Drinfeldians

Abstract

We construct two-parameter deformation of an universal enveloping algebra $U(g[u])$ of a polynomial loop algebra $g[u]$, where $g$ is a finite-dimensional complex simple Lie algebra (or superalgebra). This new quantum Hopf algebra called the Drinfeldian $D_{q\eta}(g)$ can be considered as a quantization of $U(g[u])$ in the direction of a classical r-matrix which is a sum of the simple rational and trigonometric r-matrices. The Drinfeldian $D_{q\eta}(g)$ contains $U_{q}(g)$ as a Hopf subalgebra, moreover $U_{q}(g[u])$ and $Y_{\eta}(g)$ are its limit quantum algebras when the $D_{q\eta}(g)$ deformation parameters $\eta$ goes to 0 and $q$ goes to 1, respectively. These results are easy generalized to a supercase, i.e. when $g$ is a finite-dimensional contragredient simple superalgebra.