  In this paper we study the tensor category structure of the module category
of the restricted quantum enveloping algebra associated to $\mathfrak{sl}_2$.
Indecomposable decomposition of all tensor products of modules over this
algebra is completely determined in explicit formulas. As a by-product, we show
that the module category of the restricted quantum enveloping algebra
associated to $\mathfrak{sl}_2$ is not a braided tensor category.
