  Given a finite type root datum and a primitive root of unity $q=\sqrt[l]{1}$,
G.~Lusztig has defined in [Lu] a remarkable finite dimensional Hopf algebra
$\fu$ over the cyclotomic field ${\Bbb Q}(\sqrt[l]{1})$. In this note we study
the adjoint representation $\ad$ of $\fu$ in the simplest case of the root
datum $sl_2$. The semisimple part of this representation is of big importance
in the study of local systems of conformal blocks in WZW model for $\hat{sl}_2$
at level $l-2$ in arbitrary genus. The problem of distinguishing the semisimple
part is closely related to the problem of integral representation of conformal
blocks (see [BFS]). We find all the indecomposable direct summands of $\ad$
with multiplicities. It appears that $\ad$ is isomorphic to a direct sum of
simple and projective modules. It can be lifted to a module over the (infinite
dimensional) quantum universal enveloping algebra with divided powers
$U_q(sl_2)$ which is also a direct sum of simples and projectives.
