$U_q\left({\mathfrak{s}\mathfrak{l}}_2\right)$-symmetries of the quantum disc: a complete list

This work presents a classification of $U_q({\mathfrak{s}\mathfrak{l}}_2)$-symmetries on the quantum disc. The principal invariant of such classification, the grading jump, is introduced. It turns out that, under the present subjects, the grading jump can take only 3 values: $0$, $1$, $-1$. The subcollection of the complete collection of symmetries is extracted in such a way that the selected symmetries satisfy certain compatibility condition for involutions.