On a Deformation of $sl(2)$ with Paragrassmannian Variables

Abstract

We propose a new structure ${\cal U}^{r}_{\displaystyle{q}}(sl(2)) $. This is realized by multiplying $δ$ ($q=e^δ$, $δ\in \CC$) by $θ$, where $θ$ is a real nilpotent -paragrassmannian- variable of order $r$ ($θ^{r+1}=0$) that we call the order of deformation, the limit $r\rightarrow \infty$ giving back the standard ${\cal U}_{\displaystyle {q}}(sl(2))$. In particular we show that, for $r=1$, there exists a new ${\cal R}$-matrix associated with $sl(2)$. We also proof that the restriction of the values of the parameters of deformation give nonlinear algebras as particular cases.