The invariant and quasi-invariant transformations of the stable processes with independent increments

Let $\xi(t)$, $t\in[0,1]$ be a strictly stable process with the index of stability $\alpha\in(0,2)$. By $\mathcal P_\xi$ we denote the law of $\xi$ in the Skorokhod space $\mathbb D[0,1]$. For arbitrary strictly stable process $\xi$ we construct $\mathcal P_\xi-$quasi-invariant semigroup of transformations of $\mathbb D[0,1]$. For strictly stable processes with positive and negative jumps we construct $\mathcal P_\xi-$quasi-invariant group of transformations of $\mathbb D[0,1]$. In symmetric case this group is a group of the invariant transformations.