Differential Calculus on h-Deformed Spaces

[摘要] 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 $\operatorname{Diff}_{{\bf h},\sigma}(n)$.

[关键词] dif ferential operators;Yang–Baxter equation;reduction algebras;universal enveloping algebra;representation theory;Poincar´e–Birkhof f–Witt property;rings of fractions