Quantized $SL(2)$ representations of knot groups

For a braided Hopf algebra $A$ with braided commutativity, we introduce the space of $A$ representations of a knot $K$ as a generalization of the $G$ representation space of $K$ defined for a group $G$. By rebuilding the $G$ representation space from the view point of Hopf algebras, it is extended to any braided Hopf algebra with braided commutativity. Applying this theory to $\mathrm{BSL}(2)$ which is the braided quantum $\mathrm{SL}(2)$ introduced by S. Majid, we get the space of $\mathrm{BSL}(2)$ representations. It is a non-commutative algebraic scheme which provides quantized $\mathrm{SL}(2)$ representations of $K$.