Basic Hopf algebras and quantum groups

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].