Braid group action on projective quantum $\mathfrak{s}\mathfrak{l}(2)$ modules

We define a family of the braid group representations via the action of the $R$-matrix (of the quasitriangular extension) of the restricted quantum $\mathfrak{s}\mathfrak{l}(2)$ on a tensor power of a simple projective module. This family is an extension of the Lawrence representation specialized at roots of unity. Although the center of the braid group has finite order on the specialized Laurence representations, this action is faithful for our extension.