BASIC HOPF-ALGEBRAS AND QUANTUM GROUPS

This paper investigates the structure of basic finite dimensional Hopf algebras H over an algebraically closed field k. The algebra H is bas ic provided H module its Jacobson radical is a product of the field k. In this case H is isomorphic to a path algebra given by a finite quiv er with relations. Necessary conditions on the quiver and on the coalg ebra structure are found. In particular, it is shown that only the qui vers Gamma(G)(W) given in terms of a finite group G and sequence W = ( w(1), w(2),..., w(r)) of elements of G in the following way can occur. The quiver Gamma(G)(W) has vertices {upsilon(g)}(g is an element of G ) and arrows {(a(i), g) : upsilon(g-1) --> upsilon(wig-1) \ g is an el ement of G, w(i) is an element of W}, where the set {w(1), w(2),...,w( r)} is closed under conjugation with elements in G and for each g in G , the sequences W and (gw(1)g(-l), gw(2)g(-1),..., gw(r)g(-1)) are the same up to a permutation. We show how k Gamma(G)(W) is a kG-bimodule and study properties of the left and right actions of G on the path al gebra. Furthermore, it is shown that the conditions we find can be use d to give the path algebras k Gamma(G)(W) themselves a Hopf algebra st ructure (for an arbitrary field k). The results are also translated in to the language of coverings. Finally, a new class of finite dimension al basic Hopf algebras are constructed over a not necessarily algebrai cally closed field, most of which are quantum groups. The construction is not characteristic free. All the quivers rc(W), where the elements of W generates an abelian subgroup of G, are shown to occur for finit e dimensional Hopf algebras. The existence of such algebras is shown b y explicit construction. For closely related results of Cibils and Ros so see [Ci-R].