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

SIGMA 13 (2017), 082, 28 pages      arXiv:1704.05330      https://doi.org/10.3842/SIGMA.2017.082
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

Abstract
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.

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