential {Calculus} on $\mathbf{h}${-Deformed} {Spaces}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2017},
     volume = {13},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a81/}
}
TY  - JOUR
AU  - Basile Herlemont
AU  - Oleg Ogievetsky
TI  - Differential Calculus on $\mathbf{h}$-Deformed Spaces
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2017
VL  - 13
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a81/
LA  - en
ID  - SIGMA_2017_13_a81
ER  - 
%0 Journal Article
%A Basile Herlemont
%A Oleg Ogievetsky
%T Differential Calculus on $\mathbf{h}$-Deformed Spaces
%J Symmetry, integrability and geometry: methods and applications
%D 2017
%V 13
%U http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a81/
%G en
%F SIGMA_2017_13_a81
Basile Herlemont; Oleg Ogievetsky. Differential Calculus on $\mathbf{h}$-Deformed Spaces. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a81/

[1] Alekseev A. Y., Faddeev L. D., “$(T^*G)_t$: a toy model for conformal field theory”, Comm. Math. Phys., 141 (1991), 413–422 | DOI | MR | Zbl

[2] Bergman G. M., “The diamond lemma for ring theory”, Adv. Math., 29 (1978), 178–218 | DOI | MR

[3] Bokut' L. A., “Embeddings into simple associative algebras”, Algebra Logic, 15 (1976), 73–90 | DOI | MR

[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, arXiv: q-alg/9508022 | DOI | MR | Zbl

[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, arXiv: hep-th/0003210 | DOI | MR | Zbl

[6] Gelfand I. M., Collected papers, v. II, Springer-Verlag, Berlin, 1988, 653–656 | MR | MR | Zbl

[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, arXiv: q-alg/9712026 | DOI | MR | Zbl

[8] Khoroshkin S., Nazarov M., “Mickelsson algebras and representations of Yangians”, Trans. Amer. Math. Soc., 364 (2012), 1293–1367, arXiv: 0912.1101 | DOI | MR | Zbl

[9] Khoroshkin S., Ogievetsky O., “Mickelsson algebras and Zhelobenko operators”, J. Algebra, 319 (2008), 2113–2165, arXiv: math.QA/0606259 | DOI | MR | Zbl

[10] Khoroshkin S., Ogievetsky O., “Diagonal reduction algebras of $gl$ type”, Funct. Anal. Appl., 44 (2010), 182–198, arXiv: 0912.4055 | DOI | MR | Zbl

[11] Khoroshkin S., Ogievetsky O., “Structure constants of diagonal reduction algebras of $gl$ type”, SIGMA, 7 (2011), 064, 34 pp., arXiv: 1101.2647 | DOI | MR | Zbl

[12] Khoroshkin S., Ogievetsky O., “Rings of fractions of reduction algebras”, Algebr. Represent. Theory, 17 (2014), 265–274 | DOI | MR | Zbl

[13] Khoroshkin S., Ogievetsky O., “Diagonal reduction algebra and the reflection equation”, Israel J. Math., 221 (2017), 705–729, arXiv: 1510.05258 | DOI | MR

[14] Mickelsson J., “Step algebras of semi-simple subalgebras of Lie algebras”, Rep. Math. Phys., 4 (1973), 307–318 | DOI | MR | Zbl

[15] Ogievetsky O., “Uses of quantum spaces”, Quantum Symmetries in Theoretical Physics and Mathematics (Bariloche, 2000), Contemp. Math., 294, Amer. Math. Soc., Providence, RI, 2002, 161–232 | DOI | MR | Zbl

[16] Ogievetsky O., Herlemont B., “Rings of $\bf h$-deformed differential operators”, Theoret. and Math. Phys., 192 (2017), 1218–1229, arXiv: 1612.08001 | DOI | MR

[17] Tolstoy V. N., “Fortieth anniversary of extremal projector method for Lie symmetries”, Noncommutative Geometry and Representation Theory in Mathematical Physics, Contemp. Math., 391, Amer. Math. Soc., Providence, RI, 2005, 371–384, arXiv: math-ph/0412087 | DOI | MR | Zbl

[18] van den Hombergh A., Harish–Chandra modules and representations of step algebra, Ph.D. Thesis, Katolic University of Nijmegen, 1976 http://hdl.handle.net/2066/147527

[19] Wess J., Zumino B., “Covariant differential calculus on the quantum hyperplane”, Nuclear Phys. B Proc. Suppl., 18 (1990), 302–312 | DOI | MR

[20] Zhelobenko D. P., “Classical groups. Spectral analysis of finite-dimensional representations”, Russian Math. Surveys, 17:1 (1962), 1–94 | DOI | MR

[21] Zhelobenko D. P., “Extremal cocycles on Weyl groups”, Funct. Anal. Appl., 21 (1987), 183–192 | DOI | MR | Zbl

[22] Zhelobenko D. P., Representations of reductive Lie algebras, Nauka, M., 1994 | MR