A Generalization of $U_h(sl(2))$ via Jacobian Elliptic Function

Abstract

A two-parametric generalization of the Jordanian deformation $U_h (sl(2))$ of $sl(2)$ is presented. This involves Jacobian elliptic functions. In our deformation $U_{(h,k)}(sl(2))$, for $k^2=1$ one gets back $U_h(sl(2))$. The constuction is presented via a nonlinear map on $sl(2)$. This invertible map directly furnishes the highest weight irreducible representations of $U_{(h,k)}(sl(2))$. This map also provides two distinct induced Hopf stuctures, which are exhibited. One is induced by the classical $sl(2)$ and the other by the distinct one of $U_h(sl(2))$. Automorphisms related to the two periods of the elliptic functions involved are constructed. Translations of one generator by half and quarter periods lead to interesting results in this context. Possibilities of applications are discussed briefly.