## Abstract.

This paper investigates the structure of basic finite dimensional Hopf algebras *H* over an algebraically closed field *k*. The algebra *H* is basic provided *H* modulo its Jacobson radical is a product of the field *k*. In this case *H* is isomorphic to a path algebra given by a finite quiver with relations. Necessary conditions on the quiver and on the coalgebra structure are found. In particular, it is shown that only the quivers \(\Gamma_G(W)\) given in terms of a finite group *G* and sequence \(W=(w_1,w_2,\ldots,w_r)\) of elements of *G* in the following way can occur. The quiver \(\Gamma_G(W)\) has vertices \(\{v_g\}_{g\in G}\) and arrows \(\{ (a_i,g)\colon v_{g^{-1}}\to v_{w_ig^{-1}}\mid g\in G, w_i\in W\}\), where the set \(\{ w_1,w_2,\ldots,w_r\}\) is closed under conjugation with elements in *G* and for each *g* in *G*, the sequences *W* and \((gw_1g^{-1}, gw_2g^{-1},\ldots, gw_rg^{-1})\) 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 algebra. Furthermore, it is shown that the conditions we find can be used to give the path algebras \(k\Gamma_G(W)\) themselves a Hopf algebra structure (for an arbitrary field *k*). The results are also translated into the language of coverings. Finally, a new class of finite dimensional basic Hopf algebras are constructed over a not necessarily algebraically closed field, most of which are quantum groups. The construction is not characteristic free. All the quivers \(\Gamma_G(W)\), where the elements of *W* generates an abelian subgroup of *G*, are shown to occur for finite dimensional Hopf algebras. The existence of such algebras is shown by explicit construction. For closely related results of Cibils and Rosso see [Ci-R].

## Keywords

Finite Group Hopf Algebra Related Result Quantum Group Explicit Construction## Preview

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