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Differential Calculus on $\mathbf{h}$-Deformed Spaces
Basile Herlemonta, Oleg Ogievetskybca a Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France
b On leave of absence from P.N. Lebedev Physical Institute,
Leninsky Pr. 53, 117924 Moscow, Russia
c Kazan Federal University, Kremlevskaya 17, Kazan 420008, Russia
Abstract:
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.
Received: April 18, 2017; in final form October 17, 2017; Published online October 24, 2017
Citation:
Basile Herlemont, Oleg Ogievetsky, “Differential Calculus on $\mathbf{h}$-Deformed Spaces”, SIGMA, 13 (2017), 082, 28 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1282 https://www.mathnet.ru/eng/sigma/v13/p82
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Abstract page: | 140 | Full-text PDF : | 29 | References: | 25 |
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