NOTES ON THE DRINFELD DOUBLE OF FINITE-DIMENSIONAL GROUP ALGEBRAS

Abstract. Let k be an algebraically closed field of characteristic p> 0. We characterize the finite groups G for which the Drinfeld double D(kG) of the group algebra kG has the Chevalley property. We also show that this is the case if and only if the tensor product of every simple D(kG)-module with its dual is semisimple. The analogous result for the group algebra kG is also true, but its proof requires the classification of the finite simple groups. A further result concerns the largest Hopf ideal contained in the Jacobson radical of D(kG)). We prove that this generated by the augmentation ideal of kOp(Z(G)), where Z(G) is the center of G and Op(Z(G)) the largest p-subgroup of this center. 1