Differential calculus on $\mathbf{h}$-deformed spaces

Abstract

The ring $\text{Diff}_{\mathbf{h}}(n)$ of $\mathbf{h}$-deformed differential operators appears in the theory of reduction algebras. In this thesis, we construct the rings of generalized differential operators on the $\mathbf{h}$-deformed vector spaces of $\mathfrak{gl}$-type. In contrast to the $q$-deformed vector spaces for which the ring of differential operators is unique up to an isomorphism, the general ring of $\mathbf{h}$-deformed differential operators $\text{Diff}_{\mathbf{h},\sigma}(n)$ 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. We show that the center of $\text{Diff}_{\mathbf{h},\sigma}(n)$ is a ring of polynomials in $n$ variables. We construct an isomorphism between certain localizations of $\text{Diff}_{\mathbf{h},\sigma}(n)$ and the Weyl algebra $\text{W}_n$ extended by $n$ indeterminates. We present some conditions for the irreducibility of the finite dimensional $\text{Diff}_{\mathbf{h},\sigma}(n)$-modules. Finally, we discuss difficulties for finding analogous constructions for the ring $\text{Diff}_{\mathbf{h}}(n,N)$ formed by several copies of $\text{Diff}_{\mathbf{h}}(n)$.