  To study the representation category of the triplet W-algebra W(p) that is
the symmetry of the (1,p) logarithmic conformal field theory model, we propose
the equivalent category C(p) of finite-dimensional representations of the
restricted quantum group $U_q SL2$ at $q=e^{\frac{i\pi}{p}}$. We fully describe
the category C(p) by classifying all indecomposable representations. These are
exhausted by projective modules and three series of representations that are
essentially described by indecomposable representations of the Kronecker
quiver. The equivalence of the W(p)- and $U_q SL2$-representation categories is
conjectured for all $p\ge 2$ and proved for p=2, the implications including the
identifications of the quantum-group center with the logarithmic conformal
field theory center and of the universal R-matrix with the braiding matrix.
