Summary: The authors propose a new structure \({\mathcal U}_q^r (sl(2))\). This is realized by multiplying \(\delta\) \((q=e^\delta\), \(\delta\in\mathbb{C})\) by \(\theta\), where \(\theta\) is a real nilpotent, paragrassmannian variable of order \(r\) \((\theta^{r+1}= 0)\) that they call the order of deformation, the limit \(r\to\infty\) giving back the standard \({\mathcal U}_q (sl(2))\). In particular, they show that for \(r=1\) there exists a new \({\mathcal R}\)-matrix associated with \(sl(2)\). They also prove that the restriction of the values of the parameters of deformation give nonlinear algebras as particular cases.