Decomposition of the adjoint representation of the small quantum $sl_2$

Abstract

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.